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, 674 stretching/compressing factor, 928, 929 substitution method, 1216, 1332 Index 1560 vertex, 476, 635, 810, 893, 1344, 1344, 1387, 1395 vertex form of a quadratic function, 479, 635 vertical asymptote, 582, 586, 592, 635, 951 vertical compression, 336, 380 vertical line, 140, 418, 466 vertical line test, 242, 380 v... |
ometric identities, 1080 trinomial, 68, 100 turning point, 513, 534, 635 U union of two events, 1524, 1534 unit circle, 817, 835, 851, 865, 893, 1032 unit vector, 1188, 1202 upper limit of summation, 1488, 1534 upper triangular form, 1233 V variable, 23, 100 varies directly, 625, 635 varies inversely, 627, 635 vector, ... |
expression. More Irrational Numbers The Irrational Numbers 19 2 When a square measures 1 unit on every side, its diagonal measures units. You can use a ruler to " measure the diagonal and then on a show the placement of number line. " 2 What is the value of 2? Can we find a decimal number that, when multiplied by itse... |
roximation Scientific calculators have a key that, when pressed, will place in the display a rational approximation for p that is more accurate than the ones given above. p On a graphing calculator, when the key is accessed, the screen shows the symbol p but a rational approximation is used in the calculation. On a gra... |
equal to 5. Add 1 to the digit in the hundredths place and drop all digits to the right of it. Answer: 3 ≈ 1.73 " Answer: " 0.1 < 0.32 EXAMPLE 3 The circumference C of a circle with a diameter d is found by using the formula C pd. a. Find the exact circumference of a circle whose diameter is 8. b. Find, to the nearest... |
8 " 11. 0.989989998... 15. 5.28 8. 10p 12. 0.725 16. 0.14141414... 19. 48 20. 49 " " 9. 0.12131415... 13. 17. 21. 121 " 2 5 " 0.24682 16 10. " 14. p 30 18. –p 22. p – 2 23. Determine which of the following irrational numbers are between 1 and 4. (1) p 2 (2) 5 " 2 (3) " 4 (4) 11 " (5) 2 3 " 24 Number Systems In 24–43 w... |
diagonal (the line joining opposite corners of the square). Arrange the four pieces of the squares into a larger square. a. What is the area of each of the two squares that you cut out? b. What is the area of the larger square formed by using the pieces of the smaller squares? c. What should be the length of each side... |
, we can express them in decimal form (even using rational approximations) to see which is greater. EXAMPLE 1 The number line that was first seen in Section 1-1 is repeated below. –2 13 – 6 –1 2– 0 — –0.43.8 Of the numbers shown here, tell which are: a. counting numbers b. whole numbers c. integers d. rational numbers ... |
- hundred decimal places. The digits in the places that follow the one-hundredth decimal place are random, form no pattern, and do not terminate. Is the number rational or irrational? Explain. Developing Skills 4. Twelve numbers have been placed on a number line as shown here. –2 3– – –2.7 –1 0 –0.63 1 0.5 1 3 2 π 2 6 ... |
line corresponds to a rational number. 31. Every irrational number corresponds to a point on the real number line. 32. Every point on the real number line corresponds to an irrational number. 33. Some numbers are both rational and irrational. 34. Every repeating decimal corresponds to a point on the real number line. ... |
the last digit. Significant Digits The accuracy of measurement is often indicated in terms of the number of significant digits. Significant digits are those digits used to determine the measure and excludes those zeros that are used as place holders at the beginning of a decimal fraction and at the end of an integer. ... |
the measurement as 4,500 50 feet. One number is said to be more precise than another if the place value of its last significant digit is smaller. For example, 3.40 is more precise than 3.4 because 3.40 is correct to the nearest hundredth and 3.4 is correct to the nearest tenth. 1 2 When measures are added, the sum can... |
