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Solution Manual for Beginning Algebra, 7th Edition

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    I NSTRUCTOR ’ S S OLUTIONS M ANUAL C T RIMBLE & A SSOCIATES B EGINNING A LGEBRA S EVENTH E DITION Elayn Martin-Gay University of New Orleans

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    Contents Chapter 1 1 Chapter 2 34 Chapter 3 10 7 Chapter 4 16 3 Chapter 5 2 1 5 Chapter 6 2 5 5 Chapter 7 2 9 7 Chapter 8 3 6 9 Chapter 9 4 11 Appendix A 4 6 7 Appendix B 4 6 9 Appendix C 4 71 Appendix D 4 7 3 Practice Final Exam 4 7 4

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    1 Chapter 1 Section 1.2 Practice Exercises 1. a. 5 < 8 since 5 is to the left of 8 on the number line. b. 6 > 4 since 6 is to the right of 4 on the number line. c. 16 < 82 since 16 is to the left of 82 on the number line. 2. a. 9  3 is true, since 9 > 3 is true. b. 3  8 is false, since neither 3 > 8 nor 3 = 8 is true. c. 25  25 is true, since 25 = 25 is true. d. 4  14 is true, since 4 < 14 is true. 3. a. 3 < 8 b. 15  9 c. 6  7 4 . The integer  10 represents 10 meters below sea level. 5. a. The natural number is 25. b. The whole number is 25. c. The integers are 25,  15,  99. d. The rational numbers are 25, 7 , 3  15, 3 , 4   3.7, 8.8,  99. e. The irrational number is 5. f. The real numbers are 25, 7 , 3  15, 3 , 4  5,  3.7, 8.8,  99. 6. a. 0 < 3 since 0 is to the left of 3 on a number line. b. 15 >  5 since 15 is to the right of  5 on a number line. c. 12 3 4  since 12 4 simplifies to 3. 7. a. |  8| = 8 since  8 is 8 units from 0 on a number line. b. |9| = 9 since 9 is 9 units from 0 on a number line. c. |  2.5| = 2.5 since  2.5 is 2.5 units from 0 on a number line. d. 5 5 11 11  since 5 11 is 5 11 unit from 0 on a number line. e. 3 3  since 3 is 3 units from 0 on a number line. 8. a. |8| = |  8| since 8 = 8. b. |  3| > 0 since 3 > 0. c. |  7| < |  11| since 7 < 11. d. |3| > |2| since 3 > 2. e. |0| < |  4| since 0 < 4. Vocabulary, Readiness & Video Check 1.2 1. The whole numbers are {0, 1, 2, 3, 4, ...}. 2. The natural numbers are {1, 2, 3, 4, 5, ...}. 3. The symbols  ,  , and > are called inequality symbols. 4. The integers are {...,  3,  2,  1, 0, 1, 2, 3, ...}. 5. The real numbers are {all numbers that correspond to points on a number line}. 6. The rational numbers are and are integers, 0 . a a b b b        7. The irrational numbers are {nonrational numbers that correspond to points on a number line}. 8. The distance between a number b and 0 on a number line is | b | .

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 2 9. To form a true statement: 0 < 7 . 10. Five is greater than or equal to four; 5  4 11. 0 belongs to the whole numbers, the integers, the rational numbers, and the real numbers; since 0 is a rational number, it cannot also be an irrational number. 12. The absolute value of a real number a , denoted by | a |, is the distance between a and 0 on a number line. Exercise Set 1.2 2. 9 < 15 since 9 is to the left of 15 on a number line. 4. 2.13 > 1.13 since 2.13 is to the right of 1.13 on a number line. 6. 20 > 0 since 20 is to the right of 0 on a number line. 8.  4 >  6 since  4 is to the right of  6 on a number line. 10. 0 < 100 since 0 is to the left of 100 on a number line. 12. 360  180 since 360 is to the right of 180 on a number line. 14. 4  7 is false, since 4 is to the left of 7 on a number line. 16. 17 > 16 is true, since 17 is to the right of 16 on a number line. 18. 8  8  8  7 is false, since 56 is to the left of 64 on a number line. 20. 4 < 7 is true, since 4 is to the left of 7 on a number line. 22. 0 <  15 is false, since 0 is to the right of  15 on a number line. 24. Fifteen is greater than five is written as 15 > 5. 26. Negative ten is less than or equal to thirty - seven is written as  10  37. 28. Negative seven is not equal to seven is written as  7  7. 30. The integer 535 represents 535 feet above sea level. The integer  8 represents 8 feet below sea level. 32. The integer  80,784 represents 80,784 fewer students. 34. 30 represents an ascent of feet.  50 represents a descent of 50 feet. 36 . The number 1 4 belongs to the sets of: rational numbers and real numbers. 38 . The number 1 2  belongs to the sets of: rational numbers and real numbers. 40 . The number 5 belongs to the sets of: natural numbers, whole numbers, integers, rational numbers, and real numbers. 4 2 . The number 3 belongs to the sets of: irrational numbers and real numbers. 44 . The number 5 1 9  belongs to the sets of: rational numbers and real numbers. 46 . False; negative numbers may be irrational. 48 . True 5 0 . False; irrational numbers are real. 5 2 . False; 1 2 is not an integer. 54 . False; 0 is a whole number that is not positive. 56 .  200 <  20 since  200 is to the left of  20 on a number line. 58 . 7.1 >  7 since 7.1 is to the right of  7 on a number line. 6 0 . 8 12 2 3  since 4 = 4. 6 2 . |  20| >  200 since 20 > 200. 64 . 0 = |0| since 0 = 0.

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    ISM: Beginning Algebra Chapter 1: Review of Real Numbers 3 66 . 2 2 5 5   since 2 2 . 5 5  68 .  500 < |  50| since  500 < 50. 7 0 . 24 12 2   since 24 12 . 2  7 2 . The 20 14 cranberry production in Massachusetts was 206 million pounds, while the 2 014 cranberry production in Wisconsin was 539 million pounds. 206 million < 539 million 74 . The bars for Oregon and New Jersey are closest in length, so Oregon and New Jersey had the closest cranberry crops in 20 14 . 76. The tallest bars represent 49 players admitted. 78. Look for the bar with a height less than 30; 1956  1965. 80. answers may vary 82 . 0.96 < 0.98 since 0.96 is to the left of 0.98 on the number line. 8 4 . Spica is dimmer since 0.98 > 0. 96 . 8 6 . Regulus is dimmest since 1.35 is to the right of all other numbers listed. 8 8 . 13   13 has the same meaning as  13  13. 90 . 7 3 < 7 5 has the same meaning as 7 5 > 7 3. 92 .  2 >  4 has the same meaning as  4 <  2. 9 4 . a nswers may vary Section 1.3 Practice Exercises 1. a. 36 = 4  9 = 2  2  3  3 b. 200 2 100 2 4 25 2 2 2 5 5           2. a. 63 3 3 7 7 7 72 2 2 2 3 3 2 2 2 8            b. 64 2 2 2 2 2 2 2 2 2 2 16 12 2 2 3 3 3              c. 7 7 25 5 5   There are no common factors other than 1, so 7 25 is already in lowest terms. 3. 3 7 3 7 3 7 7 7 8 9 8 9 2 2 2 3 3 2 2 2 3 24                4. a. 3 4 3 9 3 9 27 4 9 4 4 4 4 16        b. 5 5 1 5 1 5 1 15 12 12 15 12 15 12 3 5 36           c. 7 7 7 15 7 15 15 3 5 5 6 15 6 7 6 7 6 2 3 2            5. a. 8 3 8 3 5 1 5 5 5 5      b. 8 2 8 2 6 5 5 5 5     c. 3 1 3 1 4 5 5 5 5     d. 5 1 5 1 6 1 12 12 12 12 2      6. 2 2 7 2 7 14 3 3 7 3 7 21       7. a. 5 1 5 7 1 11 11 7 11 7 7 11 35 11 77 77 35 11 77 46 77             b.      5 10 21 42 1 7 6 42 1 3 14 42

