messages listlengths 1 1 | ground_truth stringlengths 1 4 | dataset stringclasses 1 value |
|---|---|---|
[
{
"content": "How many solutions does the equation (-2((-2(-1|(5x^2 + 3x + 5)| + 2) + 3)/(2(-1|(5x^2 + 3x + 5)| + 2) - 1)))/(-1((-2(-1|(5x^2 + 3x + 5)| + 2) + 3)/(2(-1|(5x^2 + 3x + 5)| + 2) - 1)) - 2) = -3|((-1(-2cos(2πx - 3)) - 3)/(3(-2cos(2πx - 3)) + 3)) - 2| have for x in the interval [-10, 10]?\n\nPresent t... | 80 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -3|(-2(-1sin(1π(-4x^2 + 5x + 3) + 2) - 2)) + 3| - 1 and g(x) = -2|(-2e^(-2((-1x + 2)/(1x))) - 1) - 1| - 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 304 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3sin(1π(-2|((-1(1x + 5) + 2)/(-3(1x + 5) + 4))|) - 1) - 3 and g(x) = -4(-3e^(-1((-3x + 3)/(-2x - 2))))^2 + 1(-3e^(-1((-3x + 3)/(-2x - 2)))) - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 24 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(1(-1e^(-1(-2|x| + 2)) - 3) - 5) + 3| - 3 and g(x) = 3((3(-4x^2 + 2x + 3))/(3(-4x^2 + 2x + 3) + 1))^2 - 5((3(-4x^2 + 2x + 3))/(3(-4x^2 + 2x + 3) + 1)) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(3π(1|(1|(1x - 4) + 2| + 3) - 1| - 3)) - 1 and g(x) = -2|((3(-1|x - 2| - 1) + 1)/(-3(-1|x - 2| - 1) + 2)) - 1| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 62 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(1|(-4(-3x - 5) - 1) + 3| + 3) + 3)/(1(1|(-4(-3x - 5) - 1) + 3| + 3) + 2) and g(x) = -5(1|(-2|x + 2|)| - 1)^2 + 4(1|(-2|x + 2|)| - 1) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|(3((-1(-2x^2 - 1x) - 3)/(3(-2x^2 - 1x)))^2 + 3((-1(-2x^2 - 1x) - 3)/(3(-2x^2 - 1x))) - 3)| + 2 and g(x) = -5(3(4x - 2)^2 - 1(4x - 2) - 1) - 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-3cos(1π(1(4x + 1) + 5) - 3) + 1) - 3)/(2(-3cos(1π(1(4x + 1) + 5) - 3) + 1) + 3) and g(x) = 3|(-1(1sin(2πx + 3) - 3) - 2) + 1| intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2((1(3(-1x^2 - 5x + 1) + 5) - 1)/(-3(3(-1x^2 - 5x + 1) + 5) - 2))^2 - 2((1(3(-1x^2 - 5x + 1) + 5) - 1)/(-3(3(-1x^2 - 5x + 1) + 5) - 2)) + 2 and g(x) = 1sin(1π(1(-1x^2 - 2x)^2 - 4(-1x^2 - 2x) + 2)) - 3 intersect in the interval [-10, 10].\n\nPresent th... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2e^(1(3(3|((-3x)/(-3x - 2))| - 2)^2 + 5(3|((-3x)/(-3x - 2))| - 2))) and g(x) = -1(1|((-2x - 2)/(-2x + 3)) + 1| - 1)^2 + 4(1|((-2x - 2)/(-2x + 3)) + 1| - 1) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your an... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(2|((3(2|x + 3|) + 2)/(1(2|x + 3|) + 1)) - 2|)^2 + 5(2|((3(2|x + 3|) + 2)/(1(2|x + 3|) + 1)) - 2|) - 4 and g(x) = (3(3(2|x + 2|) - 3) + 3)/(-3(3(2|x + 2|) - 3) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format:... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 5(1(-1((-2x - 2)/(2x - 1)) - 4) + 5) and g(x) = -4(4(-3cos(3πx - 1)) - 3)^2 + 3(4(-3cos(3πx - 1)) - 3) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(5(3sin(3π(2x - 5) + 1) - 1) + 3)^2 + 1(5(3sin(3π(2x - 5) + 1) - 1) + 3) + 5 and g(x) = 2|(-3e^(1((-1x - 3)/(2x - 1)))) - 1| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 79 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(3π((-1(-1|((1x - 2)/(3x + 2)) - 3| - 1) - 1)/(-2(-1|((1x - 2)/(3x + 2)) - 3| - 1) + 1))) - 2 and