to find the answer to an exercise in which the given numbers are thought of as exact values and the answers are given as exact values. However, in certain problems that model practical applications, when the given data are approximate measurements, you may be asked to use the precision or accuracy of the data to deter... |
a. 86.1 cm b. 456 sq cm Numbers as Measurements 33 EXERCISES Writing about Mathematics 1. If 12.5 12.50, explain why a measure of 12.50 inches is more accurate and more precise than a measurement of 12.5 inches. 2. A circular track has a radius of 63 meters. Mario rides his bicycle around the track 10 times. Mario mul... |
3, 4,...}. The whole numbers are {0, 1, 2, 3, 4,...}. The integers are {..., 4, 3, 2, 1, 0, 1, 2, 3, 4,...}. These sets of numbers form the basis for a number line, on which the length of a segment from 0 to 1 is called the unit measure of the line. a The rational numbers are all numbers that can be expressed in the f... |
whether each sentence is true of false. 7. 7 8 8. –7 2 9. 4 8 10. 9 9 In 11–16, write each rational number in the form, where a and b are integers and b 0. a b 11. 0.9 12. 0.45 13. 81 2 14. 14 15. 0.3 16. 63 17. Find a rational number between 19.9 and 20. In 18–22, tell whether each number is rational or irrational. 1... |
without using a radical sign: a. Write an irrational number. b. Write three irrational numbers that are between 5 and 6 in increasing order. c. Write three irrational numbers that are between 0.55 and 0.56 in increasing order. d. Write three irrational numbers that are between 0.556 and 0.556 in increas- ing order. OP... |
the set of rational numbers. When no set is identified, use the set of all real numbers. 2. The rule for the binary operation must be clear, such as the rules you know for addition, subtraction, multiplication, and division. 3. The order of the elements is important. Later in this chapter, we will use the notation (a,... |
” or “4 raised to the second power,” or “the second power of 4.” Exponent Base 42 = 16 Power The exponent 3 indicates that a factor is used three times. 4 4 4 64 can be written as 43 64. 43 is read as “4 cubed,” or “4 raised to the third power,” or “the third power of 4.” Exponent Base 43 = 64 Power 40 Operations and P... |
2 11 6 5 is correct. A different problem involving powers is solved in this way: 1. Simplify powers: 5 23 + 3 5 8 3 2. Multiply and divide: 3. Add and subtract: 40 3 43 Expressions with Grouping Symbols In mathematics, parentheses ( ) act as grouping symbols, giving different meanings to expressions. For example, (4 6... |
ly: (4) Subtract: 80 4(7 5) 80 4(2) 80 8 72 Calculator Solution ENTER: 80 4 ( 7 5 ) ENTER DISPLAY Answer 72 EXERCISES Writing About Mathematics 1. Explain why 2 is the only even prime. 2. Delia knows that every number except 2 that ends in a multiple of 2 is composite. Therefore, she concludes that every number except ... |
28 cents? 25. What is the cost of two chocolate chip cookies that cost 30 cents each and three peanut but- ter cookies that cost 25 cents each? 26. How many miles did Ms. McCarthy travel if she drove 30 miles per hour for hour and 55 3 4 miles per hour for 11 2 hours? 27. What is the cost of two pens at $0.38 each and... |
irrational numbers is not closed under addition. There are some pairs of irrational numbers whose sum is not an irrational number. However, p, p, 0, and each of the other numbers used in these examples are real numbers and the sum of two real numbers is a real number. The sets of whole numbers, rational numbers, and r... |
there are some pairs of irrational numbers whose quotient is not an irrational number. The set of irrational numbers is not closed under division. 5, 1, and each of the other numbers used in these examples are real However, numbers and the quotient of two nonzero real numbers is a nonzero real number. " The sets of no... |
8). This example illustrates the associative property of addition. In general, we assume that for every number a, every number b, and every number c: (a b) c a (b c) Associative Property of Multiplication In a similar way, to find a product that involves three factors, we first multiply any two factors and then multip... |
