{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -4(-2sin(1\u03c0x - 2) + 2) + 5 and g(x) = 4x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1((-1x + 3)/(-1x + 2)) + 3)/(-3((-1x + 3)/(-1x + 2)) + 2) and g(x) = 1|x + 3| intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3|(1|x + 2| + 2) + 3| + 2 and g(x) = -1x^2 - 4x - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-8", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -3(5x)^2 and g(x) = (2x - 1)/(2x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3((2x)/(-2x - 1)) - 2 and g(x) = 4x^2 + 1x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -4(-2|x - 1|) - 5 and g(x) = (-2x + 2)/(-2x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(4x^2 + 2x + 3) + 1)/(3(4x^2 + 2x + 3) - 2) and g(x) = 5x^2 + 3x - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2|(-2|x + 1| + 3) - 1| + 3 and g(x) = 4x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2(3sin(2\u03c0x - 2) - 3)^2 + 4(3sin(2\u03c0x - 2) - 3) - 4 and g(x) = (2x)/(3x - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -4(2sin(2\u03c0x + 3) + 3) - 2 and g(x) = (-2x + 1)/(-1x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 3(-2x^2 - 4x + 2)^2 - 4(-2x^2 - 4x + 2) - 5 and g(x) = -3|x - 3| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1(2x - 2) - 5 and g(x) = -2|x| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -3(3x^2 + 5x + 2) - 2 and g(x) = -2x - 5 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1(-2|x - 2| + 2) - 1 and g(x) = (-2x + 2)/(2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -4((-3x)/(1x + 1)) - 4 and g(x) = (1x - 1)/(-2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (3(-3x + 2) + 1)/(2(-3x + 2) - 1) and g(x) = 5x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3|(-1x - 5) - 2| and g(x) = -5x^2 - 5x meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -5(1e^(1x) + 1)^2 - 4(1e^(1x) + 1) - 4 and g(x) = (1x - 3)/(3x - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (2(5x^2 + 3x + 1) + 3)/(1(5x^2 + 3x + 1)) and g(x) = (-3x - 1)/(-2x + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 5(3x + 3)^2 + 5(3x + 3) - 2 and g(x) = 1x^2 - 4x + 5 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1(1e^(1x) - 1) - 1 and g(x) = -2|x| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3((1x)/(3x + 3)) + 5 and g(x) = -1|x - 1| + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2e^(-1(-4x + 1)) + 3 and g(x) = -3|x - 1| - 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3(2x^2 + 5x + 3) + 4 and g(x) = (3x)/(1x + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (1((-2x + 2)/(3x)) + 3)/(-2((-2x + 2)/(3x)) + 3) and g(x) = (3x + 3)/(1x) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(-1x^2 - 2x - 1) - 1)/(1(-1x^2 - 2x - 1) - 3) and g(x) = (-2x - 2)/(-1x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5(1x^2 + 5) + 1 and g(x) = (3x)/(-2x + 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1(-1|x - 3|) + 1 and g(x) = 3|x + 3| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1sin(2\u03c0(-2|x - 1| - 2) - 2) + 1 and g(x) = -3x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 4(1x^2 - 2x - 4) + 5 and g(x) = -3|x - 2| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2|(5x + 2) + 2| and g(x) = 3x + 5 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-1((3x + 2)/(1x)) + 3)/(-2((3x + 2)/(1x)) + 3) and g(x) = -1|x| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(-5x^2 + 2x))/(-2(-5x^2 + 2x) - 1) and g(x) = 1x + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1sin(1\u03c0(1x + 2) + 2) and g(x) = 2x^2 - 5x - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1(-1x^2 + 5x + 4) + 3 and g(x) = 1x^2 + 3x meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1|(-1sin(3\u03c0x + 1) + 3) - 1| - 1 and g(x) = -2x - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1(3|x + 3| + 1) + 2 and g(x) = 1|x + 2| - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 4(4x^2 + 4x - 5) - 4 and g(x) = (2x - 2)/(-1x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1sin(2\u03c0((-2x - 3)/(2x + 2)) - 1) - 3 and g(x) = -1x^2 - 1x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1((-3x + 1)/(-3x))^2 - 5((-3x + 1)/(-3x)) - 4 and g(x) = 5x^2 + 4x + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (2(-1x^2 - 3x) - 1)/(2(-1x^2 - 3x)) and g(x) = (-2x - 1)/(-1x - 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1(-2e^(-1x) + 1)^2 + 5(-2e^(-1x) + 1) + 2 and g(x) = -2x^2 - 1x + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (1(-3|x + 3| + 2) + 3)/(-2(-3|x + 3| + 2) + 3) and g(x) = -1x^2 + 3x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-1(2x^2 + 3x - 3) + 1)/(1(2x^2 + 3x - 3) + 2) and g(x) = -4x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -4(3|x - 3|)^2 + 3(3|x - 3|) + 2 and g(x) = -3x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3|(1x + 5) - 3| - 1 and g(x) = (-1x + 2)/(2x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1e^(-1(-2x^2 + 5x + 5)) + 2 and g(x) = (-2x + 3)/(2x + 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(1e^(-1x) - 2) - 3)/(1(1e^(-1x) - 2) + 3) and g(x) = 4x^2 + 5x - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1((2x - 1)/(1x - 2)) + 2 and g(x) = -2|x| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1((-3x + 3)/(2x))^2 - 5((-3x + 3)/(2x)) + 4 and g(x) = 1|x - 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(1sin(3\u03c0x - 1) + 3) - 1)/(3(1sin(3\u03c0x - 1) + 3)) and g(x) = (2x + 3)/(-1x + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1e^(-1(2x^2 - 5x + 4)) + 2 and g(x) = 5x + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-3(-4x - 3) - 3)/(-1(-4x - 3)) and g(x) = -4x^2 - 2x + 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1(1x^2 + 4x - 3) + 5 and g(x) = 3|x| + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(-2e^(1x)) + 3)/(-1(-2e^(1x)) - 3) and g(x) = 3x + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3|(-2|x + 3| + 3)| - 3 and g(x) = 2|x - 2| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-9", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -4(-3x^2 + 4x - 2) + 3 and g(x) = (1x - 2)/(3x + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2(5x^2 - 4x + 2)^2 - 4(5x^2 - 4x + 2) - 4 and g(x) = 4x^2 - 3x - 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-1(-1x^2 + 4x - 1) - 3)/(1(-1x^2 + 4x - 1) + 1) and g(x) = -3x^2 + 5x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2(-4x^2 - 1x + 4)^2 + 1(-4x^2 - 1x + 4) + 3 and g(x) = 4x^2 + 5x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (3(1x^2 - 5x - 3) - 2)/(1(1x^2 - 5x - 3) + 2) and g(x) = -4x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-3(1|x - 2| - 1) - 2)/(-2(1|x - 2| - 1) - 3) and g(x) = -1|x + 2| + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3((-3x - 3)/(2x - 1)) + 1 and g(x) = 3x^2 + 3x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (3(-2|x + 1| - 2) + 1)/(1(-2|x + 1| - 2) - 3) and g(x) = -4x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (2(1x - 1) - 1)/(-2(1x - 1) - 1) and g(x) = -3|x - 1| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2(5x - 3)^2 - 4(5x - 3) - 3 and g(x) = -2|x + 2| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-3(3|x + 1| + 1) - 1)/(1(3|x + 1| + 1) - 3) and g(x) = 2x + 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2cos(1\u03c0(1x + 3)) - 3 and g(x) = (2x - 3)/(1x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1(-1e^(1x) - 1) + 1 and g(x) = 2|x - 1| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1(3x^2 + 5x + 1)^2 - 1(3x^2 + 5x + 1) + 5 and g(x) = -1|x + 3| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -1|((-2x + 1)/(2x - 3)) - 1| and g(x) = -3x^2 + 5x + 5 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2|(1x - 3) + 1| + 3 and g(x) = (-2x - 2)/(-2x - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -2e^(2((2x - 1)/(3x + 1))) and g(x) = -2|x + 2| + 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-8", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 4(2cos(3\u03c0x - 2))^2 - 1(2cos(3\u03c0x - 2)) + 2 and g(x) = (1x - 2)/(-1x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2((1x + 3)/(2x - 1))^2 + 4((1x + 3)/(2x - 1)) - 3 and g(x) = (-3x + 2)/(1x + 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3|(2sin(3\u03c0x) - 3) - 3| and g(x) = (2x + 1)/(1x - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1|(-2e^(-2x) - 1) + 1| + 3 and g(x) = 3|x - 3| + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -2sin(1\u03c0((2x + 3)/(-2x - 1)) + 3) - 2 and g(x) = (-1x - 1)/(1x) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 1(1x + 2) and g(x) = 2x^2 + 2x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-3((-1x + 1)/(-1x - 1)) - 3)/(-1((-1x + 1)/(-1x - 1)) + 2) and g(x) = -1x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-3(1x + 4) + 1)/(-2(1x + 4) + 3) and g(x) = 1x^2 + 2x - 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -3|(1e^(-1x) - 3) - 2| - 2 and g(x) = 3x + 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2(-2x^2 + 4x - 1) - 2 and g(x) = (-2x + 3)/(-3x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2e^(-2(3|x + 1| + 1)) - 2 and g(x) = (-2x + 3)/(-2x - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 1|(3x - 3) + 3| - 3 and g(x) = (2x + 1)/(2x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2((2x + 2)/(1x)) + 3)/(3((2x + 2)/(1x)) + 1) and g(x) = -3x^2 + 1x + 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1e^(-1(-1x^2 - 3x - 3)) - 1 and g(x) = (3x - 3)/(-1x - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -5(3e^(2x) - 3)^2 - 4(3e^(2x) - 3) - 5 and g(x) = 5x - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2(1|x - 2| - 1)^2 - 2(1|x - 2| - 1) + 5 and g(x) = (-1x)/(-3x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 1(1cos(2\u03c0x - 3) - 3) + 3 and g(x) = -2x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1|(1cos(3\u03c0x + 2)) - 1| - 3 and g(x) = (-1x - 1)/(2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2(-2|x - 1| + 1)^2 + 3 and g(x) = -3|x| + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 3|((1x + 1)/(-1x + 2)) - 1| + 1 and g(x) = 4x - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the third x-coordinate (rounded to the nearest integer) where f(x) = (2((2x - 3)/(1x + 3)) + 3)/(2((2x - 3)/(1x + 3)) + 2) and g(x) = 1|x| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(-3x + 3) + 1)/(-1(-3x + 3) + 2) and g(x) = 3x^2 + 1x intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (2((-1x)/(2x - 1)))/(-2((-1x)/(2x - 1)) - 2) and g(x) = (1x + 2)/(-1x - 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -2sin(2\u03c0(2|x| + 2) - 1) - 1 and g(x) = -4x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2(2cos(1\u03c0x + 3) + 1) - 1 and g(x) = -4x meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-1((-3x + 2)/(-1x + 3)) + 1)/(2((-3x + 2)/(-1x + 3)) + 3) and g(x) = (-2x + 1)/(3x + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1((-2x - 2)/(-2x))^2 - 5((-2x - 2)/(-2x)) - 1 and g(x) = -2x^2 + 2x - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -4(2cos(1\u03c0x - 3) + 2)^2 + 2(2cos(1\u03c0x - 3) + 2) - 1 and g(x) = (1x + 2)/(-3x + 2) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 3e^(2(-3x)) and g(x) = (2x - 2)/(2x + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2|(2x^2 - 3x + 3) - 1| - 3 and g(x) = 5x^2 - 1x - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1e^(-2((1x - 3)/(-3x))) + 1 and g(x) = -4x^2 + 2x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 1(-3x^2 + 5x + 5) and g(x) = 2|x| - 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1|((3x - 2)/(-2x)) + 2| - 1 and g(x) = -4x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2sin(1\u03c0(-3|x - 1| - 3) + 2) - 3 and g(x) = (-1x + 1)/(-2x + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-1((2x)/(-1x - 1)) + 3)/(1((2x)/(-1x - 1))) and g(x) = -2x - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(-1e^(1x) - 2) - 2)/(-1(-1e^(1x) - 2) + 2) and g(x) = 2x intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 4(-2sin(1\u03c0x - 2) + 3) + 5 and g(x) = (-1x + 3)/(-1x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-1(1|x - 2| - 1) - 1)/(-1(1|x - 2| - 1) + 2) and g(x) = 2x - 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2|((-1x)/(1x + 2)) + 1| - 1 and g(x) = 5x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1|(-2e^(-1x) + 2)| - 2 and g(x) = 3x^2 - 5x - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3|(5x - 2) - 2| - 2 and g(x) = (-2x + 1)/(1x - 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2|((-3x - 3)/(1x + 1)) + 3| + 1 and g(x) = (1x - 1)/(-3x + 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2|((-3x - 3)/(1x - 2)) + 2| + 1 and g(x) = 1x + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3|((1x - 3)/(-2x - 2)) - 2| + 1 and g(x) = (-1x - 3)/(-3x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2(-3e^(-2x))^2 - 4(-3e^(-2x)) and g(x) = 3|x + 1| - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3|(-1|x - 3| - 1) + 2| - 3 and g(x) = (2x + 1)/(1x + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(-1x^2 + 2x - 3) + 2| and g(x) = 1x - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 4((-2x + 2)/(2x)) + 4 and g(x) = (-2x - 2)/(-1x - 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (3((1x)/(3x + 3)) + 2)/(1((1x)/(3x + 3)) - 2) and g(x) = -1x^2 + 2x meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (3(5x^2 + 2x + 4) - 2)/(-1(5x^2 + 2x + 4)) and g(x) = 1x intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3e^(1(2|x - 3| + 1)) + 2 and g(x) = (1x + 3)/(2x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1|(4x + 1) - 1| + 3 and g(x) = 1|x + 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1|((-3x - 3)/(-1x - 2)) - 2| + 1 and g(x) = 2x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2(-2x + 3)^2 + 2 and g(x) = 2x + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 5(-4x^2 - 5x) - 4 and g(x) = 2|x + 2| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 5((-1x + 3)/(-1x - 2))^2 + 5((-1x + 3)/(-1x - 2)) + 1 and g(x) = (-3x + 2)/(1x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -2(-3|x - 2| - 2) - 1 and g(x) = 4x - 4 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 5(2x^2 - 4x - 1)^2 + 2(2x^2 - 4x - 1) + 3 and g(x) = 4x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3(2x + 4) + 5 and g(x) = 4x^2 + 4x - 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (3(-4x^2 + 4x - 5) + 3)/(-2(-4x^2 + 4x - 5) - 1) and g(x) = (-1x - 2)/(1x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -4((1x + 1)/(-3x + 1)) - 1 and g(x) = 3x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2(4x - 3)^2 - 2(4x - 3) - 5 and g(x) = (1x - 1)/(1x + 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3(-1|x - 1| - 1)^2 + 5(-1|x - 1| - 1) - 4 and g(x) = 1x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (1(2x) - 3)/(3(2x) + 3) and g(x) = 2x^2 + 5x + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (-2(3x + 1))/(3(3x + 1) - 2) and g(x) = -1x meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -4(3e^(-1x) - 3) + 5 and g(x) = -5x^2 - 4x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2|(-1|x - 2| + 2) + 3| and g(x) = -2x + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -2(-1e^(-2x) + 3) + 2 and g(x) = -4x^2 - 2x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1(-2sin(1\u03c0x - 3) + 3) + 3 and g(x) = 5x^2 + 3x + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3e^(2(-1|x + 3| - 1)) - 2 and g(x) = 3x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -5((-3x - 3)/(2x - 1))^2 - 3((-3x - 3)/(2x - 1)) + 2 and g(x) = 3x + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1((-1x - 2)/(2x + 3)) - 4 and g(x) = -1|x - 3| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (1(-1|x - 3| - 3) + 2)/(2(-1|x - 3| - 3) - 3) and g(x) = 2|x - 1| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (3(-1sin(2\u03c0x - 2) - 2) - 3)/(1(-1sin(2\u03c0x - 2) - 2) + 3) and g(x) = (-2x - 1)/(-2x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -4((3x - 3)/(1x))^2 + 3((3x - 3)/(1x)) + 5 and g(x) = -3x - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -3|(2e^(-2x) - 1) - 3| + 2 and g(x) = (3x + 2)/(2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1e^(2(5x^2 + 1x - 4)) - 3 and g(x) = (2x - 2)/(-2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-1((2x + 3)/(2x - 1)))/(-1((2x + 3)/(2x - 1)) - 2) and g(x) = -2|x + 2| + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1(-3|x - 2| - 2)^2 + 3(-3|x - 2| - 2) - 5 and g(x) = -1|x - 3| - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -3((-3x)/(-3x + 4)) - 4 and g(x) = (-3x + 3)/(3x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|((2x + 2)/(2x - 3)) - 1| - 1 and g(x) = -5x - 4 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 4(-3x - 5) + 4 and g(x) = (2x - 1)/(2x + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1(2x + 3) - 3 and g(x) = -2|x - 2| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3(-1x^2 - 3x) and g(x) = (-3x - 2)/(1x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -4(-3e^(1x) + 1)^2 - 1 and g(x) = (-2x - 2)/(2x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (3(-3x^2 + 3x + 4) - 1)/(-2(-3x^2 + 3x + 4) + 3) and g(x) = (1x + 2)/(2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3(3e^(2x) - 3)^2 + 4(3e^(2x) - 3) and g(x) = 2|x - 1| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 5((-2x - 1)/(-1x - 2)) - 4 and g(x) = 2x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3e^(1(1x^2 - 4x - 5)) and g(x) = -3x^2 + 1x + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 5(2|x + 3|) - 3 and g(x) = (-2x - 3)/(-3x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1cos(1\u03c0(-1x + 2) + 3) - 1 and g(x) = -3x + 5 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (3(3e^(2x)) + 1)/(-2(3e^(2x)) + 1) and g(x) = 1x^2 + 3x + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (1((1x - 3)/(-2x + 1)) + 1)/(3((1x - 3)/(-2x + 1)) + 1) and g(x) = (-3x + 3)/(-1x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -3(1x - 3)^2 + 4(1x - 3) + 4 and g(x) = (-3x + 1)/(3x - 2) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (2(1x + 5) - 3)/(-2(1x + 5) - 1) and g(x) = -3x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (1((2x - 2)/(-3x + 2)) - 1)/(-3((2x - 2)/(-3x + 2)) - 1) and g(x) = -1|x + 1| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 3|(3x) + 3| + 1 and g(x) = -3|x + 1| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2sin(1\u03c0(-2x - 5)) - 3 and g(x) = -2x - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (2((2x - 3)/(-2x - 1)) - 2)/(3((2x - 3)/(-2x - 1))) and g(x) = (-1x - 2)/(3x - 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (2(1sin(2\u03c0x + 2) + 2) - 1)/(2(1sin(2\u03c0x + 2) + 2) + 3) and g(x) = 1|x + 2| meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -5(2e^(-1x) + 2)^2 + 5(2e^(-1x) + 2) + 2 and g(x) = -1|x| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (1(2x^2 - 5x + 5) + 1)/(1(2x^2 - 5x + 5) - 3) and g(x) = (-2x + 1)/(-3x + 4) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 4(-2x - 3)^2 - 3(-2x - 3) + 2 and g(x) = (-2x + 1)/(2x) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (1(-5x - 5) - 2)/(2(-5x - 5) - 1) and g(x) = (-1x + 3)/(3x + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -5(-1x^2 - 3x + 1)^2 - 2(-1x^2 - 3x + 1) - 2 and g(x) = (-2x + 2)/(-3x + 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2e^(-2((-3x + 2)/(-3x))) + 1 and g(x) = 2x - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2|(-2e^(1x) - 1) + 3| - 3 and g(x) = -3|x - 3| + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (1(-4x^2 + 1x + 1) + 3)/(2(-4x^2 + 1x + 1) + 3) and g(x) = 5x - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3((3x + 2)/(3x + 2)) + 2 and g(x) = (-3x)/(-2x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 4((2x + 3)/(2x + 1))^2 + 4((2x + 3)/(2x + 1)) - 2 and g(x) = 3x^2 + 3x + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -3(2cos(3\u03c0x + 2)) + 3 and g(x) = -3|x - 2| - 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -5(-1sin(1\u03c0x - 1) - 1) + 3 and g(x) = 3x^2 + 5x + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (3(-2sin(3\u03c0x + 1) - 3) - 3)/(3(-2sin(3\u03c0x + 1) - 3) + 1) and g(x) = (1x + 1)/(-2x + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1|(3|x + 2| + 3) - 1| + 3 and g(x) = 5x meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the third intersection point (counting from left to right) between the functions f(x) = -2(-4x + 5)^2 - 5(-4x + 5) and g(x) = (3x + 2)/(3x - 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3|((-1x + 2)/(-1x - 3))| - 3 and g(x) = 2x + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 1cos(2\u03c0(1|x - 2| - 3) + 3) + 1 and g(x) = 3x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1((2x + 2)/(3x + 1))^2 + 1((2x + 2)/(3x + 1)) - 5 and g(x) = -3|x| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2cos(1\u03c0(-1x^2 + 5x - 1) + 2) + 3 and g(x) = (-2x + 3)/(1x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1(1cos(3\u03c0x - 3) - 3) - 2 and g(x) = -2x + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -5(-3x^2 + 4x - 1)^2 + 2 and g(x) = (-3x - 2)/(-1x + 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (1((-3x + 3)/(2x - 1)) - 2)/(-2((-3x + 3)/(2x - 1)) + 1) and g(x) = (3x + 1)/(-2x) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1(-4x + 5) - 2 and g(x) = -3|x - 2| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3(1cos(3\u03c0x - 1) - 3) and g(x) = (-2x - 1)/(2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(1sin(1\u03c0x + 1) + 3) - 1)/(2(1sin(1\u03c0x + 1) + 3)) and g(x) = (1x - 3)/(-2x + 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1e^(-2((2x + 3)/(-3x - 1))) - 1 and g(x) = (1x - 3)/(-1x + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 3(2x - 3)^2 - 2(2x - 3) - 3 and g(x) = 3x - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(3x + 3) - 3)/(-3(3x + 3) - 3) and g(x) = 1|x - 1| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (3(1x + 5) + 1)/(-2(1x + 5) - 2) and g(x) = -2|x - 1| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 5(-1x + 5) - 2 and g(x) = (-1x + 1)/(-3x - 2) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (2(4x^2 + 5x - 1) + 1)/(-2(4x^2 + 5x - 1) + 1) and g(x) = 1x - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1(-1sin(2\u03c0x - 2) + 2)^2 + 2(-1sin(2\u03c0x - 2) + 2) + 2 and g(x) = (3x - 1)/(-2x - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -5(3e^(2x) - 2) + 3 and g(x) = (1x + 1)/(-1x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (1(-2e^(1x) + 1) - 3)/(1(-2e^(1x) + 1) + 1) and g(x) = 5x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 5(-2|x - 1| + 1)^2 - 3(-2|x - 1| + 1) + 2 and g(x) = (-3x - 2)/(3x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3(-2cos(1\u03c0x + 2) - 2)^2 + 1 and g(x) = (-2x - 3)/(2x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3(1|x - 1| + 1)^2 + 1(1|x - 1| + 1) + 5 and g(x) = -1|x - 1| + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1(1|x| - 1) + 2 and g(x) = (1x - 2)/(-1x - 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1(-2|x + 2| - 2) + 5 and g(x) = (1x + 2)/(-3x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (2(-2|x - 1| - 2) + 3)/(-2(-2|x - 1| - 2) - 2) and g(x) = -3x^2 + 4x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1e^(-1(-5x - 3)) + 1 and g(x) = (1x + 1)/(2x - 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3((3x + 1)/(3x + 3))^2 + 4((3x + 1)/(3x + 3)) + 5 and g(x) = -3x^2 + 5x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1|(-2cos(1\u03c0x - 3) - 2) - 3| + 1 and g(x) = 3|x + 2| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -4(3x + 3) - 4 and g(x) = (2x - 2)/(1x + 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -2(-5x^2 + 3x + 1) + 2 and g(x) = (-1x + 1)/(3x + 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3e^(2(4x + 3)) - 3 and g(x) = 5x - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1|(-1|x + 3| - 2) + 1| + 3 and g(x) = 4x meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 5(-3x + 1)^2 - 5(-3x + 1) - 4 and g(x) = 2|x + 3| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1(4x - 5)^2 + 1(4x - 5) + 4 and g(x) = (2x + 1)/(-3x + 4) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5(-5x - 2) + 3 and g(x) = -2|x + 3| + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -4((-1x + 1)/(-3x + 2))^2 - 4((-1x + 1)/(-3x + 2)) + 3 and g(x) = 3x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2(-2x^2 + 4x - 2) + 1 and g(x) = -1x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 3(-1e^(-2x) + 3) - 1 and g(x) = (2x)/(2x + 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2(1|x - 3| - 3) - 4 and g(x) = -4x + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1e^(2(3|x + 1|)) + 1 and g(x) = (2x + 3)/(-1x - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (1(-4x^2 + 3x - 3) - 1)/(-2(-4x^2 + 3x - 3) + 1) and g(x) = -3x + 5 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-2(-2x^2 - 2x - 1) - 2)/(-2(-2x^2 - 2x - 1) + 3) and g(x) = 4x^2 - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3|(1e^(-2x) + 3) - 1| - 3 and g(x) = (-1x + 1)/(-1x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(3x^2 - 5x + 3)| + 3 and g(x) = (-2x + 2)/(1x) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1e^(1(1x^2 + 4x - 4)) - 2 and g(x) = (-2x - 1)/(1x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1|(-2x^2 + 1x) - 3| and g(x) = -1|x - 1| - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (-3(-1|x + 3|) - 1)/(3(-1|x + 3|) + 2) and g(x) = (1x + 1)/(2x + 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2(2|x - 1|)^2 - 4(2|x - 1|) + 1 and g(x) = 2x^2 + 5x - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -3(3|x + 3| - 3) and g(x) = 1x^2 + 5x - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2|(3|x - 3| - 1) - 1| + 1 and g(x) = 1x^2 - 3x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2|(2e^(-2x) - 2)| - 3 and g(x) = -4x + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3((2x - 2)/(-2x + 1))^2 - 5((2x - 2)/(-2x + 1)) - 5 and g(x) = 1x + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 4((-2x + 2)/(1x - 2))^2 + 4((-2x + 2)/(1x - 2)) + 2 and g(x) = (1x + 2)/(2x + 1) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1|(-4x + 2) + 1| and g(x) = -3x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -5(-1|x + 3| - 2)^2 + 3(-1|x + 3| - 2) - 5 and g(x) = -4x^2 - 1x - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1sin(2\u03c0((-2x - 3)/(-1x - 1)) - 1) and g(x) = -1x - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2e^(1((2x - 2)/(2x - 3))) + 2 and g(x) = 1|x - 2| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2(-1x + 5)^2 - 4(-1x + 5) - 2 and g(x) = -1x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (2((-3x - 3)/(1x + 3)))/(2((-3x - 3)/(1x + 3)) + 1) and g(x) = -1x - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-1(3e^(-2x) - 3) + 2)/(-2(3e^(-2x) - 3) + 3) and g(x) = -3x + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1(2|x + 2| - 3) and g(x) = -1x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 1|((-1x + 2)/(-3x + 2)) - 3| and g(x) = 1|x - 2| in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -4(2|x + 1| - 3) + 3 and g(x) = (-3x + 3)/(-2x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -2e^(1(-3|x + 2| + 2)) + 3 and g(x) = -1|x - 1| + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -4(1e^(-2x) + 2)^2 - 1(1e^(-2x) + 2) - 4 and g(x) = (1x - 3)/(-3x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2(-2x^2 + 3x + 5) + 4 and g(x) = 3x^2 + 3x + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1|((-1x + 1)/(-1x + 3))| + 3 and g(x) = -1x^2 + 4x + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(2x + 2) + 2)/(-3(2x + 2) - 1) and g(x) = 3x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3|((-2x + 3)/(-1x - 3)) - 3| + 2 and g(x) = 3x + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1sin(3\u03c0((1x - 2)/(-2x - 1)) - 2) - 2 and g(x) = -1x^2 - 4x + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (3(-5x^2))/(-2(-5x^2) + 3) and g(x) = 5x^2 + 4x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 2|(-1x^2 + 3) + 2| - 2 and g(x) = 5x^2 + 1x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1((3x)/(-2x - 2)) + 2 and g(x) = 1|x| - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-2(-3x^2 + 2x + 3))/(-1(-3x^2 + 2x + 3) + 2) and g(x) = (-2x - 3)/(-1x + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1(1sin(2\u03c0x - 2) - 1)^2 - 1(1sin(2\u03c0x - 2) - 1) - 3 and g(x) = (1x + 3)/(3x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1cos(1\u03c0((3x - 2)/(-2x + 3)) + 3) + 2 and g(x) = (-3x + 2)/(-3x - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1cos(2\u03c0((-1x + 2)/(3x - 2))) - 3 and g(x) = -2x + 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -1(-4x - 5)^2 + 5(-4x - 5) + 1 and g(x) = -2|x + 1| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2|(1e^(-2x) - 3) - 1| - 1 and g(x) = (2x + 2)/(-2x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3(3x^2 - 1x - 2) and g(x) = 5x^2 - 3x - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1((1x - 2)/(1x + 1)) + 3 and g(x) = -2x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (3(3sin(1\u03c0x + 2) - 3) - 1)/(2(3sin(1\u03c0x + 2) - 3) - 1) and g(x) = 5x^2 - 1x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 4((2x + 3)/(-1x + 2))^2 + 2((2x + 3)/(-1x + 2)) + 5 and g(x) = -3x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-3(-2|x + 2| - 2))/(3(-2|x + 2| - 2) - 2) and g(x) = 1x - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 4(2e^(1x) + 1) + 4 and g(x) = 3x^2 - 4x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -5((-2x)/(-1x + 2)) + 3 and g(x) = 2x + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(-4x - 4) - 2| and g(x) = -3|x - 1| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1(-3|x - 3| - 2) - 3 and g(x) = (-2x + 3)/(2x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1(2|x - 3|)^2 + 5(2|x - 3|) - 3 and g(x) = -3x - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-3(3cos(2\u03c0x + 1) + 3) + 1)/(-1(3cos(2\u03c0x + 1) + 3) - 2) and g(x) = 2x^2 + 1x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 5(-3x - 5) + 4 and g(x) = (-1x - 3)/(-3x - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3(1e^(2x) + 1) - 3 and g(x) = 4x^2 - 5 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (3(1x - 4))/(3(1x - 4) + 1) and g(x) = -1|x| - 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2(1|x| + 1) - 2 and g(x) = -4x + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (1(-1x^2 - 1x + 3) - 1)/(3(-1x^2 - 1x + 3) - 2) and g(x) = 5x meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (3(1x - 5) + 1)/(-1(1x - 5) + 2) and g(x) = 3x^2 + 3x - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -5(-2|x + 1| + 3)^2 + 1 and g(x) = 4x + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -5(-2e^(-2x) - 1)^2 - 2(-2e^(-2x) - 1) - 4 and g(x) = (1x - 3)/(3x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -4(1sin(2\u03c0x - 2) + 3) + 3 and g(x) = (-2x + 2)/(-3x) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3|(-3sin(3\u03c0x + 3) + 1) + 3| - 1 and g(x) = (-3x - 3)/(1x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1(-1x^2 - 2x - 1) + 3 and g(x) = (3x)/(2x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 4(-3x - 1) - 2 and g(x) = -3|x + 1| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the third intersection point (counting from left to right) between the functions f(x) = 5(-3x^2 - 5x + 2)^2 - 5(-3x^2 - 5x + 2) - 4 and g(x) = (-3x - 3)/(2x - 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5(-3e^(1x) - 1)^2 + 5(-3e^(1x) - 1) + 2 and g(x) = (-2x - 2)/(-2x + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (1(-1cos(2\u03c0x + 3) - 1) + 1)/(-2(-1cos(2\u03c0x + 3) - 1) + 1) and g(x) = (-2x - 1)/(1x + 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2|(3|x - 2| + 1) - 1| + 3 and g(x) = 3x^2 - 2x - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (3(-2|x + 3| - 2) - 2)/(1(-2|x + 3| - 2) + 3) and g(x) = (3x - 2)/(2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -5(-1x + 4)^2 + 4(-1x + 4) + 3 and g(x) = -3x - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 1|(1x + 1) + 2| + 2 and g(x) = 2x^2 - 3x - 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 4((1x - 2)/(-2x))^2 - 5((1x - 2)/(-2x)) + 4 and g(x) = 3|x - 2| - 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 2|(-3|x - 1|) + 1| - 3 and g(x) = (2x - 3)/(-2x + 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2e^(-2(-2|x| + 2)) + 2 and g(x) = (1x - 3)/(-1x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (2(3x^2 + 2x + 4) - 1)/(1(3x^2 + 2x + 4) + 3) and g(x) = -1x - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -5((-3x + 2)/(-1x + 2))^2 + 5((-3x + 2)/(-1x + 2)) - 1 and g(x) = (-1x)/(-3x - 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1e^(-1(5x + 2)) - 1 and g(x) = -1|x - 3| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 5(5x - 2) + 4 and g(x) = (1x - 1)/(2x + 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2e^(2(-4x - 5)) - 3 and g(x) = -1x^2 - 1x - 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-2(-2e^(1x) - 3) + 1)/(3(-2e^(1x) - 3)) and g(x) = (1x + 2)/(-3x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1((-3x - 3)/(-3x + 1)) + 2 and g(x) = (-1x + 2)/(2x - 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -5(1x + 4)^2 + 3(1x + 4) + 2 and g(x) = 1x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(-2|x - 1| - 3) - 1)/(2(-2|x - 1| - 3)) and g(x) = (2x + 2)/(-1x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2|((-2x - 3)/(-2x - 3)) - 1| - 2 and g(x) = -2|x - 3| + 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (-1(-3|x + 1| - 3) + 3)/(2(-3|x + 1| - 3) - 3) and g(x) = -3|x + 2| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2(1e^(-1x)) + 4 and g(x) = -3x^2 - 4x + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-3(-3|x| - 3) - 3)/(-2(-3|x| - 3) - 3) and g(x) = 2|x - 2| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3(-1e^(1x) - 3)^2 - 2(-1e^(1x) - 3) - 2 and g(x) = 3x^2 - 4x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -3cos(1\u03c0(-1|x + 3| - 2) + 2) + 3 and g(x) = 2x^2 - 3x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2(1sin(1\u03c0x - 3) - 2)^2 + 4(1sin(1\u03c0x - 3) - 2) + 4 and g(x) = (-1x + 2)/(2x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 3cos(1\u03c0((-3x - 2)/(2x))) - 2 and g(x) = 2x^2 + 4x + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3e^(1(-1|x + 2| - 3)) and g(x) = (-2x + 3)/(-1x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3e^(1((1x + 1)/(3x - 2))) + 3 and g(x) = -2x + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1sin(2\u03c0(1x^2 - 2x - 3) - 2) + 2 and g(x) = -5x^2 + 2x + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3(2x + 2)^2 + 1(2x + 2) - 3 and g(x) = (2x)/(2x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 1(-1x^2 + 3x + 5)^2 - 1(-1x^2 + 3x + 5) - 5 and g(x) = (-1x - 1)/(2x + 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1(1sin(3\u03c0x + 2) + 3) + 4 and g(x) = 2x^2 - 1x + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 3cos(1\u03c0(1|x + 1| + 3) - 2) - 1 and g(x) = 2x - 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (2(4x^2 + 2x) - 3)/(2(4x^2 + 2x)) and g(x) = -3x + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1cos(1\u03c0((-1x)/(-1x - 3)) + 1) + 2 and g(x) = 4x^2 + 3x - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1|(2|x - 3|)| - 1 and g(x) = -1|x + 1| - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2(-3x + 3)^2 + 4(-3x + 3) - 3 and g(x) = (-2x + 3)/(1x + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1(2|x - 2| - 1)^2 - 3(2|x - 2| - 1) + 2 and g(x) = (-3x - 3)/(3x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (1(1e^(1x) + 1) + 2)/(2(1e^(1x) + 1)) and g(x) = -1x - 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3e^(-2(5x - 3)) + 1 and g(x) = (1x - 1)/(3x - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (1(5x^2 - 4) + 1)/(-1(5x^2 - 4) + 3) and g(x) = (-3x - 2)/(1x) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1|((-2x - 3)/(-2x - 2)) + 2| + 3 and g(x) = 3x + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3|(-2|x - 2| - 1) - 2| + 3 and g(x) = (-2x - 2)/(3x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1sin(1\u03c0(3|x - 2| + 1)) - 3 and g(x) = -5x - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 4((-1x - 2)/(3x + 1)) - 2 and g(x) = (-3x + 3)/(1x + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1e^(-2(-1|x + 2| + 3)) + 2 and g(x) = (-2x + 1)/(3x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(-2x^2 - 4x + 2) - 1)/(-2(-2x^2 - 4x + 2) - 3) and g(x) = 2|x + 3| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 4(2sin(2\u03c0x - 1) + 3) + 4 and g(x) = 5x^2 + 1x meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1|((-2x + 3)/(-2x - 3))| and g(x) = 5x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1((-3x - 2)/(1x + 2)) - 3 and g(x) = 1x^2 - 3x - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3(2e^(2x) + 2)^2 + 4(2e^(2x) + 2) + 2 and g(x) = 3x^2 - 3x + 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -3|(3|x - 2| + 1) - 3| + 2 and g(x) = -2|x| - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2(5x^2 - 2x - 2) + 5 and g(x) = 4x - 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 2cos(3\u03c0((-3x + 1)/(-1x - 3)) + 1) and g(x) = -5x + 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 3(-3|x - 1| + 3) + 2 and g(x) = -2x^2 - 4 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|(-2x + 1) - 1| + 1 and g(x) = 2|x - 3| + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 2e^(1(1|x| - 2)) - 1 and g(x) = (1x + 2)/(2x - 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 5(-2x - 3) - 1 and g(x) = 1|x - 3| + 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2sin(2\u03c0(-2x^2 - 1x + 2) + 2) - 3 and g(x) = (-2x - 1)/(-2x - 3) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1((-3x - 2)/(1x - 3)) + 4 and g(x) = 3|x - 2| + 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1((-2x - 3)/(-2x - 2)) + 2 and g(x) = -1x + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2((-1x + 1)/(2x + 2)) - 1 and g(x) = 1x^2 - 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1e^(2((2x + 3)/(2x - 1))) and g(x) = 5x meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1e^(-1((3x)/(-2x))) + 3 and g(x) = (-1x - 2)/(3x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-3((3x - 3)/(-1x + 3)) + 3)/(3((3x - 3)/(-1x + 3))) and g(x) = 1x^2 + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1(3e^(-1x) - 3)^2 - 2(3e^(-1x) - 3) + 5 and g(x) = (-1x + 1)/(-3x - 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2(-2x - 2)^2 - 1(-2x - 2) - 1 and g(x) = (-1x)/(3x - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2cos(1\u03c0(-1|x| + 3) - 1) - 1 and g(x) = 4x^2 + 2x - 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 1e^(2(-3|x + 1| - 2)) + 2 and g(x) = -5x^2 - 1x + 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (1(-3x^2 + 3x - 5) - 2)/(-1(-3x^2 + 3x - 5) - 2) and g(x) = 1x^2 - 3x - 4 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (2(1x^2 - 3x))/(3(1x^2 - 3x) - 2) and g(x) = (-3x - 2)/(-2x - 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2e^(1(3x + 4)) - 3 and g(x) = 2x + 5 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -1|(1cos(3\u03c0x + 2) - 3) + 3| + 2 and g(x) = (-1x + 2)/(2x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3|(-1e^(-2x) + 2) - 1| + 1 and g(x) = -3|x| - 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (1((2x + 3)/(-1x + 2)) - 2)/(-1((2x + 3)/(-1x + 2)) + 2) and g(x) = -1x^2 + 2x + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2|((-3x + 3)/(1x + 1))| + 2 and g(x) = -4x^2 - 5x + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3|(-2sin(1\u03c0x + 3)) + 3| and g(x) = 1x + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-8", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2cos(3\u03c0(-3|x + 2| + 2)) + 3 and g(x) = (-1x + 1)/(3x + 1) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2sin(2\u03c0(-2|x - 2| + 2) - 3) + 3 and g(x) = 4x^2 - 2x - 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (1(2x^2 + 1x + 2) + 1)/(2(2x^2 + 1x + 2) + 1) and g(x) = -2x - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2|(-1cos(3\u03c0x - 2) + 1) + 1| + 1 and g(x) = (-2x - 1)/(2x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 4(2x - 3) - 2 and g(x) = (2x + 2)/(-3x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1e^(1(-4x^2 + 3x - 2)) + 3 and g(x) = (3x + 1)/(-1x - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-2(-1x + 3) - 3)/(-1(-1x + 3) - 1) and g(x) = (-1x + 2)/(2x - 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1e^(1(-4x)) + 3 and g(x) = 1x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1e^(1((2x + 2)/(3x + 2))) and g(x) = 1x^2 - 5x - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (3(-1e^(2x) + 1) + 1)/(2(-1e^(2x) + 1)) and g(x) = 3|x - 3| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(3|x - 2| - 2) + 3| + 3 and g(x) = -2x - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-3(2cos(3\u03c0x - 1) - 3) + 3)/(2(2cos(3\u03c0x - 1) - 3) - 3) and g(x) = -1x^2 + 4x - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2e^(2(-1|x + 1| - 1)) + 2 and g(x) = 1x^2 + 3x - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1|((-1x + 2)/(1x)) - 1| + 1 and g(x) = -1x + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1e^(-1((2x - 3)/(2x - 1))) + 1 and g(x) = (-3x - 3)/(-1x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1e^(1(-4x)) - 1 and g(x) = -3|x - 3| - 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2(-1x - 1) - 3 and g(x) = -5x^2 - 4x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1(2|x| - 2) + 4 and g(x) = (-1x)/(-1x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1e^(-2(-2|x - 1| + 1)) + 1 and g(x) = 2x^2 + 3x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -5(-4x + 2) + 1 and g(x) = -5x^2 - 1x + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2(-5x - 5)^2 + 4(-5x - 5) - 1 and g(x) = (-3x - 3)/(1x + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3|((3x + 2)/(-2x + 1)) - 1| - 1 and g(x) = -3x + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (3(5x - 1) + 2)/(1(5x - 1) - 2) and g(x) = -1x^2 + 2x + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3e^(-1(4x^2 + 2x - 5)) and g(x) = (3x)/(1x - 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 5(-5x + 1)^2 + 5(-5x + 1) + 5 and g(x) = (3x - 1)/(2x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -4(-1sin(1\u03c0x + 2) + 2)^2 - 5(-1sin(1\u03c0x + 2) + 2) + 2 and g(x) = -5x^2 + 4x + 5 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (1(5x + 2) - 3)/(-3(5x + 2) - 3) and g(x) = 1x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 4((3x + 1)/(3x + 2))^2 - 5((3x + 1)/(3x + 2)) + 2 and g(x) = 2x^2 + 1x + 4 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 4(1x + 2)^2 + 2(1x + 2) - 3 and g(x) = (-2x + 3)/(-3x - 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -1((1x + 2)/(3x - 1))^2 + 4((1x + 2)/(3x - 1)) + 3 and g(x) = -2|x| - 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 3|(-2cos(1\u03c0x + 2) - 2) + 3| - 1 and g(x) = 5x^2 + 2x + 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (3(-1x^2 - 4x + 2) - 3)/(-1(-1x^2 - 4x + 2) - 3) and g(x) = -2x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2(3|x - 2|) + 1 and g(x) = (3x - 3)/(-1x - 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1e^(2(-3x - 1)) + 3 and g(x) = -3|x + 3| - 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1(-2x^2 - 1x - 3)^2 - 3(-2x^2 - 1x - 3) - 2 and g(x) = -3x^2 - 1x - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -2(-1x^2 - 1x + 2) and g(x) = 2|x + 2| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1sin(2\u03c0(-2x - 2) + 2) - 1 and g(x) = (1x)/(2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -1(-1x^2 - 5x + 5) - 3 and g(x) = -2x^2 - 4x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (-2(2x^2 - 1) - 2)/(-1(2x^2 - 1) - 1) and g(x) = -1x^2 - 1x + 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 5(1x^2 + 1x + 4) - 5 and g(x) = (2x + 1)/(3x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1sin(2\u03c0((-1x + 1)/(1x + 3)) + 2) + 3 and g(x) = (-1x + 3)/(-3x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (3(-2|x|) + 1)/(-3(-2|x|) - 1) and g(x) = 1|x - 3| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1(-1|x + 2|)^2 + 5(-1|x + 2|) + 1 and g(x) = -1x^2 + 4x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1|(-1|x| - 3)| - 2 and g(x) = -4x^2 + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1|(-2cos(1\u03c0x - 1) + 2) + 3| and g(x) = -1x - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-25", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (2(5x - 4) + 2)/(2(5x - 4) - 1) and g(x) = 1x + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-3(1x^2 - 2x + 1) - 3)/(-1(1x^2 - 2x + 1) - 3) and g(x) = (-1x + 2)/(2x) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3e^(1(-5x^2 + 1x + 3)) + 1 and g(x) = 1|x + 2| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(-2x^2 - 2x - 4) - 1)/(-2(-2x^2 - 2x - 4) - 3) and g(x) = 2|x + 1| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -3(5x^2 - 3x) + 5 and g(x) = 5x^2 + 2x - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2(3x + 5) + 3 and g(x) = (-2x - 3)/(1x + 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1(2e^(-1x) - 2) - 2 and g(x) = 3x - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1e^(-1(4x^2 + 1x + 3)) and g(x) = -3|x - 3| intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1sin(3\u03c0((-2x - 2)/(-3x - 2)) + 3) + 3 and g(x) = (-2x + 1)/(-3x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-2(-3x - 2))/(-2(-3x - 2) + 1) and g(x) = -1|x + 1| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1(2x - 5) + 1 and g(x) = (3x - 1)/(1x + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|((1x)/(-1x)) - 1| and g(x) = (-1x - 2)/(2x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -2cos(1\u03c0((1x + 1)/(2x + 3)) - 3) + 3 and g(x) = 5x^2 + 1x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -1(2cos(1\u03c0x - 3) - 3)^2 - 5(2cos(1\u03c0x - 3) - 3) + 5 and g(x) = -4x - 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1|(2x^2 + 4) + 1| + 2 and g(x) = (-1x + 3)/(-2x - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (3(4x^2 - 5x) - 1)/(3(4x^2 - 5x) - 2) and g(x) = (2x + 2)/(3x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 5(1x + 4)^2 + 5(1x + 4) + 2 and g(x) = 2|x + 2| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-9", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2|(1x^2 - 2x + 4) - 2| + 1 and g(x) = (-1x)/(-2x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (3(-3x^2 + 3x + 4) + 3)/(2(-3x^2 + 3x + 4) + 1) and g(x) = -3x + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3(-2|x| + 2)^2 - 2(-2|x| + 2) - 2 and g(x) = (-2x + 3)/(2x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1((-2x - 1)/(1x - 2))^2 + 5((-2x - 1)/(1x - 2)) + 2 and g(x) = 2|x + 2| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2|((1x + 3)/(1x - 3)) + 1| and g(x) = 5x^2 + 5x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3|((3x - 1)/(-2x - 1)) - 1| + 1 and g(x) = 1x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-8", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2|((-2x + 3)/(-1x + 3))| + 2 and g(x) = -3x + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-3(-1e^(2x) + 2) - 1)/(3(-1e^(2x) + 2) + 2) and g(x) = 2|x - 3| - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2(-1x - 2)^2 - 5(-1x - 2) - 3 and g(x) = 2|x| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (2(4x^2 - 3x + 2))/(-1(4x^2 - 3x + 2) - 2) and g(x) = -2|x - 3| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-1((-3x - 3)/(1x + 2)) + 1)/(-1((-3x - 3)/(1x + 2)) + 2) and g(x) = -2x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (-3(-1x^2 - 4x - 4) - 3)/(-1(-1x^2 - 4x - 4) - 3) and g(x) = -1x - 4 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -4(-1x^2 + 4x - 3)^2 - 4 and g(x) = -2|x| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1e^(-2((2x - 2)/(-3x + 1))) - 3 and g(x) = (1x - 1)/(-1x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3(-2|x + 1| - 3) - 1 and g(x) = (-2x + 1)/(1x + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -3e^(-1((2x + 3)/(-3x - 2))) - 2 and g(x) = -2x^2 - 3x + 4 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-1((2x + 1)/(-2x + 1)) - 1)/(-1((2x + 1)/(-2x + 1))) and g(x) = -3|x + 1| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2e^(-1(-4x + 5)) + 2 and g(x) = 4x^2 + 5x + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1|((-1x + 2)/(-3x - 1)) - 3| - 1 and g(x) = -2x^2 - 1x - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -5(4x - 4) - 3 and g(x) = 3|x| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3|(4x^2 - 3x - 1) - 2| + 1 and g(x) = 5x - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (2(5x^2 - 2) + 1)/(2(5x^2 - 2) - 1) and g(x) = 1x + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -4(-1|x - 2| + 3)^2 - 1(-1|x - 2| + 3) + 1 and g(x) = -4x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -1(3x^2 - 4x - 1)^2 + 2(3x^2 - 4x - 1) - 1 and g(x) = -2x^2 - 5x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -5(2x^2 + 4x + 2)^2 - 5 and g(x) = (-1x - 1)/(-1x - 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1sin(1\u03c0(-3x^2 - 3x) + 3) + 2 and g(x) = (-3x + 2)/(3x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -3e^(-1((-2x - 3)/(-2x))) - 2 and g(x) = -3|x - 1| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1|((-1x - 3)/(-1x - 1)) + 1| + 2 and g(x) = -2x^2 - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = (2(-4x - 4) - 3)/(2(-4x - 4) + 1) and g(x) = 1x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-2((1x)/(-3x - 3)) + 2)/(2((1x)/(-3x - 3)) + 1) and g(x) = -5x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1|(-4x + 3) + 1| - 1 and g(x) = 3|x + 2| + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1sin(3\u03c0((-2x)/(1x + 2)) - 1) - 1 and g(x) = 2|x - 2| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -3((1x + 1)/(3x - 2)) + 4 and g(x) = (2x - 2)/(-1x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (1(2|x - 1| + 3) - 3)/(-2(2|x - 1| + 3)) and g(x) = 3x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 3(-2x^2 - 1x - 1)^2 - 4(-2x^2 - 1x - 1) - 3 and g(x) = -5x + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1((-3x + 2)/(3x - 2)) - 1 and g(x) = (-1x - 3)/(1x + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1((-1x + 2)/(2x)) - 2)/(-2((-1x + 2)/(2x))) and g(x) = 2x^2 - 3x - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1e^(-1((3x - 3)/(3x - 2))) - 2 and g(x) = 2x^2 + 4x + 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3(-1|x + 2| - 3) - 1 and g(x) = -5x - 5 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -5(3x^2 + 1x - 3) - 2 and g(x) = -2x + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2|((-2x + 2)/(-2x)) + 3| - 2 and g(x) = (-1x - 1)/(-3x + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1cos(3\u03c0((-3x + 1)/(-2x - 2)) - 2) + 2 and g(x) = -3x + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 5(-1x^2 - 1x + 2) and g(x) = (3x + 2)/(3x - 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 3e^(1((-1x - 2)/(2x + 2))) + 3 and g(x) = -4x + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1((3x + 2)/(-2x)) + 3 and g(x) = 3x^2 + 3x + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -5(-2x - 2)^2 - 3(-2x - 2) - 4 and g(x) = 1x - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3e^(1(5x + 4)) + 3 and g(x) = (-3x - 2)/(-1x - 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-3((2x - 1)/(-2x - 1)) - 2)/(-3((2x - 1)/(-2x - 1)) + 2) and g(x) = (1x + 3)/(-2x + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3(5x + 2) - 1 and g(x) = -2x^2 + 1x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 1|(-2x + 1) + 3| - 2 and g(x) = (-2x + 3)/(2x + 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1(5x^2 + 5x + 1)^2 - 3(5x^2 + 5x + 1) - 5 and g(x) = 5x + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-3(-3e^(-2x) - 1) - 1)/(-3(-3e^(-2x) - 1) - 1) and g(x) = 1x + 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-3(-1x + 5) - 1)/(-2(-1x + 5) + 1) and g(x) = -5x^2 + 4x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 2e^(2((1x + 1)/(2x + 1))) + 3 and g(x) = 2x^2 - 3x meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-2(5x^2 + 1x - 2) + 1)/(1(5x^2 + 1x - 2) - 3) and g(x) = (1x + 2)/(-2x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1(2x^2 - 4x + 2) - 1 and g(x) = -3|x + 1| + 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (2(1x^2 + 1x + 4) - 2)/(1(1x^2 + 1x + 4) + 1) and g(x) = -1|x + 3| + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2((1x + 2)/(-3x + 2)) + 3 and g(x) = -2|x - 3| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2((-3x - 2)/(2x - 3)) - 2)/(1((-3x - 2)/(2x - 3)) - 3) and g(x) = 1x - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3(-5x + 3) + 2 and g(x) = 2|x + 2| - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2(-3e^(1x))^2 - 5(-3e^(1x)) - 2 and g(x) = (3x + 2)/(1x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1|(3|x + 3| - 3) - 3| - 1 and g(x) = -2x^2 + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (3(-1e^(2x)) - 3)/(1(-1e^(2x)) + 3) and g(x) = -3|x - 2| + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1|(-3cos(1\u03c0x) - 2) + 3| + 3 and g(x) = (-1x + 1)/(2x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3|(2e^(2x) - 3) + 2| + 2 and g(x) = (-1x + 1)/(-1x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3|(1x + 3) + 1| - 3 and g(x) = 2|x + 1| - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1(1e^(1x) + 2) + 4 and g(x) = -4x - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (3(1sin(2\u03c0x - 3) - 2))/(-3(1sin(2\u03c0x - 3) - 2) - 1) and g(x) = 1x + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1((-1x + 3)/(-2x + 1)) - 1 and g(x) = -1|x - 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 4(1|x| - 2) + 5 and g(x) = -5x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1|(2e^(-1x)) - 1| - 2 and g(x) = -2x - 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3e^(-1(2x^2 + 5x + 5)) - 2 and g(x) = -1x - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-3(3e^(-1x) - 1) - 2)/(2(3e^(-1x) - 1) - 3) and g(x) = -4x^2 + 2x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1sin(1\u03c0(1|x + 3| + 1) - 2) + 3 and g(x) = (-3x - 2)/(-3x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 4(2x + 1) - 1 and g(x) = 1|x + 3| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (-1(4x) + 2)/(3(4x) - 2) and g(x) = -3x^2 - 5x + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1((-2x - 3)/(-2x + 3))^2 - 3((-2x - 3)/(-2x + 3)) and g(x) = -4x - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2|(1|x - 2|) - 3| and g(x) = 3|x + 1| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1|((-3x - 1)/(-2x - 3)) - 1| - 1 and g(x) = (-2x + 1)/(-3x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 