g(x) = 3(-3(-2cos(3πx - 2) + 2) + 5)^2 + 4(-3(-2cos(3πx - 2) + 2) + 5) - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaT... | 112 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1sin(2π(5(-4(-3x - 1) + 5)^2 - 3) + 2) and g(x) = (1(2cos(1π((1x - 2)/(-1x + 3)) - 1) + 1) + 3)/(2(2cos(1π((1x - 2)/(-1x + 3)) - 1) + 1) - 1) in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 314 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(3|(-1(-2sin(1πx - 2) + 1) - 1) + 3| - 3)^2 + 1(3|(-1(-2sin(1πx - 2) + 1) - 1) + 3| - 3) - 1 and g(x) = -2|((-2(-3cos(2πx + 3) - 1) - 3)/(-2(-3cos(2πx + 3) - 1) + 3)) + 1| + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format... | 100 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(1π(-2(-1(-2|x - 3| - 1)^2 + 4(-2|x - 3| - 1) + 5) - 4) + 2) + 3 and g(x) = 5(-2cos(3π((-2x - 3)/(-1x - 1)) - 2) + 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 414 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1(2|((-1((3x)/(-2x + 3)) + 3)/(-2((3x)/(-2x + 3)) - 1)) + 1| - 1) + 2 and g(x) = 3cos(3π(-4(-1x + 4) + 3) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 3(-1|((3(3|x + 1| + 2) + 1)/(-3(3|x + 1| + 2) + 2)) + 2| + 1)^2 - 3(-1|((3(3|x + 1| + 2) + 1)/(-3(3|x + 1| + 2) + 2)) + 2| + 1) - 2 and g(x) = (-1(3(2cos(1πx + 3) + 1) - 2) + 1)/(-1(3(2cos(1πx + 3) + 1) - 2) - 1) where -10 ≤ ... | 20 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1((1(-1|(-2cos(1πx + 3)) + 2| - 3) - 1)/(-1(-1|(-2cos(1πx + 3)) + 2| - 3) + 3)) + 1)/(3((1(-1|(-2cos(1πx + 3)) + 2| - 3) - 1)/(-1(-1|(-2cos(1πx + 3)) + 2| - 3) + 3)) + 3) and g(x) = (-2(4(-2|x| + 1)^2 + 2(-2|x| + 1) - 5) + 1)/(-2... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(1sin(1π(2|(3|x - 1|) + 3| - 2) + 1) + 1) - 2)/(-2(1sin(1π(2|(3|x - 1|) + 3| - 2) + 1) + 1) - 2) and g(x) = 3e^(-2(2|(3|x + 1|) - 3| + 2)) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"rol... | 52 | MATH |
[
{
"content": "How many solutions does the equation -4(4(3sin(3π(1x^2 + 5x - 4) + 3) + 3)) + 2 = 2cos(3π(-5(4x - 4) - 3) - 2) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 108 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1((1(-4(1x^2 - 1x + 4)^2 + 2) + 2)/(3(-4(1x^2 - 1x + 4)^2 + 2) - 1)) + 1 and g(x) = 1(-1(3sin(1πx - 2) + 3) + 2)^2 - 5(-1(3sin(1πx - 2) + 3) + 2) - 4 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your an... | 20 | MATH |
[
{
"content": "How many solutions does the equation -3|(-1|(-1|(-4x) - 1| - 1) - 3| + 2) + 3| - 3 = 1(-1e^(-1(-5x^2 + 5x + 2)) + 2) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(3sin(3π(3|(5x^2 - 5x - 2) - 1| + 1) + 3)))/(-1(3sin(3π(3|(5x^2 - 5x - 2) - 1| + 1) + 3)) - 2) and g(x) = 1(1((-3x + 1)/(1x))^2 + 4((-3x + 1)/(1x)) - 5) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\box... | 465 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(1|((-3(-2|x + 1| + 2) - 3)/(-1(-2|x + 1| + 2) + 3)) - 3| - 1) - 3| + 1 and g(x) = 3(-4(3|x - 1| - 3)^2 + 2(3|x - 1| - 3) + 4) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 3|(-3e^(1(5(-1|x + 1| + 2) - 2)) + 1) + 2| - 1 and g(x) = 5(-3cos(1π((-1x + 3)/(-2x - 1)) + 3) + 3)^2 - 4(-3cos(1π((-1x + 3)/(-2x - 1)) + 3) + 3) + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your an... | 6 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2sin(3π(3|(-1(2|x + 1| - 2)^2 - 3(2|x + 1| - 2) - 2) - 1| + 2)) - 2 and g(x) = (-3(2sin(3π(-3x - 4) + 1)))/(3(2sin(3π(-3x - 4) + 1)) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "u... | 1109 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|((-2(-1(3x - 4)^2 - 2(3x - 4)) - 1)/(-3(-1(3x - 4)^2 - 2(3x - 4)) + 2)) - 1| - 1 and g(x) = -1(-5((2x - 3)/(1x + 1)) + 2)^2 + 1(-5((2x - 3)/(1x + 1)) + 2) + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your a... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1cos(2π(3|((-1(-5x - 1) - 3)/(-2(-5x - 1) - 1)) - 1| + 2) - 2) + 2 and g(x) = (-2((-1(-2sin(3πx - 1)) + 1)/(3(-2sin(3πx - 1)) + 3)) - 1)/(3((-1(-2sin(3πx - 1)) + 1)/(3(-2sin(3πx - 1)) + 3)) - 2) where -10 ≤ x ≤ 10.\n\nPresent... | 61 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(4((-1(1|x + 1| - 3))/(-2(1|x + 1| - 3) - 2))^2 - 3((-1(1|x + 1| - 3))/(-2(1|x + 1| - 3) - 2)) - 5)^2 - 5(4((-1(1|x + 1| - 3))/(-2(1|x + 1| - 3) - 2))^2 - 3((-1(1|x + 1| - 3))/(-2(1|x + 1| - 3) - 2)) - 5) + 3 and g(x) = 1|((-1((-3... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (2(-1(-2|(-3sin(3πx + 2) - 3) + 1| + 3) + 2))/(-3(-1(-2|(-3sin(3πx + 2) - 3) + 1| + 3) + 2) + 2) and g(x) = -1(-1sin(1π(1x + 4) - 1) + 3)^2 - 4(-1sin(1π(1x + 4) - 1) + 3) + 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \... | 49 | MATH |
[
{
"content": "How many solutions does the equation -5(-3sin(1π((1(-1|x - 3| + 3) - 1)/(3(-1|x - 3| + 3) - 2)) + 3))^2 - 4(-3sin(1π((1(-1|x - 3| + 3) - 1)/(3(-1|x - 3| + 3) - 2)) + 3)) = -1cos(2π(2|(-1|x - 3| + 2) - 2| + 1) + 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\b... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(2cos(3π((3(-4x^2 + 2x - 2) - 2)/(-2(-4x^2 + 2x - 2) - 1)) + 1) + 2) - 2 and g(x) = 3sin(3π(-2(-1|x - 2| + 1) - 1) - 3) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "use... | 30 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(-3cos(1π(3(-2|x| - 3)^2 + 4(-2|x| - 3) - 1) + 1) - 1)^2 + 4(-3cos(1π(3(-2|x| - 3)^2 + 4(-2|x| - 3) - 1) + 1) - 1) - 4 and g(x) = (2(-1e^(-1(3x)) - 1) + 1)/(-1(-1e^(-1(3x)) - 1) - 2) in the range -10 ≤ x ≤ 10?\n\nPresent the answe... | 977 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-5((1(-3|x|) + 1)/(-3(-3|x|) + 2)) + 1)| + 3 and g(x) = 1|((-3(2sin(2πx) - 1) + 3)/(2(2sin(2πx) - 1) - 1)) - 2| + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 42 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2(4(1|(-1x^2 - 4x + 2) + 1| - 2)^2 - 5(1|(-1x^2 - 4x + 2) + 1| - 2) + 5) + 2)/(1(4(1|(-1x^2 - 4x + 2) + 1| - 2)^2 - 5(1|(-1x^2 - 4x + 2) + 1| - 2) + 5) + 2) and g(x) = 1sin(1π((-3(-1x - 4) + 2)/(3(-1x - 4) - 2)) + 2) in the range... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1sin(1π(-3(-3((-3x - 1)/(2x + 2))^2 - 2((-3x - 1)/(2x + 2)) - 3)^2 + 2(-3((-3x - 1)/(2x + 2))^2 - 2((-3x - 1)/(2x + 2)) - 3) + 2) + 2) - 3 and g(x) = (1(-3cos(2π(-5x^2 - 3x + 3) - 1) - 3))/(-2(-3cos(2π(-5x^2 - 3x + 3) - 1) - 3) - ... | 271 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-1e^(2(1(4x^2 + 3x + 1) - 5)) - 1) - 1 and g(x) = 1|(-2(-2cos(3πx - 1) + 1)^2 - 2(-2cos(3πx - 1) + 1) + 4) + 3| - 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 16 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2((2(1(1x^2 - 4x - 1)^2 - 2(1x^2 - 