2. 3. 6.5 8 (6 0.5)8 6 8 0.5 8 48 4 52 1 9 3 9 27 3 30 3 311 3 9 B A Working backward, we can also use the distributive property to change the form of an expression from a sum or a difference to a product: 1. 5(12) 5(8) 5(12 8) 5(20) 100 2. 7(14) 7(4) 7(14 4) 7(10) 70 Addition Property of Zero and the Additive Identit... |
other. Consider these examples: 4? 1 4 5 1 1 The reciprocal of 4 is. 4 The reciprocal of 1 4 is 4. 21 The multiplicative inverse of B A The multiplicative inverse of 21 2 or 5 2 2 is. 5 2 5 5 is or 2 21 2. Since there is no number that, when multiplied by 0, gives 1, the number 0 has no reciprocal, or no multiplicativ... |
Step Reason (1) 6t t 6t 1t (6 1)t 7t (2) (3) Multiplication property of 1. Distributive property. Addition. Answer 7t Properties of Operations 53 EXERCISES Writing About Mathematics 1. If x and y represent real numbers and xy x: a. What is the value of y if the equation is true for all x? Explain your answer. b. What ... |
a correct application of the distributive property. If you believe that it is not, state your reason. 26. 6(5 8) 6(5) 6(8) 28. 5 (8 6) (5 8) (5 6) 30. 14a 4a (14 4)a 10 5 10 3 1 2 1 1 2 1 1 1 27. 5 5 B 29. 3(x 5) 3x 3 5 31. 18(2.5) 18(2) 18(0.5) A 54 Operations and Properties In 32–35: a. Tell whether each sentence is... |
OF SIGNED NUMBERS Adding Numbers That Have the Same Signs The number line can be used to find the sum of two numbers. Start at 0. To add a positive number, move to the right. To add a negative number, move to the left. EXAMPLE 1 Add 3 and 2. Addition of Signed Numbers 55 +3 +2 –1 0 +1 +2 +3 +4 +5 +6 Solution Start at ... |
Addition property of zero Addition property of opposites The sum 1 is a number whose absolute value is the difference of the absolute values of 3 and 2 and whose sign is the same as the sign of 3, the number with the greater absolute value. EXAMPLE 4 Add: 3 and 2. Solution Start at 0 and move 3 units to the left to 3;... |
: 233 4 Property of the Opposite of a Sum For all real numbers a and b: (a b) (a) (b) When adding more than two signed numbers, the commutative and associative properties allow us to arrange the numbers in any order and to group them in any way. It may be helpful to add positive numbers first, add negative numbers next... |
at that time? 30. A football team gained 7 yards on the first play, lost 2 yards on the second, and lost 8 yards on the third. What was the net result of the three plays? 31. Fay has $250 in a bank. During the month, she made a deposit of $60 and a withdrawal of $80. How much money did Fay have in the bank at the end ... |
subtrahend difference minuend: Subtract Now, consider another way in which addition and subtraction are related. In each of the following examples, compare the result obtained when a signed number is subtracted with the result obtained when the opposite of that signed number is added. Subtract Add 9 9 6 6 3 3 Subtract... |
EXAMPLE 4 How much greater than 3 is 9? Solution 9 (3) 9 3 12 Answer 9 is 12 greater than 3. +12 –3 0 +9 Note that parentheses need not be entered in the calculator in Example 2 since the additions and subtractions are to be done in the order in which they occur in the expression. Parentheses are needed, however, in E... |
operation of subtraction commutative? In other words, for all signed numbers x and y, does x y y x? 27. State whether each of the following sentences is true or false: a. (15 9) 6 15 (9 6) b. [(10) (4)] (8) (10) [(4) (8)] 28. Is the operation of subtraction associative? In other words, for all signed numbers x, y, and... |
number and the positive number is a negative number. Multiplying a Positive Number by a Negative Number CASE 3 If a girl gained 2 pounds each month, 4 months ago she was 8 pounds lighter than she is now. Using signed numbers, we may write: (2)(4) 8 The product of the positive number and the negative number is a negati... |