4(5x^2 - 4x)^2 - 4(5x^2 - 4x) - 5 and g(x) = -5x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -4(-2x - 3) - 2 and g(x) = 3|x - 1| - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3|(-3e^(-1x) - 1) + 2| + 3 and g(x) = 5x^2 + 2x + 5 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2((-1x + 1)/(3x + 3))^2 + 5((-1x + 1)/(3x + 3)) + 5 and g(x) = (2x - 3)/(-3x) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2e^(-1(2x^2 + 1x - 2)) + 3 and g(x) = -1x + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1e^(1(-3x - 2)) and g(x) = 3|x| - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1(1cos(2\u03c0x - 3)) and g(x) = (-3x + 2)/(2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1(-2sin(1\u03c0x + 1) - 3) - 2 and g(x) = 1x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 1(-2x^2 - 3x - 2)^2 + 4(-2x^2 - 3x - 2) - 1 and g(x) = -3|x + 2| + 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (2(-1x^2 + 5x + 3) + 2)/(-3(-1x^2 + 5x + 3) - 2) and g(x) = -2x^2 + 5x - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -2|(-1sin(1\u03c0x + 2) - 3)| + 2 and g(x) = -1x^2 + 5x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3|((-2x - 1)/(3x + 1)) + 3| and g(x) = -5x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3e^(-1(1x + 4)) - 3 and g(x) = (-1x - 2)/(3x + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-1((-2x - 1)/(-1x + 2)) - 2)/(1((-2x - 1)/(-1x + 2))) and g(x) = 4x - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3|((1x + 1)/(1x)) - 3| + 3 and g(x) = -3x + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the third x-coordinate (rounded to the nearest integer) where f(x) = (3(-2|x - 2| + 2) - 2)/(-1(-2|x - 2| + 2) + 2) and g(x) = (2x - 3)/(3x) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1(-4x - 1)^2 + 2(-4x - 1) - 3 and g(x) = 3x^2 - 1x + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3(2|x + 3| - 2) + 5 and g(x) = (-2x + 1)/(2x + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (1(-1cos(2\u03c0x + 2) - 3))/(-1(-1cos(2\u03c0x + 2) - 3) + 2) and g(x) = 1x + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 2(-3e^(-1x)) - 1 and g(x) = (-2x - 1)/(1x + 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2((3x - 2)/(3x + 3)) - 4 and g(x) = (2x)/(3x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (2(-3e^(-2x) + 3) - 3)/(1(-3e^(-2x) + 3) + 1) and g(x) = (-2x - 1)/(-3x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2((2x + 1)/(1x - 2)) - 1 and g(x) = (1x - 2)/(-2x + 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3(1|x + 3| - 2)^2 - 1(1|x + 3| - 2) - 4 and g(x) = 2x^2 - 2x - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1((-2x + 3)/(-1x + 3)) + 1 and g(x) = -4x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-3(-2e^(2x) + 2) + 1)/(1(-2e^(2x) + 2) + 2) and g(x) = -5x meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2e^(2(-1x^2 + 1x + 4)) - 3 and g(x) = -3x - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -4(2|x - 1| - 2)^2 + 2(2|x - 1| - 2) - 4 and g(x) = -1|x + 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-2(2sin(2\u03c0x - 3) - 2) + 3)/(1(2sin(2\u03c0x - 3) - 2) - 2) and g(x) = 2|x - 1| - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 4(-4x^2 - 5x - 1) + 1 and g(x) = -3|x - 3| meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 4(-5x)^2 + 1(-5x) + 1 and g(x) = (3x - 3)/(1x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 3(1x^2 + 4x + 4)^2 - 1(1x^2 + 4x + 4) - 3 and g(x) = 2x^2 - 2x - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1|(-2|x - 3| + 1) + 1| + 1 and g(x) = -3|x - 1| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3|(5x + 4) + 3| - 1 and g(x) = -1x - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1((-1x - 1)/(3x + 3)) - 1 and g(x) = -3x - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1((1x - 3)/(-2x - 2))^2 + 5((1x - 3)/(-2x - 2)) + 1 and g(x) = (2x)/(-1x - 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -4(4x - 5) + 4 and g(x) = 3|x + 1| + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-2(-1x))/(-1(-1x) + 2) and g(x) = 2|x + 2| meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1|(-4x) - 1| - 3 and g(x) = (2x)/(1x - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2e^(2(3x^2 - 2x + 1)) + 2 and g(x) = (1x - 1)/(3x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (1(-3|x|) - 2)/(1(-3|x|)) and g(x) = (-2x)/(-3x - 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (2(3x + 2) - 3)/(-2(3x + 2) + 3) and g(x) = 1x - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (2(2|x + 3| - 3) - 1)/(2(2|x + 3| - 3) + 2) and g(x) = 1x^2 - 1x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -4((-3x - 1)/(-1x - 2)) - 1 and g(x) = 2x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1|(1|x + 2| + 3) + 3| + 2 and g(x) = 2x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 5(3|x + 2| - 1) - 4 and g(x) = -4x - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3(-1e^(1x) + 2) + 5 and g(x) = -1x^2 + 1x - 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2sin(1\u03c0((1x + 3)/(2x + 2))) - 3 and g(x) = (1x + 1)/(-2x + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2(1cos(2\u03c0x + 2) + 1)^2 + 5(1cos(2\u03c0x + 2) + 1) and g(x) = (1x + 2)/(-2x - 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -2|(-1|x + 2| + 2) + 2| and g(x) = (2x - 3)/(-1x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (2(4x + 1) - 2)/(-3(4x + 1) - 1) and g(x) = 1|x + 2| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (3(-1e^(1x)))/(2(-1e^(1x)) - 1) and g(x) = -1x + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (2(1|x - 1| - 3) - 3)/(3(1|x - 1| - 3)) and g(x) = (3x + 2)/(-1x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 3e^(2((-3x - 3)/(3x - 2))) - 3 and g(x) = 2x^2 - 5x - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1|(3|x - 1| + 3) - 3| - 1 and g(x) = 3x^2 - 5x - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3(-1|x - 2| + 1)^2 - 2(-1|x - 2| + 1) + 1 and g(x) = 2x + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3|(1|x + 3|) + 2| and g(x) = (-2x + 2)/(-1x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 5(1e^(1x) - 1)^2 - 2(1e^(1x) - 1) and g(x) = 2x^2 + 3x + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3((-1x + 3)/(-3x - 2))^2 - 4((-1x + 3)/(-3x - 2)) - 3 and g(x) = 2x - 4 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3(-3x + 5) + 2 and g(x) = 2|x - 3| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (2((-1x - 3)/(1x + 2)) - 1)/(1((-1x - 3)/(1x + 2))) and g(x) = 4x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-1(3|x - 1| + 2) + 1)/(1(3|x - 1| + 2)) and g(x) = 2x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (1(-1e^(-2x) - 3))/(1(-1e^(-2x) - 3) - 2) and g(x) = -1x meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the third intersection point (counting from left to right) between the functions f(x) = -1sin(1\u03c0(-3|x - 2| - 2) - 1) + 1 and g(x) = -3x + 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 4(3|x - 2| + 3)^2 + 5(3|x - 2| + 3) + 2 and g(x) = (-2x - 3)/(3x - 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2e^(-1(-3x - 3)) - 3 and g(x) = 3x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3|(-1x + 3) + 3| - 2 and g(x) = 2|x - 2| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 5(2x - 1) - 4 and g(x) = 2|x + 1| - 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2|(3|x - 2| - 1)| - 2 and g(x) = 1x^2 - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 1|(3x^2 - 2x - 5) + 1| - 1 and g(x) = 1x^2 + 3x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1(-2|x - 2| + 2) - 1 and g(x) = 4x - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (3(-1|x| + 2) + 2)/(3(-1|x| + 2) + 1) and g(x) = -3|x + 1| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1((2x + 1)/(2x - 3))^2 - 3((2x + 1)/(2x - 3)) + 4 and g(x) = 3x^2 - 5x + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 4(-1x - 3) and g(x) = -2x^2 - 2x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-3(2x - 1))/(-1(2x - 1) + 2) and g(x) = -1|x - 3| + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1|(-3|x - 3| + 1) - 1| + 1 and g(x) = (2x)/(-1x + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1((-1x + 2)/(1x + 1)) - 4 and g(x) = -1|x + 2| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 5((-2x)/(1x))^2 - 4((-2x)/(1x)) - 3 and g(x) = (-1x - 2)/(-1x + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1((1x + 1)/(2x))^2 + 2((1x + 1)/(2x)) - 2 and g(x) = (2x - 1)/(1x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2|(-2x^2 + 3x - 4) - 3| - 1 and g(x) = (1x - 3)/(-2x + 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-2(4x - 3) + 2)/(-1(4x - 3) + 3) and g(x) = -3|x + 1| + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -5(-3x + 2) + 5 and g(x) = -2x^2 - 3x - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-9", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-1(-1x^2 + 1x) + 3)/(2(-1x^2 + 1x) + 2) and g(x) = (1x)/(1x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -2(-2x^2 + 4x - 3) and g(x) = -1x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1|(3e^(-2x) - 2) + 2| + 1 and g(x) = (2x)/(-3x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3(1x - 4)^2 + 1(1x - 4) + 3 and g(x) = (3x + 2)/(-3x - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (3((1x + 3)/(1x + 2)))/(-2((1x + 3)/(1x + 2))) and g(x) = -5x^2 - 4x + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3((3x + 3)/(2x + 3)) + 4 and g(x) = -3|x + 3| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2|((3x)/(-3x + 4)) - 1| - 1 and g(x) = (1x + 1)/(1x + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (2((-3x - 2)/(-2x - 3)) - 1)/(-2((-3x - 2)/(-2x - 3)) - 3) and g(x) = (2x + 2)/(1x + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2|(-1e^(1x))| - 3 and g(x) = 4x + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3((3x - 3)/(3x - 1)) + 2 and g(x) = -1|x| + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2cos(1\u03c0((-2x - 3)/(1x)) + 3) + 3 and g(x) = (1x)/(-1x + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3|((-1x + 3)/(-1x + 2)) - 2| + 2 and g(x) = 3x + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1((1x + 3)/(-1x + 3)) + 2)/(-3((1x + 3)/(-1x + 3))) and g(x) = 2|x + 2| + 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 5((-1x - 3)/(2x + 1)) + 1 and g(x) = 3|x + 2| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (1(-2e^(-1x)) + 2)/(-3(-2e^(-1x)) + 4) and g(x) = -3x^2 - 2x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -4((-2x + 3)/(-1x - 1)) and g(x) = (2x + 3)/(-2x + 3) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -3(2|x + 1| - 2) - 3 and g(x) = 1x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = -1|(-2e^(2x) - 3) + 1| + 2 and g(x) = (-1x - 1)/(2x - 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2(5x^2 - 3x + 3) - 3 and g(x) = (-2x + 2)/(-3x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -1(-2x - 3)^2 + 4(-2x - 3) - 4 and g(x) = -1|x| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -4(-3|x - 2| - 2) - 3 and g(x) = (-3x - 2)/(-2x - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-1(5x - 3) + 1)/(1(5x - 3) + 2) and g(x) = -1|x - 3| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -4(5x + 2) - 3 and g(x) = -3|x + 1| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 1(-3x - 4)^2 + 5(-3x - 4) - 1 and g(x) = 3|x| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (3(4x + 1))/(-3(4x + 1) + 4) and g(x) = -3|x| - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1|(-3cos(1\u03c0x - 3) + 2) + 1| + 1 and g(x) = 2x - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (3(-5x - 3) + 3)/(-3(-5x - 3) + 4) and g(x) = (3x - 1)/(3x) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3((1x + 3)/(3x - 1)) + 2 and g(x) = 4x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1e^(-2(1|x + 2| + 3)) + 1 and g(x) = (1x + 2)/(-2x + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -1((2x + 2)/(-2x + 3))^2 - 3((2x + 2)/(-2x + 3)) + 5 and g(x) = (2x - 3)/(1x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-1(-2x^2 - 3x - 2) - 2)/(-3(-2x^2 - 3x - 2) - 3) and g(x) = (3x - 2)/(1x) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3((2x - 2)/(-2x - 1))^2 + 1((2x - 2)/(-2x - 1)) - 1 and g(x) = -2x^2 - 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2cos(1\u03c0((-1x)/(-3x)) - 2) + 3 and g(x) = 4x^2 - 4x - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2|(3x - 4) + 3| and g(x) = (-2x - 1)/(-1x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 1|((-2x + 3)/(-2x - 2)) + 3| - 2 and g(x) = (-2x + 2)/(-2x - 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(-3x^2 + 1x - 3) + 1)/(-2(-3x^2 + 1x - 3) + 3) and g(x) = (-1x - 3)/(3x - 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 5(-1x + 1)^2 - 1 and g(x) = (-3x + 3)/(-3x - 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (2((-3x)/(-3x - 3)))/(3((-3x)/(-3x - 3)) + 1) and g(x) = -3x^2 + 4x + 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 5(-1x^2 - 3x) + 3 and g(x) = (1x + 2)/(2x - 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2sin(2\u03c0(-1|x + 2| + 2) - 1) + 2 and g(x) = 5x + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (-2((-1x)/(-1x + 2)) + 1)/(2((-1x)/(-1x + 2)) + 3) and g(x) = 1x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 4(-2x + 2) - 1 and g(x) = (-3x + 2)/(-1x + 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -4(1x + 3)^2 + 3(1x + 3) - 3 and g(x) = 3x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 4(1sin(3\u03c0x + 3) - 2)^2 + 3(1sin(3\u03c0x + 3) - 2) + 3 and g(x) = (3x + 1)/(-1x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (-3(3e^(-1x) - 1))/(-1(3e^(-1x) - 1)) and g(x) = 2x^2 + 2x meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (3(-2x + 1) - 2)/(2(-2x + 1)) and