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 4 c. 1 29 4 10 29 4 6 3 30 5 30 30 5 6 10 29 24 30 30 39 24 30 15 30 1 2               8. 1 2 31 22 5 6 5 6 5 31 22 6 5 31 2 11 2 3 5 341 15 11 22 15               9.         1 1 13 76 76 75 12 12 12 1 3 3 35 35 35 4 12 12 5 10 40 40 6 12 Vocabulary , Readiness & Video Check 1.3 1. A quotient of two numbers, such as 5 , 8 is called a fraction . 2. In the fraction 3 , 11 the number 3 is called the numerator and the number 11 is called the denominator . 3. To factor a number means to write it as a product . 4. A fraction is said to be simplified when the numerator and the denominator have no common factors other than 1. 5. In 7  3 = 21, the numbers 7 and 3 are called factors and the number 21 is called the product . 6. The fractions 2 9 and 9 2 are called reciprocals . 7. Fractions that represent the same quantity are called equivalent fractions. 8. 5, Fundamental Principle of Fractions 9. The division operation changes to multiplication and the second fraction 1 20 changes to its reciprocal 20 1 . 10. Find the LCD; two fractions must have the same or common denominator before you can subtract (or add). 11. The number 7 4 6 is not in proper mixed number form as the fraction part, 7 6 , should not be an improper fraction. Exercise Set 1.3 2. 1 of the 4 equal parts are shaded; 1 4 4. 2 of the 5 equal parts are shaded; 2 5 6 . 60 = 4  15 = 2  2  3  5 8 . 27 = 3  9 = 3  3  3 10 . 56 = 4  14 = 2  2  2  7 12 . 32 = 4  8 = 2  2  2  2  2 1 4 . 24 = 4  6 = 2  2  2  3 1 6 . 3 1 3 1 6 2 3 2     1 8 . 15 3 5 3 20 4 5 4     20 . 5 5 9 9 

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    ISM: Beginning Algebra Chapter 1: Review of Real Numbers 5 22 . 42 2 3 7 14 45 3 3 5 15       24. 360 2 2 2 3 3 5 2 3 3 18 700 2 2 5 5 7 5 7 35                2 6 .      7 3 7 3 21 11 5 11 5 55 2 8 . 7 3 7 3 7 3 1 8 21 8 21 8 3 7 8          30 . 7 1 7 2 7 2 7 2 7 12 2 12 1 12 1 2 6 1 6            32 . 3 9 3 10 3 10 3 2 5 2 5 10 5 9 5 9 5 3 3 3             34 . 3 10 3 10 3 2 5 2 2 35 63 35 63 5 7 3 3 7 3 7 7 147                3 6 . 1 19 1 19 1 19 19 4 6 4 6 2 2 2 3 24           3 8 . 1 1 5 1 5 5 Area sq m 2 2 4 2 2 4 2 16         bh 40. 2 3 2 3 6 Area square meters 5 11 5 11 55         l w 42 . 6 1 6 1 5 7 7 7 7     44 . 6 1 6 1 7 1 7 7 7 7      46 . 18 11 18 11 7 7 1 35 35 35 35 5 7 5        4 8 . 13 35 13 35 132 132 132 48 132 2 2 2 2 3 2 2 3 11 2 2 11 4 11                50 . 2 2 3 6 3 3 3 9     52 . 8 8 8 64 7 7 8 56     54 . 4 4 5 20 5 5 5 25     56 . 3 1 3 3 1 2 9 2 11 4 6 4 3 6 2 12 12 12           58 . 11 1 44 3 44 3 41 12 16 48 48 48 48       60 . 7 8 7 3 8 2 10 15 10 3 15 2 21 16 30 30 21 16 30 5 30 1 6              62 . 3 16 3 16 3 13 2 8 8 8 8 8       64 . 3 2 11 3 2 11 3 2 6 1 11 11 11 11 11 11 11          The unknown part is 6 . 11 66 . 1 3 10 1 3 2 10 1 6 3 1 10 5 10 10 5 2 10 10            The unknown part is 3 . 10 6 8 . 1 1 5 12 1 2 1 4 5 1 6 3 12 12 6 2 3 4 12 12 2 4 5 12 1 12                 The unknown part is 1 . 12

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 6 70. 3 7 11 15 165 5 2 1 5 4 8 4 8 32 32      72. 7 8 15 35 15 9 3 5 9 27 1 3 8 9 8 9 8 35 8 5 7 56            74. 11 7 11 14 26 40 26 40 20 10 20 20 25 66 20 5 67 20 1 67 4       76. 7 14 4 4 8 16 3 3 2 2 16 16 11 2 16   78 . 11 3 11 3 14 2 7 2 35 35 35 35 5 7 5         80 . 11 3 11 5 3 55 3 52 7 35 7 5 35 35 35 35         82 . 3 7 3 7 3 7 7 7 4 12 4 12 2 2 2 2 3 2 2 2 2 16                84. 3 7 3 12 3 12 3 3 4 9 4 12 4 7 4 7 4 7 7            86 . 1 70 1 70 1 71 7 10 10 10 10 10       88. 5 1 59 1 59 6 59 6 59 9 ÷ ÷ 59 6 6 6 6 6 1 6 1 1         90. 7 7 7 5 7 3 35 21 35 21 14 15 25 15 5 23 3 75 75 75 75             92 . 57 13 57 13 44 4 11 1 132 132 132 132 3 4 11 3          94 . 3 7 13 47 2 4 5 10 5 10 13 2 47 5 2 10 26 47 10 10 26 47 10 73 10 3 7 10              96 . 8 1 1 8 4 1 11 1 22 11 4 2 11 4 4 11 2 22 32 11 22 44 44 44 32 11 22 44 43 44                   98 . 1 3 2 33 2 99 2 16 2 12 2 8 1 2 1 8 33 99 1 4 33 4 99 1 4 4 132 99 4 4 132 99 4 231 4 3 57 feet 4                              100 . a nswers may vary

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    ISM: Beginning Algebra Chapter 1: Review of Real Numbers 7 102 . 1 1 1 3 7 11 17 7 3 5 2 1 2 2 8 4 2 2 8 4 7 4 11 4 17 7 2 2 4 2 4 8 4 2 28 44 17 14 8 8 8 8 28 44 17 14 8 103 8 7 12 8                            The distance for all 4 trails is 7 12 mi. 8 104. The sector for digital but not 3D is labeled 11 20 . 106. 3 3 3 15 18 9 40 8 40 40 40 20      9 20 of U.S. movie screens are analog or digital 3D. 108. The work is correct. 110. The work is incorrect. 16 2 2 2 2 4 28 2 2 7 7        Section 1.4 Practice Exercises 1. a. 3 1 1 1 1 1     b. 2 5 5 5 25    c. 2 1 1 1 1 10 10 10 100                  d. 1 9 9  e. 3 2 2 2 2 8 5 5 5 5 125                     2. a. 6 + 3  9 = 6 + 27 = 33 b. 3 4 8 3 64 8 3 8 3 11         c. 2 2 4 32 5 8 8 or 3 3 9 9 9            d. 9(14 6) 9(8) 72 36 2 2 2      e. 7 1 1 7 4 3 4 4 4 16 16 16      3. 2 6 5 36 5 31 31 3 6 5 8 3 1 8 3 8 11            4. 4[25 3(5 3)] 4[25 3(8)] 4[25 24] 4[1] 4         5. 2 36 9 5 4 5 9 25 3 22 5 3        6. a. 2 x + y = 2(2) + 5 = 4 + 5 = 9 b. 4 4(2) 8 3 3(5) 15 x y   c. 3 3 2 15 4 19 2 5 10 10 10 x x y       d. 3 2 3 2 2 5 8 25 33 x y       7. 9 6 7 9(4) 6 7(4) 36 6 28 30 28 False x x      0 0 4 is not a solution of 9 x  6 = 7 x . 8. a. Six times a number is 6 x , since 6 x denotes the product of 6 and x . b. A number decreased by 8 is x  8 because “decreased by” means subtract. c. The product of a number and 9 is x  9 or 9 x . d. Two times a number is 2 x , plus 3 is 2 x + 3. e. The sum of 7 and a number x is 7 + x .