the two numbers have the same sign. 3. Write a minus sign before this product when the two numbers have different signs. 66 Operations and Properties EXAMPLE 1 Find the product of each of the given pairs of numbers. a. (12)(4) c. (18)(3) Answers 48 54 e. (3.4)(3) 10.2 b. (13)(5) d. (15)(6) 271 8 f. A B (23) Answers 5 ... |
( 27) 6. 23(15) 7. +4(4) 8. +5.4(0.6) 9. 2.6( 0.05) 10. 231 3 112 3 A B 11. 253 4 21 2 A B 12. 262 3(7) In 13–18, find the value of each expression. 13. (18)(4)(5) 16. (12)(7 3) 14. (3.72)(0.5)(0.2) 273 1 723 8 28 17. A B 15. (4)(35 7) 18. 4 23 4 1 1 4 In 19–28, find the value of each power. 19. (3)2 24. 24 20. 32 25. ... |
following examples: CASE 1 CASE 2 CASE 3 CASE 4 16 13 26 23 26 13 16 23? implies (?)(3) 6. Since (2)(3) 6, 2 16 13? implies (?)(3) 6. Since (2)(3) 6, 2 26 23? implies (?)(3) 6. Since (2)(3) 6, 2 26 13? implies (?)(3) 6. Since (2)(3) 6, 2 16 23 In the preceding examples, observe that: 1. When the dividend and divisor a... |
4 b. b. 110 290 21 9 227 c. 23 c. 9 d. (45) 9 d. 5 e. 0 (9) f. 3 0 e. 0 f. Undefined Answers Using the Reciprocal in Dividing Signed Numbers In Section 2-2, we learned that for every nonzero number a, there is a unique number, called the reciprocal or multiplicative inverse, such that a? 1 a 5 1 1 a. 70 Operations and... |
. (100) (2.5) 21. (0.5) (0.25) 23. 25. (112) 4 17 8 4 A B A 21 3 A 221 32 B B 24. 26. A 23 4 B 211 4 A 4 (16) 221 2 4 B A B In 27–28, state whether each sentence is true or false: 27. a. [(16) (4)] (2) (16) [(4) (2)] b. [(36) (6)] (2) (36) [(6) (2)] c. Division is associative. 28. a. (12 6) 2 12 2 6 2 b. [(25) (10)] (5... |
element, point E, the point that is on line AB ( ) and on line CD 4 AB ( ). We write the intersection of the lines in the 4 d CD example shown as 4 AB 4 CD A D E C B 3. Intersection is a binary operation, are subsets of the universal set and example: 5 E. F d G 5 H, where sets F, G, and H d is the operation symbol. Fo... |
union. {1, 2, 3, 4, 5}, or A < B Operations with Sets 73 4. The union of the set of all rational numbers and the set of all irrational numbers is the set of real numbers. {real numbers} {rational numbers} {irrational numbers} Complement of a Set The complement of a set A, denoted by, is the set of all elements that be... |
Answer 74 Operations and Properties EXERCISES Writing About Mathematics 1. A line is a set of points. Can the intersection of two lines be the empty set? Explain. 2. Is the union of the set of prime numbers and the set of composite numbers equal to the set of counting numbers? Explain. Developing Skills In 3–10, A {1,... |
the complement of A ) c. B d. B B (the complement of ) 25. Let the universe U {1, 2, 3, 4, 5, 6, 7, 8}. a. Find the elements of A d B, A d B, and A < B when A and B are equal to: (1) A = {1, 2, 3, 4}; B {5, 6, 7, 8} (3) A {1, 3, 5, 7}; B {2, 4, 6, 8} (2) A {2, 4}; B {6, 8} (4) A {2}; B {4} A d B. b. When A and B are di... |
Moving to the right and moving up are regarded as movements in the positive direction. In the coordinate plane, points to the right of O on the x-axis and on lines parallel to the x-axis and points above O on the y-axis and on lines parallel to the y-axis are assigned positive values. Moving to the left and moving dow... |
, every ordered pair (x, y) consists of two negative numbers. For example, C, the graph of (2, 3), lies 2 units to the left of the origin (in the negative direction along the x-axis) and then down 3 units (in the negative direction parallel to the y-axis). Point D, the graph of (3, 5) in quadrant IV, lies 3 units to th... |
The number assigned to that point on the x-axis is the x-coordinate of the point. 2. From the point, move along a horizontal line to the y-axis.The number assigned to that point on the y-axis is the y-coordinate of the point. 78 Operations and Properties To find the coordinates of point R, from point R, move in the ver... |
inate and are on a line parallel to the y-axis. Graphing Number Pairs 79 6. Points A and D have the same x-coordinate and are on a line parallel to the y-axis. 7. Lines parallel to the y-axis are parallel to each other. 8. Lines parallel to the y-axis are perpendicular to the x-axis. Now, we know that ABCD is a rectang... |