g(x) = (1x - 2)/(-2x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 3e^(2((1x - 1)/(1x - 3))) and g(x) = 2|x| - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-8", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1(-1sin(1\u03c0x) + 3) - 4 and g(x) = 1x^2 + 4x - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1e^(-1(-3|x - 2| + 1)) + 3 and g(x) = (3x + 2)/(2x - 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|(-1cos(3\u03c0x + 1) - 1) + 1| - 3 and g(x) = -4x + 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (2(4x + 1) - 1)/(-3(4x + 1) + 4) and g(x) = 3x^2 + 4x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (3(4x^2 - 4x + 4) + 3)/(2(4x^2 - 4x + 4) - 3) and g(x) = -4x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -3e^(1(-5x^2 + 5x)) + 3 and g(x) = 2x - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 1(-1x^2 + 2x - 5)^2 + 5(-1x^2 + 2x - 5) - 1 and g(x) = -2|x - 1| + 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (-1(-1|x - 3|) + 3)/(-2(-1|x - 3|) - 2) and g(x) = -5x^2 - 4x - 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1e^(-1(5x^2 - 1x - 5)) - 2 and g(x) = 2x^2 + 5x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (3(-5x^2 + 4x + 2))/(-1(-5x^2 + 4x + 2)) and g(x) = 3x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(1x^2 + 1x) - 2)/(-1(1x^2 + 1x) + 2) and g(x) = -4x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2|(1cos(1\u03c0x) + 1) - 1| and g(x) = -3x^2 - 5x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (2(-3x - 4) - 1)/(-3(-3x - 4) + 2) and g(x) = 4x^2 - 1x - 5 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 4(3|x - 1| + 2)^2 - 5 and g(x) = (2x + 3)/(-1x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -2e^(-1(1|x| + 2)) - 2 and g(x) = 3|x + 2| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (2(-4x^2 + 1x + 4) + 2)/(2(-4x^2 + 1x + 4)) and g(x) = (-2x + 3)/(-1x - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3|((2x)/(-1x)) + 2| - 1 and g(x) = -2x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -3|(3cos(1\u03c0x + 1) - 1) - 1| - 3 and g(x) = 5x^2 - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1(3x - 2) + 5 and g(x) = 4x^2 + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3(4x + 3) - 4 and g(x) = (-2x + 1)/(-2x + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1cos(1\u03c0(3|x| - 3)) + 3 and g(x) = 3x - 4 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2|(-2|x + 1| + 1) + 1| and g(x) = 3|x - 2| + 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2|(-5x - 3) - 1| - 2 and g(x) = (1x - 2)/(2x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (1(1x + 1) + 3)/(-2(1x + 1) + 3) and g(x) = -1|x - 1| - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2|(2x - 3) - 2| - 2 and g(x) = 2|x + 3| - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2(3|x + 3|) - 4 and g(x) = -2x^2 - 3x + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3|(-3|x + 2| - 3)| - 2 and g(x) = 3|x - 2| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2e^(1((-2x - 1)/(3x - 1))) - 2 and g(x) = (2x - 2)/(-1x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3sin(2\u03c0(1x - 5) - 3) + 3 and g(x) = 4x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (-3(-3e^(-2x) + 2))/(1(-3e^(-2x) + 2) + 1) and g(x) = 4x - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 2e^(1(-3|x - 2| + 3)) and g(x) = 5x + 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2|(-2cos(1\u03c0x)) - 1| + 3 and g(x) = (-1x + 3)/(2x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2(2|x - 2|) + 1 and g(x) = 1x^2 + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (2((-1x)/(-3x + 1)))/(-1((-1x)/(-3x + 1)) + 3) and g(x) = -3|x + 2| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 4(5x + 1)^2 - 2(5x + 1) - 3 and g(x) = 1x^2 - 2x - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3e^(2(-2|x + 2|)) + 1 and g(x) = -3x + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -3(-2x^2 - 4x + 5) + 2 and g(x) = (3x - 3)/(-2x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1((3x - 3)/(1x + 2)) + 4 and g(x) = (-2x)/(2x + 1) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 4(-4x) + 2 and g(x) = 4x^2 + 4x + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (-1(2|x + 1| - 3) + 2)/(3(2|x + 1| - 3) + 2) and g(x) = 2|x + 3| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2e^(1(-1|x| - 2)) - 3 and g(x) = (2x)/(-1x - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (2((2x - 1)/(-2x - 1)) - 1)/(1((2x - 1)/(-2x - 1)) - 2) and g(x) = -1x - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(3e^(2x) - 3) - 3| - 2 and g(x) = -2x^2 - 4x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3e^(-2(3|x + 3| + 3)) + 2 and g(x) = (2x + 3)/(-1x + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2cos(2\u03c0((2x)/(1x + 2)) - 1) + 2 and g(x) = 2x - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 4(-3x^2 + 1x + 5) - 1 and g(x) = 3x + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -5(-1x + 3)^2 + 5(-1x + 3) + 5 and g(x) = -1|x - 3| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1(2cos(1\u03c0x + 1) + 3) - 4 and g(x) = -5x + 5 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2|((1x + 3)/(1x + 1)) + 2| - 2 and g(x) = (-3x + 2)/(-1x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(2|x| - 2) + 2)/(3(2|x| - 2)) and g(x) = -5x^2 - 3x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (3(-5x + 1) + 3)/(2(-5x + 1) - 1) and g(x) = 1|x + 2| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2|(-5x - 1) + 2| - 1 and g(x) = (2x + 1)/(-2x + 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (3(-3x) + 2)/(1(-3x) - 2) and g(x) = -2x^2 + 2x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 3|(1cos(3\u03c0x + 1) - 2) - 3| + 2 and g(x) = 4x^2 + 4x - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (2(-2e^(1x)) + 3)/(3(-2e^(1x)) - 2) and g(x) = (-1x - 2)/(-2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1|(-1x + 2) + 2| + 2 and g(x) = -1x^2 - 3x + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1sin(1\u03c0((3x - 1)/(1x)) - 1) + 3 and g(x) = 3x - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1(1x - 3)^2 + 5(1x - 3) - 1 and g(x) = 1|x - 3| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 2(3x^2 - 4x - 2) - 3 and g(x) = 3|x - 3| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(1x + 5))/(-3(1x + 5) + 1) and g(x) = -2|x - 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -5(3x^2 - 5x - 1) - 2 and g(x) = -1x + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 2|(1|x - 3| - 2) - 1| + 1 and g(x) = (3x + 2)/(3x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2e^(-2((1x - 3)/(-1x))) - 1 and g(x) = (-3x - 1)/(-2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2sin(1\u03c0((-1x - 3)/(-3x - 3))) + 3 and g(x) = -1x - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-9", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3|((2x + 2)/(-2x - 2))| + 2 and g(x) = -5x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 5((-1x - 3)/(1x - 3))^2 - 4((-1x - 3)/(1x - 3)) - 1 and g(x) = 2x^2 - 2x - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 4(3sin(1\u03c0x + 2) - 3)^2 + 1(3sin(1\u03c0x + 2) - 3) + 3 and g(x) = (3x - 2)/(2x + 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (2(4x) + 1)/(-2(4x) - 1) and g(x) = (2x + 1)/(1x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 1(-2|x - 1|) and g(x) = -3x^2 + 5x + 2 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2|(4x - 1) + 1| - 1 and g(x) = (-1x - 3)/(-1x + 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -2(3x^2 + 5x + 1)^2 + 4(3x^2 + 5x + 1) and g(x) = -4x - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2e^(-2(3x^2 - 4x + 2)) + 2 and g(x) = -2x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 4((3x + 3)/(-2x))^2 + 4((3x + 3)/(-2x)) and g(x) = -5x + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2((1x + 1)/(-3x - 3)) + 5 and g(x) = 2|x - 2| - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|(-3e^(2x) - 2) - 2| + 1 and g(x) = (3x + 1)/(2x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2|(3x + 2)| - 1 and g(x) = -5x^2 - 1x + 4 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1e^(-2((3x - 3)/(-2x))) and g(x) = 1x^2 - 5x - 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (2(-2cos(3\u03c0x) + 3) - 3)/(-3(-2cos(3\u03c0x) + 3) - 2) and g(x) = (-2x + 2)/(1x + 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -4((1x + 2)/(-2x - 3)) - 1 and g(x) = (2x + 1)/(-2x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2(3x^2 - 5x - 2)^2 - 3(3x^2 - 5x - 2) - 1 and g(x) = (2x + 2)/(-2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2e^(-1((2x + 2)/(3x + 3))) - 2 and g(x) = (-2x)/(1x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (-1(3|x - 1| + 1) - 1)/(3(3|x - 1| + 1)) and g(x) = 1x^2 + 4x + 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3(2x)^2 + 3(2x) + 4 and g(x) = (-1x + 2)/(-1x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 5(4x + 1)^2 + 5(4x + 1) - 2 and g(x) = 3|x - 3| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (1(-2x + 2) + 1)/(-2(-2x + 2) - 3) and g(x) = 5x - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -1(3x + 4)^2 + 3(3x + 4) + 4 and g(x) = -2x - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-2((2x)/(-1x - 3)) - 2)/(-3((2x)/(-1x - 3)) - 1) and g(x) = -5x - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1((3x - 3)/(2x + 2))^2 - 2((3x - 3)/(2x + 2)) - 5 and g(x) = -5x - 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 4(-1|x + 2| - 2)^2 + 4(-1|x + 2| - 2) + 4 and g(x) = (1x - 1)/(3x - 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3|(3e^(-1x) - 2) - 3| and g(x) = 3x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (-3(3e^(1x) - 3) - 2)/(3(3e^(1x) - 3)) and g(x) = -2x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 3|(-3x^2 + 1x + 4)| - 3 and g(x) = -2x^2 + 5x + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 4(2x + 3)^2 + 1(2x + 3) + 2 and g(x) = -1x^2 - 4x + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2|(2|x - 1| - 2) - 2| - 3 and g(x) = 2|x - 3| - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (2(-5x + 2))/(-3(-5x + 2)) and g(x) = -1x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (3(-3x^2 - 3x - 2) + 1)/(2(-3x^2 - 3x - 2) + 1) and g(x) = 3|x + 3| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2(-3x^2 + 3x - 3) - 5 and g(x) = -3x^2 - 1x + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(-3x^2 - 5x + 2) + 3)/(-2(-3x^2 - 5x + 2) - 2) and g(x) = (1x + 3)/(-1x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(1cos(2\u03c0x + 2) + 3) + 1)/(1(1cos(2\u03c0x + 2) + 3) + 3) and g(x) = 3|x + 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-3(-5x - 2) + 1)/(-1(-5x - 2) + 3) and g(x) = -4x^2 + 4x + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -5((-3x - 1)/(-1x + 3))^2 - 2((-3x - 1)/(-1x + 3)) + 1 and g(x) = 2x^2 + 4x - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2sin(3\u03c0((-2x)/(-1x))) - 1 and g(x) = 2x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1|(-5x^2 + 2x - 4) + 1| - 2 and g(x) = (1x - 2)/(-3x - 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (-2(-5x + 3) + 2)/(1(-5x + 3)) and g(x) = 2x + 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3((-1x - 1)/(-3x + 4))^2 + 4((-1x - 1)/(-3x + 4)) - 4 and g(x) = 2x^2 + 1x + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3|(1x^2 - 2x - 5) + 1| - 3 and g(x) = -2|x| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -5(-2cos(1\u03c0x - 3) - 2) - 1 and g(x) = -4x^2 - 2x + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -1|((2x)/(-3x + 1)) - 3| + 3 and g(x) = -4x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(3|x + 3| - 3) + 3| + 3 and g(x) = -3x^2 + 3x + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1|(2x^2 + 2x) + 1| - 1 and g(x) = 1x^2 - 2x - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3cos(1\u03c0(-3x - 5) + 1) - 3 and g(x) = (-3x)/(-2x - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2((-2x - 1)/(-2x))^2 - 4 and g(x) = (-3x + 3)/(3x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1(2sin(2\u03c0x + 3) + 1) - 5 and g(x) = 2x + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 4((-1x - 1)/(-3x)) + 5 and g(x) = 2|x| + 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3e^(1(-5x - 2)) - 2 and g(x) = -5x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2|(3e^(-2x) + 3)| + 1 and g(x) = -3x + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3(3|x + 1| - 1) + 3 and g(x) = -4x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1(-1|x - 3| - 3)^2 + 1(-1|x - 3| - 3) - 2 and g(x) = (2x - 3)/(1x + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2sin(3\u03c0((-1x - 2)/(1x + 3)) - 2) and g(x) = -5x^2 + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 4(-3x - 2)^2 + 5(-3x - 2) + 2 and g(x) = (2x - 2)/(3x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -3|(1x - 2) + 3| and g(x) = -5x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2e^(2(4x - 3)) + 1 and g(x) = -4x + 4 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1(3e^(1x) - 2) and g(x) = -3|x - 1| - 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (-1((2x - 1)/(-2x - 3)) - 1)/(-2((2x - 1)/(-2x - 3)) + 3) and g(x) = (3x - 2)/(1x + 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -1|(3x^2 + 4x + 3) - 1| + 2 and g(x) = (-3x)/(3x - 2) meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1(3x - 4)^2 - 3(3x - 4) + 4 and g(x) = -4x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -2e^(-1(-1|x - 2| + 2)) + 2 and g(x) = 1x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|(1cos(2\u03c0x - 1) - 2) - 3| - 3 and g(x) = (-1x + 2)/(1x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1cos(1\u03c0((2x - 1)/(2x - 1)) + 3) + 2 and g(x) = 2|x + 2| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3(1x^2 + 1x + 5) + 5 and g(x) = (-1x + 3)/(2x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (3(-2|x - 3| + 2) + 2)/(-2(-2|x - 3| + 2)) and g(x) = (-1x)/(-3x - 1) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (-2(-1x - 1))/(3(-1x - 1) - 2) and g(x) = (3x + 3)/(1x - 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 3|(2x^2 - 2x - 1)| and g(x) = -3x + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5(-4x - 1) + 3 and g(x) = 1x^2 - 5x - 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (1(-1sin(1\u03c0x + 3) + 2) + 3)/(1(-1sin(1\u03c0x + 3) + 2) + 1) and g(x) = 5x + 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1(-2|x + 1| - 3) + 2 and g(x) = 3x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 1cos(1\u03c0(-4x - 4) - 2) and g(x) = -2x + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3|(-3e^(1x) - 2) + 1| - 2 and g(x) = -4x^2 - 1x + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2e^(-1(-2|x - 3| - 2)) - 1 and g(x) = (-3x + 2)/(-1x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1((-1x + 2)/(3x - 2)) + 3 and g(x) = 2x^2 - 3x + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = 4((3x + 1)/(-2x + 3))^2 - 4((3x + 1)/(-2x + 3)) - 3 and g(x) = -2x^2 + 3x - 4 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1cos(1\u03c0(-1|x + 1| - 1) + 3) - 2 and g(x) = 1x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -3e^(1((1x - 2)/(1x))) + 2 and g(x) = 1x - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1(3x + 5) - 1 and g(x) = -2|x + 3| + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (3((-2x - 2)/(-2x + 3)) - 1)/(1((-2x - 2)/(-2x + 3)) + 1) and g(x) = 1x^2 + 2x - 5 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1((2x + 1)/(-3x - 2))^2 - 3((2x + 1)/(-3x - 2)) - 1 and g(x) = -1|x + 1| + 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(3|x - 1|) - 3)/(1(3|x - 1|) + 2) and g(x) = 5x^2 + 2x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (3(1cos(2\u03c0x + 3) - 3) + 1)/(3(1cos(2\u03c0x + 3) - 3) + 2) and g(x) = (-2x - 3)/(-3x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2((-2x + 1)/(-2x))^2 + 3((-2x + 1)/(-2x)) + 5 and g(x) = 1x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|((2x + 2)/(-1x - 1)) - 2| - 2 and g(x) = -2|x - 3| + 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2|((1x + 2)/(1x + 1)) - 3| - 1 and g(x) = -3x^2 + 2x + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (1(3x - 5) - 2)/(-2(3x - 5) + 1) and g(x) = 2|x - 2| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1(-2|x| + 1)^2 + 3(-2|x| + 1) + 3 and g(x) = -5x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 4(-1|x - 2| - 1) - 4 and g(x) = -5x + 4 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2cos(1\u03c0(-2|x - 3| + 2) + 1) + 1 and g(x) = -4x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1((-1x)/(-3x - 3))^2 - 5((-1x)/(-3x - 3)) - 1 and g(x) = 3x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2(-1x^2 + 2x + 1) + 3)/(2(-1x^2 + 2x + 1) - 3) and g(x) = -3|x - 2| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1sin(2\u03c0(3|x + 2| - 1) - 3) + 3 and g(x) = (1x + 1)/(-2x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -1(-1cos(3\u03c0x - 2) + 1) - 1 and g(x) = -4x^2 - 2x - 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1|(-2cos(2\u03c0x + 3) + 2) - 2| + 1 and g(x) = 1x^2 - 5x - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3sin(1\u03c0((2x - 1)/(-2x + 1)) + 1) - 1 and g(x) = -1x - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (3(3x^2 + 5x + 4) + 1)/(2(3x^2 + 5x + 4) + 2) and g(x) = (-3x + 1)/(-1x - 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3(2|x + 2| + 1) + 5 and g(x) = (3x + 3)/(-1x + 3) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1|(1sin(2\u03c0x + 3) - 3) - 3| - 2 and g(x) = (-1x - 2)/(1x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1((-2x - 3)/(-3x + 4)) + 3 and g(x) = -3|x + 3| + 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 5((-3x - 3)/(-2x + 1))^2 - 1((-3x - 3)/(-2x + 1)) - 4 and g(x) = 5x + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|(3cos(1\u03c0x + 3) + 2) + 2| + 3 and g(x) = (2x)/(2x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3|(2|x + 1| - 1)| and g(x) = 4x - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1(-2e^(1x) + 3) + 5 and g(x) = 4x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2e^(-1(2x^2 + 5x + 2)) + 1 and g(x) = (3x + 3)/(-1x - 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(-3|x + 2| - 2) + 2)/(1(-3|x + 2| - 2)) and g(x) = (-1x + 1)/(-3x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -5((-2x + 3)/(1x + 1))^2 - 4((-2x + 3)/(1x + 1)) + 1 and g(x) = -1x - 5 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -4(5x - 1)^2 + 1(5x - 1) + 2 and g(x) = (-2x)/(2x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1sin(1\u03c0((3x - 2)/(2x - 3)) + 2) - 2 and g(x) = 3x + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -2(5x^2 + 4x + 2)^2 + 1(5x^2 + 4x + 2) - 2 and g(x) = -3|x - 3| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (3(-1x + 4))/(2(-1x + 4) - 3) and g(x) = 3|x - 2| + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (-2(2|x + 3|))/(3(2|x + 3|)) and g(x) = -1|x - 3| + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2cos(1\u03c0((-3x + 2)/(3x - 2)) + 2) - 1 and g(x) = 5x - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (1(-5x))/(-3(-5x) + 2) and g(x) = 1x + 4 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2|(2e^(2x) + 2) + 1| + 1 and g(x) = 5x^2 - 3x - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2((2x - 3)/(-1x - 1))^2 + 5((2x - 3)/(-1x - 1)) + 3 and g(x) = (3x + 3)/(-3x + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (2(-1x^2 - 2x + 3))/(-1(-1x^2 - 2x + 3) + 2) and g(x) = (-3x + 1)/(1x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -3(2e^(1x) + 1) + 4 and g(x) = 3x^2 + 2x - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (1(1|x|))/(-2(1|x|) - 2) and g(x) = (3x + 3)/(1x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-3((2x + 3)/(2x + 3)) - 1)/(3((2x + 3)/(2x + 3)) + 3) and g(x) = (-3x + 3)/(-1x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2(4x + 3)^2 + 3(4x + 3) - 5 and g(x) = 2x^2 - 5x - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3(-1e^(-1x) - 1) + 1 and g(x) = -3|x + 1| + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1|((1x + 2)/(-1x + 2))| - 1 and g(x) = (3x + 2)/(3x + 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1cos(1\u03c0(-3|x - 3| + 2) - 2) + 3 and g(x) = -5x - 5 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -2((2x - 2)/(1x + 2)) - 4 and g(x) = -3x + 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 3e^(2(-1x^2 - 1x - 1)) + 1 and g(x) = 2|x + 2| meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (-1(-1x + 5) + 2)/(-3(-1x + 5) - 2) and g(x) = 3x^2 - 3x intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 3(2x^2 - 3x - 2) + 2 and g(x) = -3|x - 1| - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2(-2e^(-1x) - 1)^2 + 2(-2e^(-1x) - 1) and g(x) = 1x^2 + 2x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -3e^(2(-1|x + 2| + 1)) + 3 and g(x) = (-1x + 2)/(-3x) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(5x + 2) - 3)/(-1(5x + 2) - 2) and g(x) = 3|x - 2| + 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-1(3e^(-1x) + 2) - 2)/(1(3e^(-1x) + 2) + 1) and g(x) = (-2x - 2)/(-1x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2|((-3x - 2)/(1x - 2)) + 2| - 3 and g(x) = (-2x + 2)/(1x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1(-2|x| + 2)^2 + 4(-2|x| + 2) + 3 and g(x) = (-2x - 2)/(-3x + 4) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the third intersection point (counting from left to right) between the functions f(x) = -3|(-2sin(3\u03c0x - 2) - 2) + 3| - 1 and g(x) = (1x + 3)/(2x) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2(1x - 3)^2 + 5(1x - 3) + 3 and g(x) = (-3x + 3)/(3x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the third intersection point (counting from left to right) between the functions f(x) = 3cos(1\u03c0(-2x + 4) - 3) + 3 and g(x) = (2x + 1)/(-2x + 3) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 4(-2e^(-1x) + 2)^2 + 2(-2e^(-1x) + 2) - 4 and g(x) = -3x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 2|((-1x - 2)/(1x - 3)) + 2| and g(x) = 3|x - 1| - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3((-2x + 1)/(-1x)) - 5 and g(x) = 1x + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3|(-2sin(3\u03c0x + 1)) + 3| + 3 and g(x) = (-2x)/(-3x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1(1cos(2\u03c0x + 1) - 1) - 1 and g(x) = (-2x + 3)/(-3x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -2e^(1(2x^2 - 4x + 1)) - 2 and g(x) = -3x - 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2((1x + 1)/(-3x - 1))^2 + 4((1x + 1)/(-3x - 1)) + 5 and g(x) = (-1x - 1)/(-1x - 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = -1(-2|x - 3| + 3)^2 + 2(-2|x - 3| + 3) + 1 and g(x) = -4x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3|((1x - 3)/(1x - 3)) - 2| + 2 and g(x) = 2x - 4 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2e^(2(-1x^2 + 2x - 4)) - 3 and g(x) = 2x - 4 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2(2|x - 1| - 1) - 5 and g(x) = -4x^2 + 3x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -2(1e^(-2x) - 1) + 5 and g(x) = -1|x - 3| - 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 4((2x - 3)/(-2x))^2 + 2((2x - 3)/(-2x)) + 5 and g(x) = 3|x| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1((-3x - 2)/(-2x))^2 - 5((-3x - 2)/(-2x)) + 2 and g(x) = (1x - 1)/(-2x - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5(-2x + 3) + 4 and g(x) = 1|x - 1| - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2|(3e^(2x) + 3) + 3| - 3 and g(x) = -3|x - 3| - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -4(-3e^(2x) + 3)^2 - 2(-3e^(2x) + 3) and g(x) = -3x - 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -4(2e^(2x) - 3)^2 + 3(2e^(2x) - 3) + 1 and g(x) = -1|x - 1| - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 3(4x^2 + 5x + 1)^2 + 3(4x^2 + 5x + 1) - 3 and g(x) = 2|x - 3| intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2|(4x^2 - 2) + 3| - 1 and g(x) = 3|x - 1| meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 3(2e^(1x))^2 - 2(2e^(1x)) - 1 and g(x) = (-3x - 1)/(1x - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1sin(3\u03c0(-1x^2 + 3x + 1) + 1) - 2 and g(x) = (-1x - 1)/(2x - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1((-1x - 3)/(1x - 3))^2 + 5 and g(x) = -2x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = (2(-2e^(1x) + 3))/(2(-2e^(1x) + 3) - 3) and g(x) = (2x - 1)/(1x + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 1|(1sin(1\u03c0x) + 2) - 1| and g(x) = 4x + 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (-3((3x + 2)/(-3x - 3)) + 1)/(-2((3x + 2)/(-3x - 3)) + 3) and g(x) = -5x + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 2(-1x + 1)^2 - 5(-1x + 1) - 5 and g(x) = 4x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (2(-2|x + 3| - 1) + 2)/(3(-2|x + 3| - 1) - 1) and g(x) = 2|x| - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1((-3x + 3)/(3x - 1)))/(3((-3x + 3)/(3x - 1)) - 1) and g(x) = 2x^2 + 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = (3(1x + 1) - 2)/(3(1x + 1) + 2) and g(x) = (3x)/(-1x + 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 1(5x - 3)^2 + 3(5x - 3) - 3 and g(x) = -3x - 4 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = (2(5x^2 + 2x) - 2)/(3(5x^2 + 2x) - 2) and g(x) = -4x^2 - 3x - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -4(5x + 4) + 2 and g(x) = (3x)/(3x - 2) meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -4(5x^2 + 3x - 5) + 4 and g(x) = -2|x - 3| - 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -1((3x)/(-1x - 2))^2 - 1((3x)/(-1x - 2)) - 1 and g(x) = -1x^2 - 1x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (1((2x + 1)/(-3x - 2)))/(1((2x + 1)/(-3x - 2)) + 1) and g(x) = 3|x - 1| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2(2cos(1\u03c0x - 1))^2 - 5(2cos(1\u03c0x - 1)) - 4 and g(x) = 2|x + 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-2(-3e^(1x) + 3) + 1)/(-2(-3e^(1x) + 3) - 2) and g(x) = 2|x - 1| + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3((2x - 3)/(-3x - 3))^2 + 5((2x - 3)/(-3x - 3)) + 4 and g(x) = 2|x| - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -4(-1|x| + 3) - 1 and g(x) = (2x - 2)/(1x) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = (2((-1x - 3)/(-3x - 3)))/(-1((-1x - 3)/(-3x - 3)) - 1) and g(x) = -4x^2 - 2x - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1(1x^2 + 4x + 4)^2 + 4(1x^2 + 4x + 4) + 2 and g(x) = 3x^2 + 4x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = (1(1|x + 2| - 1) - 2)/(2(1|x + 2| - 1) - 1) and g(x) = (1x)/(3x - 2) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|(5x^2 + 3x + 4) - 3| + 3 and g(x) = (-2x - 1)/(-3x + 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1(-3|x - 3|)^2 + 3(-3|x - 3|) and g(x) = (2x)/(-1x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 5(-1cos(1\u03c0x - 3) + 1)^2 + 5(-1cos(1\u03c0x - 3) + 1) + 3 and g(x) = (-1x - 3)/(1x + 2) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -1|(-1x) + 2| - 3 and g(x) = 5x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (1(-1|x| + 3) - 2)/(-3(-1|x| + 3) + 4) and g(x) = (-2x + 3)/(3x - 1) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -1e^(1((-3x + 2)/(-1x + 2))) - 1 and g(x) = 2x^2 - 3x - 2 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -1(2x^2 - 4x - 5) - 4 and g(x) = 4x^2 - 5 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -3e^(-2(-1x + 1)) + 2 and g(x) = -2x^2 + 2x - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3((1x + 1)/(3x - 1)) and g(x) = -2x^2 + 5x - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = -2|(4x^2 + 3x + 4) - 3| + 3 and g(x) = -1|x + 2| + 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1((-2x + 3)/(3x + 2)) - 1 and g(x) = -1x^2 + 1x + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -2(-5x^2 + 1x + 2)^2 + 4(-5x^2 + 1x + 2) + 3 and g(x) = 2|x + 1| - 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(2|x + 3|)| + 1 and g(x) = -1|x + 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -3e^(2(-5x + 4)) - 1 and g(x) = (-3x - 3)/(1x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 2|(5x^2 + 1x - 5) - 2| - 2 and g(x) = 5x - 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = -5((-3x - 2)/(3x))^2 - 4((-3x - 2)/(3x)) and g(x) = -1|x + 2| + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1|(3|x + 2|) - 1| - 3 and g(x) = 4x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2|(-1sin(1\u03c0x - 3) - 2) - 1| + 1 and g(x) = -4x^2 - 4x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|(1e^(2x) - 2) + 1| - 1 and g(x) = (-1x + 2)/(1x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (-1(-1e^(2x) + 2) + 3)/(-2(-1e^(2x) + 2) - 3) and g(x) = 2|x| intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = (-1(-1cos(1\u03c0x) - 2) - 3)/(1(-1cos(1\u03c0x) - 2) - 2) and g(x) = (2x - 2)/(2x + 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 2|(3cos(1\u03c0x - 3) - 3) - 3| - 3 and g(x) = 5x^2 + 1x + 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1e^(2(-1|x - 2| - 1)) + 1 and g(x) = (-1x + 3)/(1x + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -4(-1e^(2x) + 1)^2 + 4(-1e^(2x) + 1) and g(x) = (2x)/(-2x + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 3(-2sin(1\u03c0x - 3) - 2) - 2 and g(x) = (1x - 2)/(2x + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1e^(1((-3x - 1)/(-3x - 1))) - 1 and g(x) = (2x - 2)/(2x + 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -2|(-1sin(3\u03c0x + 1) + 2) - 2| - 3 and g(x) = 5x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 3(1|x + 3| + 1)^2 - 2(1|x + 3| + 1) - 3 and g(x) = (2x - 3)/(2x - 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 1|(-2cos(1\u03c0x + 1) + 2) + 2| - 3 and g(x) = -1x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (3((-2x - 2)/(1x - 3)) + 3)/(-2((-2x - 2)/(1x - 3)) - 3) and g(x) = (-2x + 1)/(-3x - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1((3x + 3)/(3x - 1))^2 + 5((3x + 3)/(3x - 1)) - 5 and g(x) = 1x + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 3((3x + 2)/(-1x + 2))^2 - 1 and g(x) = -4x + 5 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1(-2x^2 - 2x + 3)^2 - 3(-2x^2 - 2x + 3) - 1 and g(x) = (-2x - 1)/(2x - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1((3x)/(1x - 2)) + 4 and g(x) = (1x + 3)/(-1x - 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 4((2x - 1)/(2x + 1))^2 - 2((2x - 1)/(2x + 1)) - 2 and g(x) = -2|x + 1| + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -2e^(1(2x^2 + 4x + 1)) - 2 and g(x) = 5x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -4(-2|x + 2| - 3)^2 + 3 and g(x) = (-3x + 1)/(-1x - 1) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 5(2|x + 2| + 3) + 5 and g(x) = (2x + 1)/(1x + 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 5(2|x - 3| - 3) and g(x) = 3|x| + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 5(5x + 3) - 5 and g(x) = -5x^2 - 3x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2(1sin(3\u03c0x + 2) - 3) - 3 and g(x) = (-2x - 3)/(-2x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(-3x^2 + 3x + 1) + 2)/(-3(-3x^2 + 3x + 1) + 1) and g(x) = 1|x - 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1|(2|x - 1| - 2) - 1| + 1 and g(x) = (-3x - 1)/(-1x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1cos(2\u03c0((-2x - 2)/(2x - 3)) - 3) and g(x) = (-3x - 3)/(1x) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 3e^(1((3x)/(2x + 3))) + 2 and g(x) = 1|x - 3| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 2((-3x + 3)/(-2x - 3)) + 5 and g(x) = -2x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (-2(-3|x - 2| - 3) + 3)/(-1(-3|x - 2| - 3)) and g(x) = 1|x - 3| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (3((-2x - 1)/(2x - 1)))/(-3((-2x - 1)/(2x - 1)) - 1) and g(x) = -1|x - 1| - 3 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1cos(3\u03c0(1|x + 3| - 3) - 1) - 1 and g(x) = -1x^2 - 5x + 4 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-6", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (-2(2e^(-1x) - 3) + 3)/(-2(2e^(-1x) - 3) - 3) and g(x) = -3x^2 + 1x - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (1(1e^(1x)) + 1)/(2(1e^(1x))) and g(x) = -3|x - 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = (1(2sin(1\u03c0x - 3) + 3) + 3)/(-2(2sin(1\u03c0x - 3) + 3) - 2) and g(x) = (2x)/(2x + 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1|(4x + 3) - 1| and g(x) = -1|x + 2| + 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = -1sin(2\u03c0(2|x - 3|) - 2) - 3 and g(x) = -3x - 2 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = 2cos(1\u03c0(1x^2 + 1) - 1) + 3 and g(x) = -5x - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = (1((2x - 3)/(-3x + 2)) + 2)/(-1((2x - 3)/(-3x + 2)) - 2) and g(x) = (-2x - 1)/(-2x - 3) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(2|x + 2| - 1) - 3| - 1 and g(x) = -2x - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1((-1x + 3)/(1x - 2))^2 - 1((-1x + 3)/(1x - 2)) + 4 and g(x) = (-1x - 2)/(1x) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 1e^(1(-2|x + 2| - 3)) - 2 and g(x) = -2x^2 + 4x + 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 1|(3x^2 + 1x - 4) - 1| - 2 and g(x) = -1|x - 2| - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = -4(1x^2 + 1x - 2) - 4 and g(x) = 3x^2 + 3x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 5(-2x^2 + 1x - 3) - 3 and g(x) = (3x - 2)/(-2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -5((3x + 3)/(1x + 3)) - 5 and g(x) = -3x^2 + 4x - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 1e^(-2(5x^2 + 5x + 2)) and g(x) = -2x^2 - 5x - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-3(1cos(1\u03c0x - 2) - 2) - 2)/(-3(1cos(1\u03c0x - 2) - 2) - 2) and g(x) = -1x + 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 1(-4x + 5)^2 + 2(-4x + 5) and g(x) = 2x^2 + 1x - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (2(-5x^2 - 3x - 1) - 1)/(-2(-5x^2 - 3x - 1) - 3) and g(x) = (-2x - 2)/(2x + 1) in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = (3(2sin(1\u03c0x - 2) + 2) - 2)/(-1(2sin(1\u03c0x - 2) + 2) - 3) and g(x) = -5x^2 + 2x + 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -3(3x + 2)^2 + 5(3x + 2) + 2 and g(x) = -1x^2 - 3x - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = -3(-3x - 4) - 4 and g(x) = -5x^2 - 4x + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 4(2e^(2x)) - 5 and g(x) = 2x^2 - 5x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3e^(2(1|x + 2| - 1)) + 1 and g(x) = -1x^2 + 4x meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-4", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -4(-3x - 3) + 1 and g(x) = (1x + 2)/(3x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (1(3x + 4) - 1)/(-3(3x + 4) - 1) and g(x) = -5x^2 - 3x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = -1(-3|x - 1| + 3)^2 + 5(-3|x - 1| + 3) + 2 and g(x) = -1x^2 + 4x - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 3|(-3sin(1\u03c0x - 1) + 3)| - 2 and g(x) = -2|x - 1| - 1 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the second intersection point (counting from left to right) between the functions f(x) = (2(3|x| + 1))/(-1(3|x| + 1) - 3) and g(x) = 3|x - 1| - 3 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -1(3|x| + 2) - 4 and g(x) = (2x - 2)/(-2x - 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -2sin(1\u03c0(1x - 3) - 2) - 2 and g(x) = (-1x + 2)/(-3x - 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = 1|(-4x^2 - 1x + 2) - 1| + 3 and g(x) = -4x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1e^(1(1|x + 1| - 2)) - 2 and g(x) = -5x^2 + 5 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2(2e^(-1x) - 2) + 1 and g(x) = 3x^2 + 5 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = -2sin(1\u03c0(2|x - 1| + 3) - 2) + 2 and g(x) = -3|x - 2| + 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (2(-1e^(1x) - 2) + 2)/(-3(-1e^(1x) - 2) + 2) and g(x) = (2x)/(1x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the third x-coordinate (rounded to the nearest integer) where f(x) = (1(4x) + 3)/(-3(4x) - 2) and g(x) = 2x^2 - 2x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = -1(-2x + 4)^2 + 5(-2x + 4) + 1 and g(x) = (3x)/(2x - 1) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 1|(2x - 5) + 3| + 1 and g(x) = (-1x + 2)/(2x) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (1(1x^2 - 1x - 4) - 2)/(3(1x^2 - 1x - 4) + 1) and g(x) = -4x + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the third x-coordinate (rounded to the nearest integer) where f(x) = -2|(1|x - 2| - 1)| + 2 and g(x) = 1x - 1 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1|(-1|x + 3| + 3) + 2| + 2 and g(x) = (-3x - 1)/(-2x - 3) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "If you add up all the x-coordinates where the functions f(x) = 1|(5x + 3) + 2| + 1 and g(x) = -1x^2 - 4x + 3 meet in the interval [-10, 10] and round to the nearest integer, what do you get?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = -3sin(1\u03c0(1|x + 3| - 3) - 3) + 2 and g(x) = 1x^2 + 5x + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-5", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = 3|(1sin(1\u03c0x + 1) - 1) - 1| - 2 and g(x) = 3x^2 + 5x - 1 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the rightmost x-coordinate where f(x) = 1sin(2\u03c0(1x + 4) + 3) and g(x) = 1x^2 - 3x - 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2|(-1|x - 1| - 1) - 3| - 1 and g(x) = 4x - 2 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = (3((-2x - 3)/(3x + 3)) - 2)/(1((-2x - 3)/(3x + 3)) - 2) and g(x) = -5x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-2((2x - 1)/(-2x - 3)) + 2)/(3((2x - 1)/(-2x - 3)) + 3) and g(x) = -5x - 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = 2e^(1(-3|x - 3| - 3)) - 2 and g(x) = 5x + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 4(-3x - 3)^2 + 5(-3x - 3) - 1 and g(x) = 2|x - 3| - 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 1cos(3\u03c0(3x) - 3) and g(x) = -5x intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = -1|(4x - 4)| and g(x) = -1|x| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Find the average of all x-coordinates (rounded to the nearest integer) where f(x) = (2((-1x)/(3x + 1)) + 2)/(3((-1x)/(3x + 1)) + 2) and g(x) = (-3x + 3)/(3x + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 2(2|x - 3| + 1) - 2 and g(x) = (-1x + 3)/(1x - 2) intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the leftmost x-coordinate where f(x) = 3((-2x - 2)/(-2x - 3)) - 2 and g(x) = -3|x - 3| + 3 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the sum of all x-coordinates (rounded to the nearest integer) where f(x) = 1e^(1(1x - 5)) + 1 and g(x) = 3|x - 1| - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10", "dataset": "MATH"}
{"messages": [{"content": "What is the mean value of the x-coordinates (rounded to the nearest integer) where the functions f(x) = (3(-2|x + 3| - 1) + 1)/(-1(-2|x + 3| - 1) - 1) and g(x) = -1|x + 2| - 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2((3x + 2)/(-2x - 1))^2 + 2((3x + 2)/(-2x - 1)) + 3 and g(x) = -4x + 3 meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = (-2((1x + 3)/(-2x - 1)) - 2)/(3((1x + 3)/(-2x - 1))) and g(x) = (1x + 1)/(1x) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-1", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the smallest x-value where the functions f(x) = 1|(-1|x + 3| + 1) - 1| + 1 and g(x) = 2|x + 3| - 3 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = 2(-1|x - 1| - 1) and g(x) = (3x + 1)/(-2x + 1) meet in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}
{"messages": [{"content": "Find the first intersection point (counting from left to right) between the functions f(x) = 2e^(2(-3x^2 - 3x - 4)) + 1 and g(x) = 3|x + 2| - 1 in the interval [-10, 10], rounded to the nearest integer.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-3", "dataset": "MATH"}
{"messages": [{"content": "Find the integer value closest to the x-coordinate where f(x) = 1e^(1(-2x + 3)) + 3 and g(x) = 3|x + 3| + 1 intersect in the range -10 \u2264 x \u2264 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the first x-coordinate (rounded to the nearest integer) where f(x) = 4(-1x) - 5 and g(x) = (2x + 3)/(1x + 2) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-2", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -2|((1x - 3)/(-1x - 1)) - 1| + 3 and g(x) = (-1x - 2)/(3x + 3) intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0", "dataset": "MATH"}
{"messages": [{"content": "At what integer value (rounded) do the functions f(x) = -5(2e^(2x) + 1)^2 + 2(2e^(2x) + 1) - 5 and g(x) = -2|x + 2| + 2 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "-7", "dataset": "MATH"}
{"messages": [{"content": "Find the integer closest to the largest x-value where the functions f(x) = 3e^(2(-3|x - 1| - 2)) - 3 and g(x) = -2|x - 3| + 1 meet in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5", "dataset": "MATH"}
{"messages": [{"content": "When ordered from left to right, what is the second x-coordinate (rounded to the nearest integer) where f(x) = -1|(2|x - 3| - 1) + 3| - 2 and g(x) = -5x^2 + 4x - 5 intersect in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1", "dataset": "MATH"}