4x - 1) - 5) - 2)/(2(1(1x^2 - 4x - 1)^2 - 2(1x^2 - 4x - 1) - 5) + 3))^2 - 2((2(1(1x^2 - 4x - 1)^2 - 2(1x^2 - 4x - 1) - 5) - 2)/(2(1(1x^2 - 4x - 1)^2 - 2(1x^2 - 4x - 1) - 5) + 3)) - 4 and g(x... | 82 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -4(-2(2cos(2π(4x^2 + 1x + 1)) - 2)^2 + 2(2cos(2π(4x^2 + 1x + 1)) - 2) - 5)^2 - 1(-2(2cos(2π(4x^2 + 1x + 1)) - 2)^2 + 2(2cos(2π(4x^2 + 1x + 1)) - 2) - 5) - 5 and g(x) = -1(-2|(3e^(-1x) + 3) + 3| - 1)^2 - 5(-2|(3e^(-1x) + 3) + 3| - 1) - 5 intersect in th... | 316 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -2|(2|(1|(-1|x - 1| + 2)| - 3) + 3| - 3)| + 1 and g(x) = 1|(1((-1x + 1)/(2x + 3)) + 2) - 1| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 6 | MATH |
[
{
"content": "How many solutions does the equation (1((-1(-1cos(2π(3|x - 2| + 3) + 1)) - 2)/(1(-1cos(2π(3|x - 2| + 3) + 1)))) - 3)/(1((-1(-1cos(2π(3|x - 2| + 3) + 1)) - 2)/(1(-1cos(2π(3|x - 2| + 3) + 1)))) + 3) = 2sin(1π((-2((1x - 1)/(1x)) + 1)/(2((1x - 1)/(1x)))) + 1) + 2 have for x in the interval [-10, 10]?\... | 124 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-3sin(2π(-3(3|x - 1|)^2 - 5(3|x - 1|) - 3) + 3) - 3) - 1)/(3(-3sin(2π(-3(3|x - 1|)^2 - 5(3|x - 1|) - 3) + 3) - 3) + 1) and g(x) = -1|(1(2|x - 2| + 3)^2) - 3| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{You... | 16 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1|(-2sin(1π(3(-3|x + 2| - 2)^2 - 5) + 3) + 1) + 1| and g(x) = -2(-3(2x) - 2) - 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 26 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-2sin(1π(-2((-1x)/(1x - 3))^2 - 2((-1x)/(1x - 3)) - 5) - 2) - 2)^2 + 1(-2sin(1π(-2((-1x)/(1x - 3))^2 - 2((-1x)/(1x - 3)) - 5) - 2) - 2) + 2 and g(x) = 3|((1(2|x| + 3) + 3)/(-3(2|x| + 3)))| - 2 where -10 ≤ x ≤ 10.\n\nPresen... | 168 | MATH |
[
{
"content": "How many solutions does the equation -2(-4(-1(-3x^2 - 1x + 5) - 5) - 3) + 4 = (3(-2|(-2x - 5) + 1| + 2) - 2)/(-3(-2|(-2x - 5) + 1| + 2) - 2) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -3(3|((-2(-2cos(1πx + 1)) - 2)/(-3(-2cos(1πx + 1)) - 1)) + 1| - 2)^2 - 4(3|((-2(-2cos(1πx + 1)) - 2)/(-3(-2cos(1πx + 1)) - 1)) + 1| - 2) + 1 and g(x) = (-2(1(2cos(2πx + 1) + 2) - 2) + 2)/(2(1(2cos(2πx + 1) + 2) - 2) - 3) intersect in the interval [-10,... | 93 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -5(-1((3(3cos(1πx - 1) + 1) + 1)/(1(3cos(1πx - 1) + 1) - 3)) + 2)^2 - 3(-1((3(3cos(1πx - 1) + 1) + 1)/(1(3cos(1πx - 1) + 1) - 3)) + 2) - 5 and g(x) = -1(-1(3x + 5) + 4)^2 + 5(-1(3x + 5) + 4) + 2 where -10 ≤ x ≤ 10.\n\nPresent... | 22 | MATH |
[
{
"content": "How many solutions does the equation 2|((-1(4((2x + 2)/(-2x - 2))) - 1)/(-2(4((2x + 2)/(-2x - 2))))) + 1| + 3 = 4(1sin(3π(-5x^2 + 1x - 4) - 3) - 2)^2 - 3(1sin(3π(-5x^2 + 1x - 4) - 3) - 2) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"r... | 1254 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2(1|(1sin(1π(1x + 4) + 2) + 2) + 1| - 3) - 2 and g(x) = -2|(1sin(3π(-5x - 1) + 3) + 3) - 2| + 3 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 100 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(3|(-3sin(3π(1|x + 1| + 2) + 1) + 3) - 3| + 3)^2 - 3(3|(-3sin(3π(1|x + 1| + 2) + 1) + 3) - 3| + 3) + 1 and g(x) = (-3(-1|(1x + 3) - 3| - 3) - 3)/(-2(-1|(1x + 3) - 3| - 3) - 2) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format:... | 70 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-1(3sin(1π(2|(-1|x + 2|) - 1| + 2) + 1)) + 2)/(3(3sin(1π(2|(-1|x + 2|) - 1| + 2) + 1)) + 2) and g(x) = -1(3|(-2cos(3πx - 3)) - 2| + 2)^2 - 5(3|(-2cos(3πx - 3)) - 2| + 2) + 5 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTe... | 8 | MATH |