2, 0), and G(0, 4) on the coordinate plane and joined the points in order. Explain how Phyllis can find the area of this polygon, then find the area. Developing Skills 3. Write as ordered number pairs the coordinates of points A, B, C, D, E, F, G, H, and O in the graph. B y F O 2 1 G –2 C –1 –1 –2 H A E x 21 D In 4–15,... |
E(5, 3), N(2, 0) 30. B(3, 2), A(2, 2), R(2, 2), N(3, 2) 32. R(4, 2), A(0, 2), M(0, 7) 29. F(5, 1), A(5, 5), R(0, 5), M(2, 1) 31. P(3, 0), O(0, 0), N(2, 2), D(1, 2) 33. M(1, 1), I(3, 1), L(3, 3), K(1, 3) 34. Graph points A(1, 1), B(5, 1), and C(5, 4). What must be the coordinates of point D if ABCD is a rectangle? 35. ... |
parentheses or other grouping symbols; (2) simplify powers; (3) multiply and divide from left to right; (4) add and subtract from left to right. Many properties are used in operations with real numbers, including: • closure under addition, subtraction, and multiplication; • commutative properties for addition and mult... |
the plane are called the coordinates of the point. The first number in the pair is called the x-coordinate or abscissa, and the second number is the y-coordinate or ordinate. The coordinates of a point are represented as (x, y). VOCABULARY 2-1 Binary operation • Factor • Prime • Composite • Base • Exponent • Power • O... |
B 17. A 22. A < B 18. 23. A d B A d B 19. A d A 24. A < B In 25–32, find each sum or difference. 25. 6 6 29. 23 0 26. 6 6 30. 54 52 27. 3.2 4.5 31. 100 25 28. 4 5 32. 0 7 84 Operations and Properties In 33–40, to each property named in Column I, match the correct application of the property found in Column II. Column ... |
on his team’s 30-yard line. On the first play, one of his linemen was offsides for a loss of 5 yards. On the next play, Doug gave the ball to the runningback who made a gain of 8 yards. He then made a 17-yard pass. Then Doug was tackled, for a loss of 3 yards. Where was Doug on the field after he was tackled? Cumulati... |
ATIVE REVIEW CHAPTERS 1–2 Part I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. Which number is not an integer? (1) 7 (2) 2 (3) 0.2 (4) 9 " 86 Operations and Properties 2. Which inequality is false? (1) 4 3 (2) 3 3 3. What is the opposite of 4? (1) 2... |
least 12 players. What is the largest possible number of players on any one team? Cumulative Review 87 Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questi... |
ENTENCES An express delivery company will deliver a letter or package locally, within two hours.The company has the following schedule of rates. In addition to the basic charge of $25, the cost is $3 per mile or part of a mile for the first 10 miles or less and $4.50 per mile or part of a mile for each additional mile ... |
a b may be used to represent several different verbal phrases, such as: a minus b b subtracted from a a decreased by b a diminished by b b less than a a reduced by b the difference between a and b Verbal Phrases Involving Multiplication The algebraic expressions a b, a b, (a)(b) and ab may be used to represent several... |
Symbols 91 EXERCISES Writing About Mathematics 1. Explain why the sum of a and 4 can be written as a 4 or as 4 a. 2. Explain why 3 less than a can be written as a 3 but not as 3 a. Developing Skills In 3–20, use mathematical symbols to translate the verbal phrases into algebraic language. 3. y plus 8 6. x times 7 4. 4... |
. EXAMPLE 1 Represent each phrase by an algebraic expression. a. a distance that is 20 meters shorter than x meters b. a bill for n baseball caps, each costing d dollars c. a weight that is 40 pounds heavier than p pounds d. an amount of money that is twice d dollars Solution a. How to Proceed (1) Think of a similar pr... |
represent the cost of n apples. c. Do the algebraic expressions in parts a and b always represent whole numbers? Explain your answer. Developing Skills In 3–18, represent each answer in algebraic language, using the variable mentioned in the problem. 3. The number of kilometers traveled by a bus is represented by x. I... |