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 8 9. a. A number x increased by 7 is x + 7, so x + 7 = 13. b. Two less than a number x is x  2, so x  2 = 11. c. Double a number x is 2 x , added to 9 is 2 x + 9, so 2 x + 9  25. d. Five times 11 is 5(11), so 5(11)  x , where x is an unknown number. Calculator Explorations 1. 4 5 625  2. 4 7 2401  3. 5 9 59, 049  4. 6 8 262,144  5. 2(20  5) = 30 6. 3(14  7) + 21 = 3(7) + 21 = 21 + 21 = 42 7. 24(862  455) + 89 = 9857 8. 99 + (401 + 962) = 1462 9. 4623 129 2376 36 34    10. 956 452 168 89 86    Vocabulary, Readiness & Video Check 1.4 1. In the expression 2 5 , the 5 is called the base and the 2 is called the exponent . 2. The symbols ( ), [ ], and { } are examples of grouping symbols. 3. A symbol that is used to represent a number is called a variable . 4. A collection of numbers, variables, operation symbols, and grouping symbols is called an expression . 5. A mathematical statement that two expressions are equal is called an equation . 6. A value for the variable that makes an equation a true statement is called a solution . 7. Deciding what values of a variable make an equation a true statement is called solving the equation. 8. The order in which we perform operations does matter! We came up with an order of operations to avoid getting more than one answer when evaluating an expression. 9. The replacement value for z is not used because it’s not needed  there is no variable z in the given algebraic expression. 10. No; the variable was replaced with 0 in the equation to see if a true statement occurred, and it did not. 11. We translate phrases to mathematical expressions and sentences to mathematical equations. Exercise Set 1.4 2. 5 2 2 2 2 2 2 32       4. 4 4 4 4 4 4 256      6. 8 1 1 1 1 1 1 1 1 1 1          8. 1 8 8  1 0 . 2 9 9 9 81    12. 2 6 6 6 6 6 36 11 11 11 11 11 121                     14. 5 1 1 1 1 1 1 2 2 2 2 2 2 1 1 1 1 1 2 2 2 2 2 1 32                                    16. 2 1 5 1 5 1 5 2 25   ( . ) ( . )( . ) . 18. 3 0 03 0 03 0 03 0 03 0 000027 ( . ) ( . )( . )( . ) .   20. 8 + 5  3 = 8 + 15 = 23

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    ISM: Beginning Algebra Chapter 1: Review of Real Numbers 9 22. 12  5  3  6 = 60  18 = 42 24. 5(6  2) = 5(4) = 20 26. 5 6 2 2 2 6 2 2 32 6 4 32 34            28. 2 2 5 2 25 50     30. 3 1 2 3 2 9 16 25 4 2 3 8 3 24 24 24        3 2 . 3[4 3(6 4)] 3[4 3(2)] 3[4 6] 3[10] 30         3 4 . 4 3 2 12 2 14 7 4 3 2 4 6 10 5          3 6 . 15 3 1 15 2 15 2 13 12 3 2 12 6 6 6           38 . 2 2 3 6(8 5) 3 6(3) 4 2 4 2 3 6(3) 16 2 3 18 16 2 21 18 7 6               4 0 . 2 2 2 16 13 5 4 16 8 4 17 5 17 5 16 8 4 17 5 16 8 16 17 5 40 12 4 10 4 3 10 3                      42. 3 4[8(5 5 20) 39] 3 4[8(25 20) 39] 3 4[8(5) 39] 3 4[40 39] 3 4[1] 3 4 7                    44. 2 3 1 1 3 9 1 1 3 8 4 8 2 64 4 8 2 9 1 3 64 4 16 9 16 12 64 64 64 37 64                     46. a. (1 + 4)  6  3 = 5  6  3 = 30  3 = 27 b. 1 + 4  (6  3) = 1 + 4  3 = 1 + 12 = 13 c. 1 + 4  6  3 = 1 + 24  3 = 22 d. (1 + 4)  (6  3) = 5  3 = 15 48. Let x = 1. 4 x = 4(1) = 4 50. Let y = 3 and z = 5. 3 3 2 2(5) 10 y z   52. Let y = 3. 6 y  8 = 6(3)  8 = 18  8 = 10 54. Let y = 3 and z = 5. 5 2 5(5) 2(3) 25 6 19 19 z y        56. Let x = 1, y = 3, and z = 5. 3 5 1 15 1 14 yz x        5 8 . Let z = 5. 2 2 2 2(5) 2(25) 50 z    60 . Let x = 12, y = 8 , and z = 4. 8 8 8(12) 2 96 98 4 y x z       6 2 . Let x = 12 and y = 8. 2 2 3 (8) 3(12) 8 64 36 8 36 y x y         

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 10 6 4 . Let x = 12 and y = 8. 2 2 2 2 (8) 12 64 12 76 19 144 24 168 42 3 (12) 3(8) y x x y           6 6 . No; answers may vary. 6 8 . Let x = 6. 2 7 3 2(6) 7 3(6) 12 7 18 19 18, false x x      0 0 6 is not a solution of the equation. 70 . Let x = 2. 4 2 8 4(2) 2 2 8 8 2 10 10 10, true x x        0 0 2 is a solution of the equation. 7 2 . Let x = 6. 3 10 8 3(6) 10 8 18 10 8 8 8, true x      0 0 6 is a solution of the equation. 7 4 . Let x = 10. 6 6 10 6 10 6 16 16, true x x       0 1 0 is a solution of the equation. 7 6 . Let x = 1. 4 1 4 1 1 4 0, false x     0 1 is not a solution of the equation. 78. A number increased by 9 is written as x + 9. 80. Five decreased by a number is written as 5  x . 82. The quotient of a number and 9 is written as . 9 x 84. Twice a number, decreased by 72 is written as 2 x  72. 86. Four subtracted from eight is equal to two squared is written as 2 8 4 2 .   88. The difference of sixteen and four is greater than ten is written as 16  4 > 10. 90. Seven subtracted from a number is 0 is written as x  7 = 0.

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    ISM: Beginning Algebra Chapter 1: Review of Real Numbers 11 92. 9.1 times a number equals 4 is written as 9.1 x = 4. 94. Eight added to twice a number is 42 is written as 2 x + 8 = 42. 96. To simplify the expression (1 + 3)  6, first add . 98. To simplify the expression 20  4  2, first divide . 100. Yes; answers may vary . Length, l Width, w Perimeter of Rectangle: 2 l + 2 w Area of Rectangle: lw 102. 6 in. 1 in. 2 l + 2 w = 2(6 in.) + 2(1 in.) = 12 in. + 2 in. = 14 in. lw = (6 in.)(1 in.) = 6 sq in. 104. 4.6 in. 2.4 in. 2 l + 2 w = 2(4.6 in.) + 2(2.4 in.) = 9.2 in. + 4.8 in. = 14 in. lw = (4.6 in.)(2.4 in.) = 11.04 sq in. 106. answers may vary 108. 2 2 (5 3 ) 2 (5 9) 2 14 28         110. a. 2 3 26 x  is an expression since it does not contain an equal sign. b. 2 3 26 1 x   is a n equation since it contains an equal sign . c. 2 x  5 = 7 x  5 is an equation since it contains an equal sign . d. 9 y + x  8 is an expression since it does not contain an equal sign . e. 2 3 4(5 3)   is an expression since it does not contain an equal sign . 112. answers may vary 114. answers may vary, for example, 2(10  7) + 1: 2(10  7) + 1 = 2(3) + 1 = 6 + 1 = 7 116 . Let h = 5, B = 15, and b = 7. 1 1 1 ( ) (5)(15 7) (5)(22) 55 2 2 2 A h B b       The area is 55 sq in. 118. Let P = 650, T = 3, and I = 126.75. 126.75 126.75 0.065 6.5% (650)(3) 1950 I PT    