[
{
"content": "How many solutions does the equation 4(3|(1|(-2cos(3πx)) + 1| - 3)| + 1)^2 + 1(3|(1|(-2cos(3πx)) + 1| - 3)| + 1) + 2 = 2e^(2(-1((-1x + 2)/(2x))^2 + 3((-1x + 2)/(2x)))) + 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-5(1((-3x + 3)/(-3x - 2))^2 - 3((-3x + 3)/(-3x - 2)) + 2)^2 + 1(1((-3x + 3)/(-3x - 2))^2 - 3((-3x + 3)/(-3x - 2)) + 2) + 1) + 3 and g(x) = -4(3sin(2π(-1x + 3) - 1) + 1) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaT... | 34 | MATH |
[
{
"content": "How many solutions does the equation 2cos(2π(1|((2(-3|x - 1| + 3) - 3)/(-1(-3|x - 1| + 3) - 1)) + 3| - 1) - 1) = 2(1cos(1π((-1x + 3)/(-2x - 1)) + 3) + 3) - 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 95 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = (-2((-1(-2(-1x + 4) - 5) - 2)/(-2(-2(-1x + 4) - 5) - 2)) - 3)/(1((-1(-2(-1x + 4) - 5) - 2)/(-2(-2(-1x + 4) - 5) - 2)) - 2) and g(x) = 4(5(2|x + 2| - 2) + 3)^2 - 3(5(2|x + 2| - 2) + 3) + 1 in the range -10 ≤ x ≤ 10?\n\nPresent the a... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(-5(2(-1x^2 + 2x + 3) - 2) - 1)^2 + 1(-5(2(-1x^2 + 2x + 3) - 2) - 1) - 3 and g(x) = 2|((1(-1x - 4))/(-1(-1x - 4) + 3)) + 2| - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3(2cos(1π(-3(3x^2 - 4x - 4) - 2) - 2) + 1)^2 + 1 and g(x) = -5(-3sin(2π((1x)/(3x + 1)) - 1) + 1)^2 - 5(-3sin(2π((1x)/(3x + 1)) - 1) + 1) + 4 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": ... | 154 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2(1((3(-2e^(1x) - 3) - 2)/(1(-2e^(1x) - 3) + 1)) + 5) - 3)/(-1(1((3(-2e^(1x) - 3) - 2)/(1(-2e^(1x) - 3) + 1)) + 5) - 2) and g(x) = (-3(5(1|x + 1| - 3)^2 + 1(1|x + 1| - 3) - 4) - 3)/(1(5(1|x + 1| - 3)^2 + 1(1|x + 1| - 3) - 4)... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2((1((-3((1x - 3)/(-1x - 3)) - 2)/(2((1x - 3)/(-1x - 3)) + 1)) - 3)/(-2((-3((1x - 3)/(-1x - 3)) - 2)/(2((1x - 3)/(-1x - 3)) + 1))))^2 - 5((1((-3((1x - 3)/(-1x - 3)) - 2)/(2((1x - 3)/(-1x - 3)) + 1)) - 3)/(-2((-3((1x - 3)/(-1x - 3)... | 10 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (2((-2(3(2x^2 + 5x)^2 - 2(2x^2 + 5x) - 2))/(-1(3(2x^2 + 5x)^2 - 2(2x^2 + 5x) - 2) + 3)) - 3)/(3((-2(3(2x^2 + 5x)^2 - 2(2x^2 + 5x) - 2))/(-1(3(2x^2 + 5x)^2 - 2(2x^2 + 5x) - 2) + 3)) - 1) and g(x) = 1|(1sin(1π(4x - 5) - 1) - 2)... | 4 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2(-3|(2(-5x^2 + 2)^2 + 4(-5x^2 + 2)) + 3| - 3) - 2 and g(x) = -5(1cos(2π(-5x^2 + 2x - 2) - 3) - 2)^2 + 5(1cos(2π(-5x^2 + 2x - 2) - 3) - 2) + 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role"... | 13 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1(-5((-2(3x^2 + 5x + 1) - 2)/(2(3x^2 + 5x + 1) - 3))^2 + 3((-2(3x^2 + 5x + 1) - 2)/(2(3x^2 + 5x + 1) - 3)) + 3)^2 - 1(-5((-2(3x^2 + 5x + 1) - 2)/(2(3x^2 + 5x + 1) - 3))^2 + 3((-2(3x^2 + 5x + 1) - 2)/(2(3x^2 + 5x + 1) - 3)) + 3) + 2 and g(x) = (-1(1|(-... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(1(1(-3cos(3πx) + 3)^2 + 1(-3cos(3πx) + 3))^2 - 3(1(-3cos(3πx) + 3)^2 + 1(-3cos(3πx) + 3)) - 3)^2 - 4(1(1(-3cos(3πx) + 3)^2 + 1(-3cos(3πx) + 3))^2 - 3(1(-3cos(3πx) + 3)^2 + 1(-3cos(3πx) + 3)) - 3) + 4 and g(x) = -2|((3(3x - 1) + ... | 60 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-1(2|(-5(-1x^2 + 4x - 3)^2 + 1(-1x^2 + 4x - 3) + 5) + 1| + 3) + 2)/(3(2|(-5(-1x^2 + 4x - 3)^2 + 1(-1x^2 + 4x - 3) + 5) + 1| + 3) - 2) and g(x) = -2((-3(3sin(1πx + 1) + 1) + 1)/(-2(3sin(1πx + 1) + 1)))^2 - 5((-3(3sin(1πx + 1)... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -1|(-3cos(3π(-2|(-4x - 3) - 3| - 3)) - 3) - 1| + 2 and g(x) = -5((2((-1x - 3)/(2x + 2)) + 3)/(1((-1x - 3)/(2x + 2)))) - 5 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 30 | MATH |
[
{
"content": "How many solutions does the equation -1((-2(-1(-2sin(3πx - 2))^2 - 5(-2sin(3πx - 2)) + 2) - 3)/(-2(-1(-2sin(3πx - 2))^2 - 5(-2sin(3πx - 2)) + 2) + 3)) + 3 = (2((1(3e^(1x) + 3) - 3)/(-3(3e^(1x) + 3) - 2)) - 1)/(2((1(3e^(1x) + 3) - 3)/(-3(3e^(1x) + 3) - 2)) + 3) have for x in the interval [-10, 10]?... | 60 | MATH |
[
{
"content": "How many solutions does the equation 2(1(-2sin(2π(-3x - 4) + 1) + 2)^2 + 3(-2sin(2π(-3x - 4) + 1) + 2))^2 - 1(1(-2sin(2π(-3x - 4) + 1) + 2)^2 + 3(-2sin(2π(-3x - 4) + 1) + 2)) + 1 = 2|(-1(2x^2 + 5x + 5)^2 + 1(2x^2 + 5x + 5) + 4) + 3| - 2 have for x in the interval [-10, 10]?\n\nPresent the answer i... | 42 | MATH |
[
{
"content": "How many solutions does the equation -2(-3sin(2π(-3(1x - 1)^2 + 1(1x - 1) - 4) - 3) - 2)^2 - 5 = -2(-4((3x - 2)/(2x + 2))^2 + 4((3x - 2)/(2x + 2)) + 3)^2 + 4(-4((3x - 2)/(2x + 2))^2 + 4((3x - 2)/(2x + 2)) + 3) - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed... | 381 | MATH |
[
{
"content": "How many solutions does the equation -1(-4(-1cos(3π(-2|x + 2| + 1)) + 1)^2)^2 - 3(-4(-1cos(3π(-2|x + 2| + 1)) + 1)^2) + 1 = (-1(1(-2e^(-1x) + 2)^2 - 5(-2e^(-1x) + 2) - 3) + 3)/(-1(1(-2e^(-1x) + 2)^2 - 5(-2e^(-1x) + 2) - 3)) have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex forma... | 194 | MATH |
[
{
"content": "How many solutions does the equation -3(-2e^(-2(1(-1x + 3)^2 + 4(-1x + 3) + 3)) - 1)^2 - 4(-2e^(-2(1(-1x + 3)^2 + 4(-1x + 3) + 3)) - 1) + 1 = 1|(2sin(2π(-2|x + 3| - 2) + 1) + 3) + 3| - 3 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 68 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -4(-1|(2e^(-1(-3x + 2)) + 3) - 1| + 3)^2 - 3(-1|(2e^(-1(-3x + 2)) + 3) - 1| + 3) + 4 and g(x) = -2cos(2π(4(5x^2 - 5x - 1)^2 + 1(5x^2 - 5x - 1) - 5) + 2) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ... | 34 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|(1|((2(-3x + 3) - 1)/(-1(-3x + 3) - 2))| - 1) - 2| + 2 and g(x) = 3|(1(-5x^2 + 3x + 3) - 1) + 3| where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "How many solutions does the equation 1(-1sin(1π(-3|(-2x - 3) + 2| - 2) - 3) - 2) = -2(3sin(3π(1x) + 2) - 3)^2 + 1(3sin(3π(1x) + 2) - 3) + 4 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 60 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = (1(-2e^(-2(-1|((-2x - 1)/(3x - 2)) - 1|)) + 3) - 3)/(-1(-2e^(-2(-1|((-2x - 1)/(3x - 2)) - 1|)) + 3) + 2) and g(x) = 1sin(3π(4(-1x^2 - 4x - 3) - 5)) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 1562 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 1(-2|(-2(2e^(2x) + 1) + 3) - 3| - 1) + 2 and g(x) = 3cos(1π(-1|(3x^2 - 1x + 1) - 2| + 1) - 1) - 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 305 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -2sin(1π(-5(-2(2|x + 3| - 3)^2 + 4(2|x + 3| - 3) - 4)^2 - 4(-2(2|x + 3| - 3)^2 + 4(2|x + 3| - 3) - 4) + 4) - 1) + 3 and g(x) = -2|(-2e^(-2((-2x - 2)/(-1x + 3))) + 1)| + 3 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex fo... | 517 | MATH |