days in w weeks and 5 days. b. Represent the total number of days in w weeks and d days. Applying Skills 20. An auditorium with m rows can seat a total of c people. If each row in the auditorium has the same number of seats, represent the number of seats in one row. 21. Represent the total number of calories in x pean... |
ic term is called the coefficient of the remaining factor, or product of factors, of that term. For example, consider the algebraic term 3xy: 3 is the coefficient of xy 3y is the coefficient of x 3x is the coefficient of y xy is the coefficient of 3 When an algebraic term consists of a number and one or more variables,... |
(d)4 1(d)(d)(d)(d) is always a positive number since the exponent is even. EXAMPLE 1 For each term, name the coefficient, base, and exponent. a. 4x5 b. w8 c. 2pr coefficient 4 coefficient –1 coefficient 2p Answers base x base w base r exponent 5 exponent 8 exponent 1 Note: Remember that coefficient means numerical coe... |
3 2a4 40. 37. (ax)5 41. (b)3 42. If x represents the cost of a can of soda, what could 5x represent? 43. If r represents the speed of a car in miles per hour, what could 3r represent? 44. If n represents the number of CDs that Alice has, what could n 5 represent? 45. If d represents the number of days until the end of... |
2 Molly earned d dollars in July and the number of dollars that Molly earned in August. 1 2d 1 10 dollars in August. Describe in words Answer In August, Molly earned 10 more than half the number of dollars that she earned in July. EXAMPLE 3 Describe a situation in which x and 12 x can be used to represent variable qua... |
a pen. h 6 hours driving to and from work. h 3 11. Seema’s essay for English class had w words and Dominic’s had 12. Virginia read r books last month and Anna read 3r 5 books. 3 4 w 1 80 words. 13. Mario and Pete are playing a card game where it is possible to have a negative score. Pete’s score is s and Mario’s score... |
21 29 Answer 29 EXAMPLE 2 Evaluate 2x2 5x 4 when: a. x = 7 b. x = 1.2 Solution How to Proceed a. (1) Write the expression: (2) Replace the variable by the value 7: (3) Evaluate the power: (4) Multiply: (5) Add: b. (1) Write the expression: (2) Replace the variable by the value 1.2: (3) Evaluate the power: (4) Multiply... |
3 2(0.40)3 0.512 2(0.064) 0.512 0.128 0.384 Evaluating Algebraic Expressions 103 EXERCISES Writing About Mathematics 1. Explain why, in an algebraic expression such as 12ab, 12 is called a constant and a and b are called variables? 2. Explain why, in step 2 of Example 1, parentheses were needed when x was replaced by i... |
the yearly cost for each of the following: a. Tiffany is an amateur potter who fired 35 pounds of work this year. b. Nia sells her pottery in a local craft shop and fired 485 pounds of work this year. 30. If a stone is thrown down into a deep gully with an initial velocity of 30 feet per second, the distance it has fa... |
Six more than x is 9.” This sentence can be written in symbols as x 6 9. Every sentence that contains a variable is called an open sentence. x 6 9 An open sentence is neither true nor false. x 5 8 3y 12 2n 0 The sentence will be true or false only when the variables are replaced by numbers from a domain or a replacemen... |
is the empty set or the null set, written in symbols as { } or as. Answer: a. The solution set is { } or. b. Procedure: Of course, you cannot replace y with every whole number, but you can use multiplication facts learned previously. You know that 3(4) 12. Let y 4. Then 3(4) 12 is true. No other whole number would mak... |
$0.99 a can but limits the number of cans that a customer may buy at the sale price to no more than 5. a. The domain for this problem is the number of cans of juice that a customer may buy at the sale price. Write the domain. b. If Mrs. Dajhon does not want to spend more than $10, the number of cans that she might buy... |