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 12 Section 1.5 Practice Exercises 1. 2 + 4 = 6 2.  2 + (  3) =  5 3. a.  5 + (  8) Add the absolute values. 5 + 8 = 13 The common sign is negative, so  5 + (  8) =  13. b.  31 + (  1) Add the absolute values. 31 + 1 = 32 The common sign is negative, so  31 + (  1) =  32. 4.  3 + 8 = 5 5. a. 15 + (  18) Subtract the absolute values. 18  15 = 3 Use the sign of the number with the largest absolute value. 15 + (  18) =  3 b.  19 + 20 = 20  19 = 1 c.  0.6 + 0.4 =  (0.6  0.4) =  (0.2) =  0.2 6. a. 3 2 5 1 5 5 5              b. 3 + (  9) =  6 c. 2.2 + (  1.7) = 0.5 d. 2 3 20 21 1 7 10 70 70 70       7. a. 8 + (  5) + (  9) = 3 + (  9) =  6 b. [ 8 5] 5 2 [ 3] [ 5 2] 3 [ 3] 6                      8.  7 + 4 + 7 =  3 + 7 = 4 The temperature at 8 A . M . was 4  F. 9. a. The opposite of 5 9  is 5 . 9 b. The opposite of 8 is  8. c. The opposite of 6.2 is  6.2. d. The opposite of  3 is 3. 10. a. Since |  15| = 15,  |  15| =  15. b. 3 3 5 5          c.  (  5 y ) = 5 y d.  (  8) = 8 Vocabulary, Readiness & Video Check 1.5 1. Two numbers that are the same distance from 0 but lie on opposite sides of 0 are called opposites . 2. If n is a number, then n + (  n ) = 0 . 3. If n is a number, then  (  n ) = n . 4. The sum of two negative numbers is always a negative number . 5. To add two numbers with the same sign, add their absolute values and use their common sign as the sign of the sum. 6. Negative; when you add two numbers with different signs, the sign of the sum is the same as the sign of the number with the larger absolute value and  8.4 has a larger absolute value than 6.3. 7. Negative temperatures; the high temperature for the day was  6  F. 8. Example 13 is an example of the opposite of the absolute value of  a , not the opposite of  a . The absolute value of  a is positive, so its opposite is negative, therefore the answers to Examples 12 and 13 have different signs.

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    ISM: Beginning Algebra Chapter 1: Review of Real Numbers 13 Exercise Set 1.5 2. 9 + (  12) =  3 4.  6 + (  14) =  20 6. 6 + (  4) = 2 8.  10 + 5 =  5 10.  7 + (  4) =  11 12. 7 + (  5) = 2 14.  5 + 9 = 4 16. 8 + (  6) = 2 18. 3 + (  6) =  3 20. 23 + (  23) = 0 22. 53 + (  37) = 16 24.  26 + 14 =  12 26.  18 + (  26) =  44 28. 9.2 + (  11.4) =  2.2 30. |  6| + (  61) = 6 + (  61) =  55 32. 144 + (  88) = 56 34.  6.7 + (  7.6) =  14.3 36. 5 7 2 1 12 12 12 6     38. 5 1 5 1 3 5 3 2 9 3 9 3 3 9 9 9             40. 5 2 5 2 2 6 3 6 3 2 5 4 6 6 9 6 3 2                                    42.  9 + 15 + (  5) = 6 + (  5) = 1 44.  18 + (  6) + (  40) =  24 + (  40) =  64 46.  14 + (  3) + 11 =  17 + 11 =  6 48. |7 + (  17)| = |  10| = 10 50. 8 + (  2) + 7 = 6 + 7 = 13 52. [  2 + (  7)] + [  11 + 22] = [  9] + [11] = 2 54. |43 + (  73)| + |  20| = |  30| + 20 = 30 + 20 = 50 56. 3.7 [0.1 ( 0.6) 8.1] 3.7 [ 0.5 8.1] 3.7 [7.6] 3.9               58.  4 + 49 = 45 It rose to a temperature of 45  F. 60.  52 + 439 = 387 Your elevation is 387 feet. 62. 3 0 3 2 26 4 63 9 6 2 26 4 63 9 20 2 63 9 84 1 . ( . ) . . . . . . . .             The total net income for fiscal year 2013 was $84.1 million. 64.  6 + (  4) + (  5) + (  6) =  21 His score was  21 or 21 under par. 66. The opposite of 4 is  4. 68. The opposite of  8 is 8. 70. The opposite of 1 4  is 1 . 4 72. Since |  11| is 11, the opposite of |  11| is  11. 7 4 .  (  3) = 3 7 6 . 2 2 3 3   78 .  (  7) = 7 8 0 . Let x = 10. 7 3 7 (10) 3 7 7, false x        0 10 is not a solution of the equation.

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 14 8 2 . Let y =  6. 1 7 1 6 7 1 1, true y      0  6 is a solution of the equation. 8 4 . Look for the bar that extends farthest in the negative direction . The month for which the temperature is the lowest is February. 86 . Look for the bar whose length has a negative value closest to 0; November 88 . [( 19.3) 27.0 8.8] 3 [7.7 8.8] 3 [16.5] 3 5.5           The average was 5.5  F. 90.  4 + 14 = 10 92.  15 + (  17) =  32 94. Since b is a negative number,  b is a positive number. 96. Since b is a negative number, b + b is a negative number. 98. True 100. True 102. a nswers may vary 104. a nswers may vary Section 1.6 Practice Exercises 1. a.  7  6 =  7 + (  6) =  13 b.  8  (  1) =  8 + 1 =  7 c. 9  (  3) = 9 + 3 = 12 d. 5  7 = 5 + (  7) =  2 2. a. 8.4  (  2.5) = 8.4 + 2.5 = 10.9 b. 5 1 5 1 4 1 8 8 8 8 8 2                 c. 3 1 3 1 4 5 4 5 15 4 20 20 19 20                         3.  2  5 =  2 + (  5) =  7 4. a. 15 2 ( 4) 7 15 ( 2) 4 7 6              b. 3.5 ( 4.1) ( 6.7) 3.5 ( 4.1) 6.7 6.1          5. a. 4 [( 8 3) 5] 4 [( 8 ( 3)) 5] 4 [( 11) 5] 4 [ 11 ( 5)] 4 [ 16] 20                              b. 2 13 3 [2 ( 7)] 13 9 [2 7] 13 9 9 13              6. a. 7 7 ( 3) 7 3 10 2 2 2(4) ( 3) 8 ( 3) 5 x y x              b. 2 2 (4) ( 3) 16 ( 3) 13 y x         7. 282  (  75) = 282 + 75 = 357 The overall change was $357. 8. a. x = 90   62  = 28  b. y = 180   43  = 137  Vocabulary, Readiness & Video Check 1.6 1. 7 minus a number 7  x 2. 7 subtracted from a number x  7 . 3. A number decreased by 7 x  7 4. 7 less a number 7  x 5. A number less than 7 7  x 6. A number subtracted from 7 7  x