[
{
"content": "How many solutions does the equation 2cos(3π(-1|(-3|(2x^2 + 2x + 5) + 1|)| - 3)) + 3 = -5(2cos(3π(1x^2 - 1x - 2) + 2) - 1) - 5 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 601 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 5(2cos(1π(2(1x - 4) + 1) + 3) - 2)^2 - 4(2cos(1π(2(1x - 4) + 1) + 3) - 2) - 2 and g(x) = -1cos(1π((-1((2x + 3)/(1x)) + 2)/(-3((2x + 3)/(1x)))) - 2) + 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your ... | 40 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = -2|((3(1(5x^2 + 4x - 5) + 2) - 2)/(1(1(5x^2 + 4x - 5) + 2))) + 1| + 2 and g(x) = -1|((-3(-2e^(1x) + 3) + 3)/(2(-2e^(1x) + 3) + 2)) - 1| - 2 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "... | 4 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(2e^(1((-2(2|x - 3| + 1) + 1)/(3(2|x - 3| + 1)))) - 3) + 1 and g(x) = -2sin(3π(-5(-1|x - 1| + 2) + 2) + 1) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 300 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2e^(-1(4(-3|(-2x^2 + 2x - 5)| - 3)^2 - 3(-3|(-2x^2 + 2x - 5)| - 3) + 1)) and g(x) = -2sin(2π(-4(1|x - 2| + 3) + 2) - 2) + 1 intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 160 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (1(-3|((1(2|x - 1| + 3) + 3)/(-1(2|x - 1| + 3) + 2)) + 2| + 3) + 3)/(1(-3|((1(2|x - 1| + 3) + 3)/(-1(2|x - 1| + 3) + 2)) + 2| + 3) + 1) and g(x) = (-2((2(-1sin(3πx - 1) - 1))/(-1(-1sin(3πx - 1) - 1) - 2)) - 2)/(-2((2(-1sin(3π... | 56 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3|(1|((1(1|x + 2| - 3) + 3)/(3(1|x + 2| - 3) - 1)) - 3| - 2) + 1| - 2 and g(x) = -3e^(1(-5(-3x + 1)^2 - 5(-3x + 1) - 5)) - 3 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user... | 8 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 3((2((3(-2e^(2x) + 3) + 1)/(3(-2e^(2x) + 3) - 2)) + 3)/(-1((3(-2e^(2x) + 3) + 1)/(3(-2e^(2x) + 3) - 2)))) - 3 and g(x) = (-2(4(-2sin(3πx - 3)) + 1) + 1)/(1(4(-2sin(3πx - 3)) + 1)) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex form... | 17 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1((-2(-5(-1|x - 1| + 1)^2 - 1(-1|x - 1| + 1) + 2))/(2(-5(-1|x - 1| + 1)^2 - 1(-1|x - 1| + 1) + 2) + 1))^2 - 3((-2(-5(-1|x - 1| + 1)^2 - 1(-1|x - 1| + 1) + 2))/(2(-5(-1|x - 1| + 1)^2 - 1(-1|x - 1| + 1) + 2) + 1)) - 2 and g(x) ... | 8 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 4(-5(-5(3sin(1πx + 1) + 1)^2 + 1(3sin(1πx + 1) + 1) + 2) + 3)^2 + 3(-5(-5(3sin(1πx + 1) + 1)^2 + 1(3sin(1πx + 1) + 1) + 2) + 3) + 2 and g(x) = 1e^(-1((1(2|x - 2| - 3) + 2)/(1(2|x - 2| - 3) - 3))) - 3 in the range -10 ≤ x ≤ 10?