less than the number of sides. Solution a. Let s represent the length of each side of a square. b. 8% (or 8 percent) means 8 hundredths, written as 0.08 or 8 100. P 4s Answer 100p c. “2 less than the number of sides” means (n 2). C p 0.08p or C 5 p 1 8 Answer S 180(n 2) Answer 108 Algebraic Expressions and Open Senten... |
cost c plus the margin of profit m. 5. The perimeter P of a rectangle is equal to the sum of twice its length l and twice its width w. 6. The average m of three numbers, a, b, and c is their sum divided by 3. 7. The area A of a triangle is equal to one-half the length of the base b multiplied by the length of the alti... |
than 3 minutes (m 3). b. Find the cost of a 2.5 minute telephone call if x $0.25 and y $0.05. c. Find the cost of a 10 minute telephone call if x $0.25 and y $0.05. 20. The cost D in dollars of sending a fax of p pages is a dollars for sending the first page and b dollars for each additional page. a. Write two formula... |
formula for E when c 36. c. Find the number of cookies Mrs. Lucy sold if she makes a profit of $2.00. CHAPTER SUMMARY Chapter Summary 111 An algebraic expression, such as x 6, is an expression or a phrase that contains one or more variables, such as x. The variable is a placeholder for numbers. To evaluate an expressi... |
and q quarters. 16. In the term 2xy3 what is the coefficient? 17. In the term 2xy3 what is the exponent of y? 18. In the term 2xy3, what is the base that is used 3 times as a factor? 19. If distance is the product of rate and time, write a formula for distance, d, in terms of rate, r, and time, t. 20. What is the smal... |
cost as much as five bananas. One orange costs the same as a banana and an apple. How many apples cost the same as three bananas? Exploration STEP 1. Write a three-digit multiple of 11 by multiplying any whole number from 10 to 90 by 11. Add the digits in the hundreds and the ones places. If the sum is greater than or... |
,...} (3) {3, 2, 1} (4) {2, 1} 5 3 (3) 1.666666667 2. The exact value of the rational number can be written as (1) 1.6 (2) 1.6 (4) 1.666666666 3. Rounded to the nearest hundredth, (1) 2.23 (2) 2.236 5 is approximately equal to (3) 2.24 (4) 2.240 " 4. Which of the following numbers is rational? (1) p (2) (3) 1.42 2 " (4... |
How many students had read both fiction and nonfiction? 12. What is the largest number that is the product of three different two-digit primes? Part III Answer all questions in this part. Each correct answer will receive 3 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diag... |
raphing the Solution Set of an Inequality 4-9 Using Inequalities to Solve Problems Chapter Summary Vocabulary Review Exercises Cumulative Review 116 FIRST DEGREE EQUATIONS AND INEQUALITIES IN ONE VARIABLE An equation is an important problem-solving tool. A successful business person must make many decisions about busin... |
equation is called an identity. Thus, 5 x x (5) is an identity when the domain is the set of real numbers because every element of the domain makes the sentence true. Two equations that have the same solution set are equivalent equations. To solve an equation is to find its solution set. This is usually done by writin... |
3 (3)x x 3 5 5 3 5 (3)5 x 15 In the equation 2x 3 15, there are two operations in the left side: multiplication and addition. In forming the left side of the equation, x was first multiplied by 2, and then 3 was added to the product. To solve this equation, we must undo these operations by using the inverse elements i... |
we do not need to write each of the steps, as shown in the examples that follow. 120 First Degree Equations and Inequalities in One Variable EXAMPLE 1 Solve and check: 7x 15 71 Solution How to Proceed (1) Write the equation: (2) Add 15, the opposite of 15 to each side: (3) Since multiplication and division are inverse... |
his next step? 7 b. In this section you learned to solve the equation 7x 15 71 by first adding the opposite of 15, 15, to both sides of the equation. Which method do you think is better? Explain your answer. Developing Skills In 3 and 4, write a complete solution for each equation, listing the property used in each st... |