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    ISM: Beginning Algebra Chapter 1: Review of Real Numbers 15 7. To evaluate x  y for x =  10 and y =  14, we replace x with  10 and y with  14 and evaluate  10  (  14) . d 8. The expression  5  10 equals  5 + (  10) . c 9. To subtract two real numbers, change the operation to addition and take the opposite of the second number. 10.  10 + (8) + (  4) + (  20); it’s rewritten to change the subtraction operations to addition and turn the problem into an addition of real numbers problem. 11. There’s a minus sign in the numerator and the replacement value is negative (notice parentheses are used around the replacement value), and it’s always good to be careful when working with negative signs. 12. This means that the overall vertical altitude change of the jet is actually a decrease in altitude from when the Example started. 13. In Example 9, you have two supplementary angles and know the measure of one of them. From the definition, you know that the two supplementary angles must sum to 180  . Therefore you can subtract the known angle measure from 180  to get the measure of the other angle. Exercise Set 1.6 2.  12  8 =  12 + (  8) =  20 4. 8  11 = 8 + (  11) =  3 6. 12  (  5) = 12 + 5 = 17 8. 3 7 3 7 4 8 4 8 3 2 7 4 2 8 6 7 8 8 1 8                                 10.  20  (  48) =  20 + 48 = 28 12.  8  4 =  8 + (  4) =  12 14. 3  (  6) = 3 + 6 = 9 16.  4  (  16) =  4 + 16 = 12 18. 15  (  33) = 15 + 33 = 48 20. 8.3  11.2 = 8.3 + (  11.2) =  2.9 22.  36  51 =  36 + (  51) =  87 24.  17  (  17) =  17 + 17 = 0 26.  6.1  (  5.3) =  6.1 + 5.3 =  0.8 28. 4 1 4 1 3 7 7 7 7 7               30. 1 7 1 7 10 8 10 8 1 4 7 5 10 4 8 5 4 35 40 40 39 40                                       32. 4.3  (  0.87) = 4.3 + 0.87 = 5.17 34.  2  3 =  2 + (  3) =  5 36. 1  17 = 1 + (  17) =  16 38.  4  9 =  4 + (  9) =  13 40. 11  (  14) = 11 + 14 = 25 4 2 . 16 ( 3) ( 11) 14 16 3 ( 11) ( 14) 13 ( 11) ( 14) 24 ( 14) 38                          4 4 . 7 12 ( 5) 2 ( 2) 7 ( 12) ( 5) ( 2) ( 2) 5 ( 5) ( 2) ( 2) 10 ( 2) ( 2) 12 ( 2) 14                                    4 6 .  9  (3  8) =  9  (  5) =  9 + 5 =  4 48 . 3 2 6 3 8 6 3 8 18 8 ( 18) 10             5 0 . 4 6(7 3) 4 6(4) 4 24 4 ( 24) 20           

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 16 5 2 . 2 2 2 (2 3) 5 [2 ( 3)] 5 [ 1] 5 [ 1] 25 24              5 4 . 5 [(4 15) ( 6) 8] 5 [(4 ( 15)) 6 ( 8)] 5 [( 11) 6 ( 8)] 5 [( 5) ( 8)] 5 [ 13] 18                                  5 6 . 2 2 2 2 6 ( 3 8) 2 6 [ 3 ( 8)] 2 6 [ 11] 2 36 [ 11] 38 [ 11] 27                        58 . Let x =  5 and y = 4. y  x = 4  (  5) = 4 + 5 = 9 6 0 . Let x =  5, y = 4, and t = 10. 7 5 10 7(4) 5 10 28 5 10 ( 28) 5 ( 28) 23 23 x t y                      6 2 . Let x =  5 an y = 4. 15 15 ( 5) 15 5 20 2 10 10 2 4 2 4 2 6 2 3 3 x y               6 4 . Let x =  5 and t = 10. 2 2 10 ( 5) 100 5 105 t x        6 6 . Let x =  5, y = 4 and t = 10. 5 5(4) ( 5) 6 6(10) 20 5 6(10) 25 6(10) 25 60 5 5 12 5 5 12 y x t             68 . 5 + (  12) =  7 The new elevation is  7 feet or 7 feet below sea level. 70. 134  (  80) = 134 + 80 = 214 134  F is 214  F warmer than  80  F. 72. 93 18 26 93 ( 18) ( 26) 75 ( 26) 49            She owes $49 on her account. 74. 15  24 = 15 + (  24) =  9 Her new score is  9. 7 6 . The difference in elevation is the higher elevation  the lower elevation. 4101  (  226) = 4101 + 226 = 4327 The Bishop airport is 4327 feet higher . 78 . x = 90  50 = 90 + (  50) = 40 The complementary angle is 40  . 8 0 . y = 180  105 = 180 + (  105) = 75 The supplementary angle is 75  . 8 2 . Let x = 3. 10 7 3 10 7 7 7, true x         0 3 is a solution of the equation. 8 4 . Let x =  10. 6 1 ( 10) 6 ( 10) 1 10 ( 6) 10 ( 1) 4 9, false x x                 0 0  10 is not a solution of the equation.

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    ISM: Beginning Algebra Chapter 1: Review of Real Numbers 17 8 6 . Let x = 5. 4 1 4 1 5 4 1 ( 5) 4 4, false x        0 0 5 is not a solution of the equation. 88. The difference of  3 and a number is  3  x . 90. Add a number and  36 is x + (  36). 92. Month Monthly Increase or Decrease July 33.6  29.7 = 33.6 + (  29.7) = 3.9  August 33.3  33.6 = 33.3 + (  33.6) =  0.3  September 27.0  33.3 = 27.0 + (  33.3) =  6.3  October 8.8  27.0 = 8.8 + (  27.0) =  18.2  November  6.9  8.8 =  6.9 + (  8.8) =  15.7  December  17.2  (  6.9) =  17.2 + 6.9 =  10.3  94. The largest negative number corresponds to October. 96. answers may vary 98.  4  8 =  4 + (  8) =  12 100.  3  (  10) =  3 + 10 = 7 102. true; answers may vary 104. true; answers may vary 106. Since 4.362 is less than 7.0086, the answer is negative. 4.362  7.0086 =  2.6466 Integrated Review 1. The opposite of a positive number is a negative number. 2. The sum of two negative numbers is a negative number. 3. The absolute value of a negative number is a positive number. 4. The absolute value of zero is 0 . 5. The reciprocal of a positive number is a positive number. 6. The sum of a number and its opposite is 0 . 7. The absolute value of a positive number is a positive number. 8. The opposite of a negative number is a positive number.

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 18 Number Opposite Absolute Value 9. 1 7 1 7  1 7 10. 12 5  12 5 12 5 11. 3  3 3 12. 9 11  9 11 9 11 13.  19 + (  23) =  42 14. 7  (  3) = 7 + 3 = 10 15.  15 + 17 = 2 16.  8  10 =  8 + (  10) =  18 17. 18 + (  25) =  7 18.  2 + (  37) =  39 19.  14  (  12) =  14 + 12 =  2 20. 5  14 = 5 + (  14) =  9 21. 4.5  7.9 = 4.5 + (  7.9) =  3.4 22.  8.6  1.2 =  8.6 + (  1.2) =  9.8 23. 3 1 21 4 21 4 25 4 7 28 28 28 28 28                  24. 2 7 16 21 16 21 5 3 8 24 24 24 24 24               25. 9 ( 7) 4 6 9 7 4 6 9 7 4 ( 6) 4                   26. 11 20 ( 3) 12 11 ( 20) ( 3) ( 12) 9 ( 3) ( 12) 12 ( 12) 24                        27. 24 6(14 11) 24 6[14 ( 11)] 24 6(3) 24 18 24 ( 18) 6               28. 30 5(10 8) 30 5[10 ( 8)] 30 5(2) 30 10 30 ( 10) 20               29. 2 2 (7 17) 4 [7 ( 17)] 4 ( 10) 16 6           30. 2 2 9 (10 30) 9 [10 ( 30)] 81 ( 20) 61           31. 2 9 3 ( 4 20) 9 9 [ 4 ( 20)] 9 9 ( 24) 18 ( 24) 6                     32. 2 2 4 5 5 ( 50) 4 ( 5) 5 ( 50) 9 25 ( 50) 9 25 ( 50) 34 ( 50) 16                           33. 7 [(1 2) ( 2 9)] 7 [( 1) ( 11)] 7 [ 12] 19                   34. 6 [( 3 7) (4 15)] 6 [(4) ( 11)] 6 ( 7) 13                  35. 1  5 = 1 + (  5) =  4 36.  3  (  2) =  3 + 2 =  1 37. 1 2 1 2 5 8 13 4 5 4 5 20 20 20              38. 5 1 25 4 25 4 29 8 10 40 40 40 40 40                        39. 3 2 3 2 3 2 2(19 17) 3( 7 9) 2[19 ( 17)] 3( 7 9) 2(2) 3(2) 2(8) 3(4) 16 12 16 ( 12) 4                    