\n\nP... | 6 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -3(-2sin(3π((-1((2x - 3)/(2x - 1)) + 2)/(-1((2x - 3)/(2x - 1)) - 1)) + 2) + 2) + 5 and g(x) = (2(3sin(3π(-1x^2 + 1x - 4) + 1) - 1) - 2)/(1(3sin(3π(-1x^2 + 1x - 4) + 1) - 1)) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex ... | 246 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -1(-2cos(2π(-3|(-5x^2 + 1x - 2)|) + 3) - 3) - 3 and g(x) = -5(3(-1e^(-2x) + 1)^2 + 4(-1e^(-2x) + 1) - 4) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2sin(1π(-1|((-1(5x^2 - 5x - 3) - 2)/(-3(5x^2 - 5x - 3) + 1)) + 1| - 2) + 3) and g(x) = (3(-4(4x^2 - 2)^2 + 4(4x^2 - 2) + 5) - 2)/(-2(-4(4x^2 - 2)^2 + 4(4x^2 - 2) + 5) - 3) where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex fo... | 9 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = 1(-2|(2(-3cos(2πx - 3) - 1) + 5) - 3|)^2 - 5(-2|(2(-3cos(2πx - 3) - 1) + 5) - 3|) - 3 and g(x) = -3cos(1π((-3(1|x + 1| - 3) - 1)/(-2(1|x + 1| - 3) + 3))) - 2 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed... | 41 | MATH |
[
{
"content": "Find the number of times the graphs of f(x) = 2|(3((1((-1x - 2)/(2x - 1)) + 2)/(-2((-1x - 2)/(2x - 1)) - 2))) + 2| - 3 and g(x) = (-1(-1sin(3π(-5x^2 + 4x - 4) - 3) - 3) - 2)/(3(-1sin(3π(-5x^2 + 4x - 4) - 3) - 3) + 3) intersect in the interval [-10, 10].\n\nPresent the answer in LaTex format: \\box... | 62 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = -1(-1sin(2π(3|(2|x - 3| + 3)| + 3) + 2) - 3)^2 + 4(-1sin(2π(3|(2|x - 3| + 3)| + 3) + 2) - 3) + 2 and g(x) = 4(-3|(5x^2 + 4x - 2) - 1| - 2) + 4 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
... | 12 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 2e^(1(-5(-2(3x)^2 - 1(3x) - 5)^2 - 4(-2(3x)^2 - 1(3x) - 5) - 3)) and g(x) = 2sin(3π(5(2x^2 - 1x - 3)^2 - 3(2x^2 - 1x - 3)) + 1) + 2 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role":... | 1209 | MATH |
[
{
"content": "How many points of intersection exist between the functions f(x) = -4(3|(2sin(3π(-2x - 1) - 3) + 1)| + 2) + 4 and g(x) = 3((3(4x^2 + 2x - 3) - 1)/(-1(4x^2 + 2x - 3) - 2)) - 1 in the range -10 ≤ x ≤ 10?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] | 216 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = (-3(3sin(1π((-2((2x + 1)/(-1x - 1)) + 2)/(3((2x + 1)/(-1x - 1)) + 3)) + 3) - 2) - 1)/(1(3sin(1π((-2((2x + 1)/(-1x - 1)) + 2)/(3((2x + 1)/(-1x - 1)) + 3)) + 3) - 2) + 1) and g(x) = (-3(3|(1cos(1πx) - 2) + 1| - 2) - 2)/(-3(3|(1... | 36 | MATH |
[
{
"content": "How many solutions does the equation (2(3(5((1x + 1)/(-1x - 3))^2 - 1((1x + 1)/(-1x - 3)) - 4) + 1) - 1)/(2(3(5((1x + 1)/(-1x - 3))^2 - 1((1x + 1)/(-1x - 3)) - 4) + 1) + 1) = 3sin(2π(5(4x) + 3) + 2) + 1 have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answe... | 767 | MATH |
[
{
"content": "How many solutions does the equation (-1((-2(-2|(2x^2 + 4x - 3)| + 3) + 1)/(1(-2|(2x^2 + 4x - 3)| + 3) - 2)) - 2)/(-1((-2(-2|(2x^2 + 4x - 3)| + 3) + 1)/(1(-2|(2x^2 + 4x - 3)| + 3) - 2)) + 2) = -3|(-2|(1|x| - 3) - 2| - 1)| have for x in the interval [-10, 10]?\n\nPresent the answer in LaTex format:... | 4 | MATH |
[
{
"content": "Determine the number of intersection points between the functions f(x) = 1|((3((-3(3|x| - 1) + 2)/(-2(3|x| - 1) + 1)) + 2)/(3((-3(3|x| - 1) + 2)/(-2(3|x| - 1) + 1)) - 2))| - 1 and g(x) = 1sin(3π(-1|(1x + 5) - 2| - 3) + 2) + 1 where -10 ≤ x ≤ 10.\n\nPresent the answer in LaTex format: \\boxed{Your ... | 60 | MATH |
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