paid $4.30 to mail a package and bought some 39-cent stamps. He paid a total of $13.66. Find s, the number of stamps that he bought, by solving the equation 0.39s 4.30 13.66. 4-2 SIMPLIFYING EACH SIDE OF AN EQUATION An equation is often written in such a way that one or both sides are not in simplest form. Before star... |
4 6 2x 3x 4 6 26 2(22) 1 3(22) 1 4 5? combining like terms: (3) Add 4, the additive inverse of 4, to each side: (4) Multiply by, the 1 5 multiplicative inverse of 5: (5) Simplify each side. Answer 2 24 2 6 1 4 5? 26 6 6 ✔ 4 4 10 5x 5(5x) 5 1 1 5(210) x 2 124 First Degree Equations and Inequalities in One Variable Note... |
5? 6 6 6 ✔ Simplifying Each Side of an Equation 125 Representing Two Numbers with the Same Variable Problems often involve finding two or more different numbers. It is useful to express these numbers in terms of the same variable. For example, if you know the sum of two numbers, you can express the second in terms of ... |
sufficient since the equation formed may not be correct. The sum of the numbers is 43: 24 19 43. The larger number minus the smaller number is 5: 24 19 5. Alternate Solution Reverse the way in which the facts are used. (1) Represent each number in terms of the same variable using Fact 2: the larger number is 5 more th... |
9) 30 12. 6(3c 1) 42 14. 4(c 1) 32 16. 18 6x 4(2x 3) 18. 5(x 3) 30 10 20. 5(3c 2) 8 43 22. 8b 4(b 2) 24 24. 28y 6(3y 5) 40 26. 0.04(2r 1) 0.03(2r 5) 0.29 27. 0.3a (0.2a 0.5) 0.2(a 2) 1.3 28. 3 4(8 1 4x) 2 1 3(6x 1 3) 5 9 Applying Skills In 29–33, write and solve an equation for each problem. Follow these steps: a. Lis... |
an equation without changing the solution set. For instance, to solve 8x 30 5x, write an equivalent equation that has only a constant in the right side. To do this, eliminate 5x from the right side by adding its opposite, 5x, to each side of the equation. METHOD 1 8x 30 5x 5x 5x 3x 30 x 10 METHOD 2 8x 30 5x 8x (5x) 30... |
Equations and Inequalities in One Variable To check that y 5 is the solution to the equation 3y 7 5y 3, first store 5 as the value of y. then enter the equation to be checked. ENTER: 5 STO ALPHA Y ENTER 3 ALPHA Y 7 2nd TEST ENTER 5 ALPHA Y 3 ENTER DISPLAY The calculator displays 1 which indicates that the statement of... |
__| ↓ (3) Solve the equation to find Clara’s share. x 5,000 x 500 x x 500 2x 500 2x 5,000 500 4,500 2,250 x Clara’s share is x $2,250. (4) Find Emma’s share: 5,000 x 5,000 2,250 $2,750. Alternate Solution (1) Use the fact that Emma’s share is $500 more than Clara’s share to express each share in terms of a variable. Le... |
9. 0.8m 0.2m 24 23 4x 1 24 5 3x 12. 15. x 9x 72 18. 7r 10 3r 50 21. x 4 9x 4 24. c 20 55 4c 27. 3m (m 1) 6m 1 3t 2 11 5 4(16 2 t) 2 1 2 3t 30. 32. 8c 1 7c 2(7 c) 34. 4(3x 5) 5x 2( x 15) 36. 5 3(a 6) a 1 8a 4. 9x 44 2x 7. 2d 36 5d 10. 8y 90 2y 13. 5a 40 3a 16. 0.5m 30 1.1m 19. 4y 20 5y 9 22. 9x 3 2x 46 25. 2d 36 3d 54 ... |
as the third, find the three numbers. Applying Skills In 43–50, use an algebraic solution to solve each problem. 43. It took the Gibbons family 2 days to travel 925 miles to their vacation home. They traveled 75 miles more on the first day than on the second. How many miles did they travel each day? 44. During the fir... |
70 centimeters and h 3.20 centimeters, substitute the given values in the formula for the area of a triangle: 1 2bh 1 2(4.70 cm) A 7.52 cm2 (3.20 cm) A is the subject of the formula. Now that you can solve equations, you will be able to find the value of any variable in a formula when the values of the other variables ... |
feet. EXAMPLE 3 The perimeter of a rectangle is 52 feet. The length is 2 feet more than 5 times the width. Find the dimensions of the rectangle. Solution Use the formula for the perimeter of a rectangle, P 2l 2w, to solve this problem. Let w the width, in feet, of the rectangle. Then 5w 2 the length, in feet, of the r... |
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