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    ISM: Beginning Algebra Chapter 1: Review of Real Numbers 19 40. 2 3 2 3 2 3 3(10 9) 6(20 19) 3[10 ( 9)] 6[20 ( 19)] 3(1) 6(1) 3 6 9               41. x  y =  2  (  1) =  2 + 1 =  1 42. x + y =  2 + (  1) =  3 43. y + z =  1 + 9 = 8 44. z  y = 9  (  1) = 9 + 1 = 10 45. 5 5(9) ( 2) 45 2 47 47 1 ( 2) 1 2 1 z x y x               46. ( 2) ( 1) 9 2 2(9) 2 1 9 18 12 18 12 18 2 3 x y z z                Section 1.7 Practice Exercises 1. a. 8(  5) =  40 b. (  3)(  4) = 12 c. (  6)(9) =  54 2. a. (  1)(  5)(  6) = 5(  6) =  30 b. (  3)(  2)(4) = 6(4) = 24 c. (  4)(0)(5) = 0(5) = 0 d. ( 2)( 3) ( 4)(5) 6 ( 20) 6 20 26           3. a. (0.23)(  0.2) =  [(0.23)(0.2)] =  0.046 b. 3 4 3 4 12 4 5 9 5 9 45 15                       c. 7 7 24 ( 24) 7 2 14 12 12 1               4. a. 2 ( 6) ( 6)( 6) 36      b. 2 6 (6 6) (36) 36         c. 3 ( 4) ( 4)( 4)( 4) 16( 4) 64          d. 3 4 (4 4 4) [16(4)] 64          5. a. The reciprocal of 8 3 is 3 8 since 8 3 1. 3 8   b. The reciprocal of 15 is 1 15 since 1 15 1. 15   c. The reciprocal of 2 7  is 7 2  since 2 7 1. 7 2             d. The reciprocal of  5 is 1 5  since 1 ( 5) 1. 5          6. a. 16 1 16 8 2 2            b. 1 24 ( 6) 24 4 6             c. 35 35 5 7 5 7 7 7       7. a. 18 18 3 6 3 6 6 6       b. 48 48 3 16 16 3 3 3         c. 3 1 3 6 ( 2) 5 2 5 5              d. 4 4 1 4 1 1 8 9 9 8 9 4 2 9 2 18              

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 20 8. a. 0 0 2   b. 4 0  is undefined. c. 5 5 6(0) 0    is undefined. 9. a. ( 8)( 11) 4 88 4 84 84 9 ( 4) 9 4 5 5               b. 3 3( 2) 9 3( 8) 9 6 3 3 24 9 3 33 3 11                 10. a. 7 y  x = 7(  2)  (  5) =  14 + 5 =  9 b. 2 3 2 3 ( 5) ( 2) 25 ( 8) 25 8 33 x y            c. 2 2( 5) 10 5 3 3( 2) 6 3 x y        11. total score = 4  (  13) =  52 The card player’s total score was  52. Calculator Explorations 1.  38(26  27) = 38 2.  59(  8) + 1726 = 2198 3. 134 + 25(68  91) =  441 4. 45(32)  8(218) =  304 5. 50(294) 163.3 175 265    6. 444 444.8 1.76 181 324      7. 5 9 4550 54, 499   8. 8 5 6259 384,366   9. 2 ( 125) 15,625   10. 2 125 15, 625    Vocabulary , Readiness & Video Check 1.7 1. If n is a real number, then n  0 = 0 and 0  n = 0 . 2. If n is a real number, but not 0, then 0 0 n  and we say 0 n is undefined . 3. The product of two negative numbers is a positive number. 4. The quotient of two negative numbers is a positive number. 5. The quotient of a positive number and a negative number is a negative number. 6. The product of a positive number and a negative number is a negative number. 7. The reciprocal of a positive number is a positive number. 8. The opposite of a positive number is a negative number. 9. The parentheses, or lack of them, determine the base of the expression. In Example 6, 4 2  ( ) , the base is  2 and all of  2 is raised to the four th power . In Example 7, 4 2  , the base is 2 and only 2 is raised to the four th power . 10. Remember, the product of a number and its reciprocal is 1, not  1. 2 3 1 3 2   , as needed. 11. Yes; because division of real numbers is defined in terms of multiplication. 12. The replacement values are negative and both will be squared. Therefore they must be placed in parentheses so the entire value, including the negative, is squared.

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    ISM: Beginning Algebra Chapter 1: Review of Real Numbers 21 13. The football team lost 4 yards on each play and a loss of yardage is represented by a negative number . Exercise Set 1.7 2.  8(5) =  40 4. 7(  4) =  28 6.  6(  11) = 66 8.  2  8 =  16 10.  6  0 = 0 12. 3(  5) =  15 14. 1 1 1 1 1 8 3 8 3 24             16. 5 3 5 3 15 15 1 6 10 6 10 60 15 4 4                18. 6(  2.5) =  15 20.  0.5(  0.3) = 0.15 22.  20(60) =  1200 24. 5(  9) =  45 26. (  7)(  7) = 49 28. 2 2 2 2 4 7 11 7 11 77              30.  12(12) =  144 32. 25 6 25 6 150 5 36 15 36 15 480 18                34. ( 2)( 3)( 4)( 2) 6( 4)( 2) 24( 2) 48            36. 8( 3) 4( 5) 24 ( 20) 24 20 4             38. 20 ( 4)(3)( 2) 20 (24) 20 ( 24) 4           40. True; example: 2  2  2 = 8 42. True; example: 2  2  2  2 = 16 44. 4 2 (2 2 2 2) 16         46. 5 ( 1) ( 1)( 1)( 1)( 1)( 1) 1          48. 2 5 (5 5) 25       50. 2 ( 7) ( 7)( 7) 49      52. Reciprocal of 100 is 1 100 since 1 100 1. 100   54. Reciprocal of 1 7 is 7 since 1 7 1. 7   56. Reciprocal of  8 is 1 8  since 1 8 1. 8     58. Reciprocal of 6 13  is 13 6  since 6 13 1. 13 6     60. Reciprocal of 1.5 is 1 1.5 since 1 1.5 1. 1.5   62. Reciprocal of 1 8.9  is  8.9 since 1 8.9 1. 8.9     64. 20 1 20 2 10 10       66. 18 1 18 3 6 6        68. 60 1 60 12 5 5       70. 0 1 0 0 9 9      72. 24 1 24 3 8 8       74. 3 0 is undefined. 76. 45 1 45 5 9 9        78. 14 1 14 7 2 2      

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 22 80. 4 1 4 2 4 2 8 5 2 5 1 5 1 5                        82. 1 8 1 11 1 11 11 10 11 10 8 10 8 80                        84. 5 5 5 12 5 12 1 12 12 12 5 12 5            86. 6( 3) 18 9 4 4 2        88. 15 15 5 1 4 3       90. 2 3 4 9 4 13 5 5 5     92. 2 6 ( 2) 6 4 10 2 4 9 4 9 5           94. 20 ( 4)(3) 20 ( 12) 32 8 1 5 1 5 4              96. 2 2 4 2 16 2 ( 16) 18 1 3( 6) 3( 6) 18 18                 98. 8 3( 2) 8 ( 6) 8 6 14 7 2 5( 4) 2 ( 20) 2 20 22 11               100. 4 8( 2) 4 ( 16) 4 16 12 4 9 2( 3) 9 ( 6) 9 6 3                       102. 3 6 2 7 3 5 3 5 8 2 2 2 4 4 4               104. Let x =  5 and y =  3. 4 x + 5 y = 4(  5) + 5(  3) =  20 + (  15) =  35 106. Let x =  5 and y =  3. 2 2 2 2 2 ( 5) 2( 3) 25 2(9) 25 ( 18) 7 x y            108. Let x =  5 and y =  3. 3 3 3 ( 3) 3( 5) 27 ( 15) 42 y x            110. Let x =  5 and y =  3. 2 12 2( 3) 12 6 12 18 2 4 5 4 5 4 9 y x                 112. Let x =  5 and y =  3. 4 2 4 2( 5) 4 10 is undefined. 3 3 3 0 x y          114. The quotient of  8 and a number is 8 x  or  8  x . 116. The sum of a number and  12 is x + (  12). 118. The difference of a number and  10 is x  (  10). 120. Multiply a number by  17 is x  (  17) or  17 x . 122. A loss of $400 is represented by  400. 7  (  400) =  2800 His total loss was $2800. 124. A drop of 5 degrees is represented by  5. 6  (  5) =  30 The total drop in temperature was 30 degrees. 1 2 6. Let x =  4. 2 1 2( 4) 4 1 8 5, false x x         0  4 is not a solution of the equation. 1 2 8. Let x =  3. 45 15 45 15 3 15 15, true x        0  3 is a solution of the equation. 1 3 0. Let x =  4. 2 4 8 2( 4) 4 4 8 8 4 4 4 4, false x x            0 0  4 is not a solution of the equation. 132. 330 330 1    The surface temperature of Neptune is  330  F. 1 34 . answers may vary

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    ISM: Beginning Algebra Chapter 1: Review of Real Numbers 23 1 36 . a nswers may vary 1 38 . Since q is negative, r is negative, and t is positive, then 2 q r t   is negative. 1 40 . Not possible to determine whether t + r is positive or negative. 1 42 . Since q is negative, r is negative, and t is positive, then r ( q  t ) is positive. 1 44 . 1 + (  8)(  5) = 1 + 40 = 41 1 46 . 0 7 0 7 7 5      Section 1.8 Practice Exercises 1. a. x  8 = 8  x b. x + 17 = 17 + x 2. a. (2 + 9) + 7 = 2 + (9 + 7) b.  4  (2  7) = (  4  2)  7 3. a. (5 ) 9 ( 5) 9 (5 9) 14 x x x x           b. 5(  6 x ) = [5  (  6)] x =  30 x 4. a. 5( x  y ) = 5( x )  5( y ) = 5 x  5 y b.  6(4 + 2 t ) =  6(4) + (  6)(2 t ) =  24  12 t c. 2(3 4 ) 2(3 ) 2( 4 ) 2( ) 6 8 2 x y z x y z x y z           d. (3  y )  (  1) = 3(  1) + (  y )(  1) =  3 + y e. ( 7 2 ) ( 1)( 7 2 ) ( 1) ( 1)( 7) ( 1)(2 ) 7 2 x s x s x s x s                   f. 1 1 1 2 4 9 2 4 9 2 2 2 2 9 11 x x x x           ( ) ( ) ( ) 5. a. 5  w + 5  3 = 5( w + 3) b. 9 w + 9 z = 9  w + 9  z = 9( w + z ) 6. a. (7  3 x )  4 = (3 x  7)  4; commutative property of multiplication b. 6 + (3 + y ) = (6 + 3) + y ; associative property of addition c. 8 + ( t + 0) = 8 + t ; identity element for addition d. 3 4 1; 4 3           multiplicative inverse property e. (2 + x ) + 5 = 5 + (2 + x ); commutative property of addition f. 3 + (  3) = 0; additive inverse property g. (  3 b )  7 = (  3  7)  b ; commutative and associative properties of multiplication Vocabulary, Readiness & Video Check 1.8 1. x + 5 = 5 + x is a true statement by the commutative property of addition . 2. x  5 = 5  x is a true statement by the commutative property of multiplication . 3. 3( y + 6) = 3  y + 3  6 is true by the distributive property . 4. 2  ( x  y ) = (2  x )  y is a true statement by the associative property of multiplication . 5. x + (7 + y ) = ( x + 7) + y is a true statement by the associative property of addition . 6. The numbers 2 3  and 3 2  are called reciprocals or multiplicative inverses . 7. The numbers 2 3  and 2 3 are called opposites or additive inverses . 8. order; grouping 9. 2 is outside the parentheses, so the point is made that you should only distribute the  9 to the terms within the parentheses and not also to the 2.

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 24 10. The identity element for addition is 0 because if we add 0 to any real number, the result is that real number. The identity element for multiplication is 1 because any real number times 1 gives a result of that original real number. Exercise Set 1.8 2. 4 + y = y + 4 4.  2  x = x  (  2) 6. ab = ba 8. 19 + 3 y = 3 y + 19 10. 3  ( xy ) = (3 x )  y 12. ( y + 4) + z = y + (4 + z ) 14. (  3 y )  z =  3  ( yz ) 16. 6 + ( r + s ) = (6 + r ) + s 18. ( r + 3) + 11 = r + (3 + 11) = r + 14 20. 2(42 x ) = (2  42) x = 84 x 22. 1 1 (8 ) 8 1 8 8 z z z z            24. 7 + ( x + 4) = 7 + (4 + x ) = (7 + 4) + x = 11 + x 26.  3(12 y ) = (  3  12) y =  36 y 28. 2 7 2 7 1 7 2 7 2 r r r r                 30. 7 2 7 2 9 9 9 9 9 9 1 y y y y                32. 7( a + b ) = 7 a + 7 b 34. 11( y  4) = 11 y  11  4 = 11 y  44 36. 5(7 + 8 y ) = 5(7) + 5(8 y ) = 35 + 40 y 38. 3(8 x  1) = 3(8 x )  3(1) = 24 x  3 40. 2( x + 5) = 2( x ) + 2(5) = 2 x + 10 42.  3( z  y ) =  3 z  (  3) y =  3 z + 3 y 44.  5(2 r + 11) =  5(2 r ) + (  5)(11) =  10 r  55 46. 8(3 y + z  6) = 8(3 y ) + 8 z  8(6) = 24 y + 8 z  48 48. 4(4 2 5 ) 4(4) ( 4)(2 ) ( 4)(5 ) 16 8 20              p q p q p q 50. (9 5) 1(9 5) 1(9 ) ( 1)(5) 9 5 r r r r             52. ( 2 6 ) 1( 2 6 ) 1 ( 1)(2) ( 1)(6 ) 2 6 q r q r q r q r                  54. 1 1 1 (4 2) (4 ) (2) 4 4 4 1 1 4 2 4 4 1 2 x x x x                      56. 1 1 1 (10 25 ) (10 ) (25 ) 5 5 5 1 1 10 25 5 5 2 5 2 5 a b a b a b a b a b                                       58. 10(4 6) 40 10(4 ) 10(6) 40 40 60 ( 40) 40 20 s s s s            60. 11(5 3) 10 11(5 ) ( 11)(3) 10 55 33 10 55 23 x x x x                62. 6(2 1) 1 6(2 ) ( 6)(1) 1 12 ( 6) 1 12 7 x x x x                 64. 14  z + 14  5 = 14( z + 5) 66. 9 a + 9 b = 9( a + b ) 68. (  3) a + (  3) b =  3( a + b ) 70. 25 x + 25 y = 25( x + y ) 72. 4(3 + 8) = 4  3 + 4  8; distributive property

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    ISM: Beginning Algebra Chapter 1: Review of Real Numbers 25 74. ( x + 9) + 3 = (9 + x ) + 3; commutative property of addition 76. 1  9 = 9; identity element for multiplication 78. 1 6 1; 6   multiplicative inverse property 80. ( a + 9) + 6 = a + (9 + 6); associative property of addition 82. (11 + r ) + 8 = ( r + 11) + 8; commutative property of addition 84. r + 0 = r ; identity element for addition Expression Opposite Reciprocal 86. 2 3  2 3 3 2  88. 4 y  4 y 1 4 y 90.  7 x 7 x 1 7 x  92. False; the reciprocal of 2 a  is 2 . a  The opposite of 2 a  is . 2 a 94. “Putting on your shoes” and “putting on your socks” are not commutative, since the order in which they are performed affects the outcome. 96. “Reading the sports section” and “reading the comics section” are commutative, since the order in which they are performed does not affect the outcome. 98. “Baking a cake” and “eating the cake” are not commutative, since the order in which they are performed affects the outcome. 100. “Feeding the dog” and “feeding the cat” are commutative, since the order in which they are performed does not affect the outcome. 102. a. The property illustrated is the associative property of addition since the grouping of addition changed. b. The property illustrated is the commutative property of addition since the order in which they are added changed. c. The property illustrated is the commutative property of addition since the order in which they are added changed. 104. answers may vary 106. a nswers may vary Chapter 1 Vocabulary Check 1. The symbols  , <, and > are called inequality symbols . 2. A mathematical statement that two expressions are equal is called an equation . 3. The absolute value of a number is the distance between that number and 0 on a number line. 4. A symbol used to represent a number is called a variable . 5. Two numbers that are the same distance from 0 but lie on opposite sides of 0 are called opposites . 6. The number in a fraction above the fraction bar is called the numerator . 7. A solution of an equation is a value for the variable that makes the equation a true statement. 8. Two numbers whose product is 1 are called reciprocals . 9. In 3 2 , the 2 is called the base and the 3 is called the exponent . 10. The number in a fraction below the fraction bar is called the denominator . 11. Parentheses and brackets are examples of grouping symbols . 12. A set is a collection of objects. Chapter 1 Review 1. 8 < 10 since 8 is to the left of 10 on a number line. 2. 7 > 2 since 7 is to the right of 2 on a number line. 3.  4 >  5 since  4 is to the right of  5 on a number line.

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 26 4. 12 8 2   since 6 >  8. 5. |  7| < |  8| since 7 < 8. 6. |  9| >  9 since 9 >  9. 7.  |  1| =  1 since  1 =  1. 8. |  14| =  (  14) since 14 = 14. 9. 1.2 > 1.02 since 1.2 is to the right of 1.02 on a number line. 10. 3 3 2 4    since 3 2  is to the left of 3 4  on a number line. 11. Four is greater than or equal to negative three is written as 4   3. 12. Six is not equal to five is written as 6  5. 13. 0.03 is less than 0.3 is written as 0.03 < 0.3. 14. 155 < 400 or 400 > 155 15. a. The natural numbers are 1 and 3. b. The whole numbers are 0, 1, and 3. c. The integers are  6, 0, 1, and 3. d. The rational numbers are  6, 0, 1, 1 1 , 2 3, and 9.62. e. The irrational number is  . f. The real numbers are all numbers in the given set. 16. a. The natural numbers are 2 and 5. b. The whole numbers are 2 and 5. c. The integers are  3, 2, and 5. d. The rational numbers are  3,  1.6, 2, 5, 11 , 2 and 15.1. e. The irrational numbers are 5 and 2  . f. The real numbers are all numbers in the given set. 17. Look for the negative number with the greatest absolute value. The greatest loss was on Friday. 18. Look for the largest positive number. The greatest gain was on Wednesday. 19. 36 = 4  9 = 2  2  3  3 20. 120 = 8  15 = 2  2  2  3  5 21. 8 27 8 27 2 4 3 3 3 12 15 30 15 30 3 5 2 3 5 25               22. 7 21 7 32 7 32 7 8 4 4 8 32 8 21 8 21 8 3 7 3             23. 7 5 7 2 5 5 15 6 15 2 6 5 14 25 30 30 14 25 30 39 30 3 13 3 10 13 10                 24. 3 3 3 5 3 4 20 4 5 20 15 3 20 20 15 3 20 12 20 3 4 5 4 3 5              

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    ISM: Beginning Algebra Chapter 1: Review of Real Numbers 27 25. 3 5 11 53 2 6 4 8 4 8 11 2 53 4 2 8 22 53 8 8 22 53 8 75 8 3 9 8              26. 1 2 43 8 7 2 6 3 6 3 43 8 2 6 3 2 43 16 6 6 43 16 6 27 6 9 3 2 3 9 2 1 4 2                  27. 1 3 5 5 15 3 1     28. 3 35 2 35 35 1 2 8 2 17 4 4 2 2 2 2         29. 1 1 12 1 2 1 3 1 6 4 12 6 2 4 3 12 2 3 12 12 12 12 2 3 12 7 12                 The unknown part is 7 . 12 30. 1 1 10 5 2 10 5 2 3 1 2 5 10 10 10 10 10          The unknown part is 3 10 . 3 1 . 2 2 P l w   1 7 2 1 2 3 8 2 4 2 7 1 3 1 8 8 14 3 8 8 8 14 3 3 8 8 3 64 42 24 24 64 42 24 106 24 10 4 24 5 4 meters 12 P                                  A lw  1 7 1 3 8 4 7 3 8 4 7 3 2 4 7 6 1 1 sq meters 6 A           3 2 . the sum of the lengths of the sides P  5 8 3 3 2 5 26 4 2 in. 11 11 11 11 11 11 11 11 P         the sum of the two areas, each given by A lw  5 5 3 3 25 9 34 sq in. 11 11 11 11 121 121 121 A       

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    Chapter 1: Review of Real Numbers ISM: Beginning Algebra 28 33. 1 1 3 9 13 7 2 3 1 2 1 2 2 16 4 16 16 16 5 49 7 41 29 39 2 16 4 16 16 16 40 49 28 41 29 39 16 16 16 16 16 16 226 16 2 14 16 1 14 lb 8                     34. 1 3 17 35 34 35 69 5 2 2 4 lb 8 16 8 16 16 16 16 16        35. Total weight weight of boys weight of girls 226 69 16 16 295 16 7 18 lb 16       36. Look for the largest number. Baby C weighed the most. 37. Look for the smallest number. Baby E weighed the least. 38. 1 3 49 7 49 28 21 5 3 1 1 lb 16 4 16 4 16 16 16 16        39. 2 6 3 2 8 6 9 2 8 54 16 70           The answer is c. 40. 3 68 5 2 68 5 8 68 40 68 ( 40) 28            The answer is b. 4 1 . 2 2 2 2 4 7 7 7 49          4 2 . 3 3 3 3 3 27 4 4 4 4 64           4 3 . 3(1 2 5) 4 3(1 10) 4 3(11) 4 33 4 37            4 4 . 8 3(2 6 1) 8 3(12 1) 8 3(11) 8 33 41            4 5 . 2 4 6 2 8 4 4 64 4 6 4 4 24 4 4 64 4 24 72 28 4 18 4 7 18 7                   46 . 5[3(2 5) 5] 5[3(7) 5] 5[21 5] 5[16] 80         47 . The difference of twenty and twelve is equal to the product of two and four is written as 20  12 = 2  4. 48 . The quotient of nine and two is greater than negative five is written as 9 5. 2   49 . Let x = 6 and y = 2. 2 x + 3 y = 2(6) + 3(2) = 12 + 6 = 18 5 0 . Let x = 6, y = 2, and z = 8. ( 2 ) 6[2 2(8)] 6[2 16] 6[18] 108 x y z        5 1 . Let x = 6, y = 2, and z = 8. 6 8 6 8 3 2 5 2 2 2(2) 2 4 x z y y         5 2 . Let x = 6 and y = 2. 2 2 2 2 3 (6) 3(2) 36 3(4) 36 12 36 ( 12) 24 x y            53. Replace a with 37 and b with 80. 180 180 37 80 180 ( 37) ( 80) 143 ( 80) 63 a b               The measure of the unknown angle is 63  .

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