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Solution Manual for Elementary and Intermediate Algebra, 4th Edition

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    S OLUTIONS M ANUAL C T RIMBLE & A SSOCIATES E LEMENTARY & I NTERMEDIATE A LGEBRA F OURTH E DITION Michael Sullivan, III Joliet Junior College Katherine R. Struve Columbus State Community College Janet Mazzarella Southwestern College with contributions by Jessica Bernards Portland Community College Wendy Fresh Portland Community College

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    Contents Chapter 1 1 Chapter 2 50 Chapter 3 124 Chapter 4 202 Chapter 5 267 Chapter 6 315 Getting Ready for Intermediate Algebra: A Review of Chapters 1 − 6 362 Chapter 7 365 Getting Ready for Intermediate Algebra: A Review of Chapters 1 − 7 443 Chapter 8 448 Chapter 9 530 Chapter 10 606 Chapter 11 747 Chapter 12 812 Chapter 13 894 Appendix A 938 Appendix B 941 Appendix C 949 Appendix E 994

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    1 Chapter 1 Section 1.2 1.2 Quick Checks 1. A natural number is prime if its only factors are 1 and itself. 2. In the statement 6 ⋅ 8 = 48, 6 and 8 are called factors and 48 is called the product. 3. 12 2 6 2 2 3 · · The prime factorization of 12 is 2 ⋅ 2 ⋅ 3. 4. 120 2 15 2 2 2 · · · 3 5 · 2 2 4 30 The prime factorization of 120 is 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5. 5. 31 is a prime number. 6. 117 / \ 9 13 / \ \ 3 ⋅ 3 ⋅ 13 The prime factorization of 117 is 3 ⋅ 3 ⋅ 13. 7. The least common multiple of two or more natural numbers is the smallest number that is a multiple of each of the numbers. 8. 6 = 2 · 3 8 = 2 · · 2 · 2 2 · 3 · 2 · 2 The LCM is 2 ⋅ 3 ⋅ 2 ⋅ 2 = 24. 9. 45 = 3 · 3 · 5 72 = 3 · 3 · 2 · 2 · 2 3 · 3 · 5 · 2 · 2 · 2 The LCM is 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 360. 10. 14 = 2 · 7 9 = 3 · 3 2 · 7 · 3 · 3 The LCM is 2 ⋅ 7 ⋅ 3 ⋅ 3 = 126. 11. 18 = 2 · 3 · 3 30 = 2 · 3 · 5 2 · 2 · 3 · 3 · 5 12 = 2 · 2 · 3 The LCM is 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 180. 12. In the fraction 7 12 , 7 is called the numerator and 12 is called the denominator. 13. Fractions which represent the same portion of a whole are called equivalent fractions. 14. Multiply the numerator and denominator of 1 2 by 5. 1 1 5 5 2 2 5 10 ⋅ = = ⋅ 15. Multiply the numerator and denominator of 5 8 by 6. 5 5 6 30 8 8 6 48 ⋅ = = ⋅ 16. The least common denominator is the least common multiple of the denominators of a group of fractions. 17. The denominators are 4 and 6. 4 = 2 · 2 6 = 2 · 3 LCD = 2 · 2 · 3 = 12 1 1 3 3 4 4 3 12 ⋅ = = ⋅ 5 5 2 10 6 6 2 12 ⋅ = = ⋅

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    Chapter 1: Operations on Real Numbers and Algebraic Expressions ISM: Elementary & Intermediate Algebra 2 18. The denominators are 20 and 16. 20 = 2 · 2 · 5 16 = 2 · 2 · 2 · 2 LCD = 2 · 2 · 5 · 2 · 2 = 80 9 9 4 36 20 20 4 80 ⋅ = = ⋅ 11 11 5 55 16 16 5 80 ⋅ = = ⋅ 19. A fraction is written in lowest terms if the numerator and denominator share no common factor other than 1. 20. 45 3 3 5 3 3 5 9 80 2 2 2 2 5 2 2 2 2 5 16 ⋅ ⋅ ⋅ ⋅ = = = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 21. 4 2 2 4 9 3 3 9 ⋅ = = ⋅ 22. 16 2 2 2 2 2 2 2 2 2 56 2 2 2 7 2 2 2 7 7 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = = = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 23. The 1 is two places to the right of the decimal; this is the hundredths place. 24. The 2 is one place to the right of the decimal; this the tenths place. 25. The 8 is four places to the left of the decimal; this is the thousands place. 26. The 9 is three places to the right of the decimal; this is the thousandths place. 27. The number 1 is in the tenths place. The number to its right is 7. Since 7 is greater than 5, we round to 0.2. 28. The number 3 is in the hundredths place. The number to its right is 2. Since 2 is less than 5, we round to 0.93. 29. The number 9 is in the hundredths place. The number to its right is 6. Since 6 is greater than 5, we round to 1.40. 30. The number 0 is in the hundredths place. The number to its right is 4. Since 4 is less than 5, we round to 690.00. 31. The number 9 is in the tenths place. The number to its right is 8. Since 8 is greater than 5, we round to 60.0. 32. 0 4 5 2 0 2 0 0 . . 2 0 4 5 . = 33. 0 833 6 5 000 4 8 20 18 20 18 2 . . 5 0 83 6 . = 34. 1 375 8 11 000 8 3 0 2 4 60 56 40 40 0 . . 11 1 375 8 . = 35. 0 428571 7 3 000000 2 8 20 14 60 56 40 35 50 49 10 7 3 . . 3 0 428571 7 . =

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    ISM: Elementary & Intermediate Algebra Chapter 1: Operations on Real Numbers and Algebraic Expressions 3 36. 6 2 3 3 0 6 10 2 5 5 . ⋅ = = = ⋅ 37. 17 0 17 100 . = 38. 625 5 125 0 625 1000 . ⋅ = = 8 125 ⋅ 5 8 = 39. The word percent means parts per hundred, so 35% means 35 parts out of 100 parts or 35 100 . 40. 1 23 23 23 0 23 100 100 % % . % = ⋅ = = 41. 1 1 1 1 0 01 100 100 % % . % = ⋅ = = 42. 1 72 4 72 4 72 4 0 724 100 100 . . % . % . % = ⋅ = = 43. 1 127 127 127 1 27 100 100 % % . % = ⋅ = = 44. To convert a decimal to a percent, multiply the decimal by 100 1 % . 45. 100 0 15 0 15 15 1 % . . % = ⋅ = 46. 100 0 8 0 8 80 1 % . . % = ⋅ = 47. 100 1 372 1 372 137 2 1 % . . . % = ⋅ = 48. 100 0 004 0 004 0 4 1 % . . . % = ⋅ = 1.2 Exercises 50. 9 3 3 · The prime factorization of 9 is 3 ⋅ 3. 52. 100 4 25 2 2 5 · · · 5 The prime factorization of 100 is 2 ⋅ 2 ⋅ 5 ⋅ 5. 54. 35 5 7 · The prime factorization of 35 is 5 ⋅ 7. 56. 54 6 9 2 3 3 · · · 3 The prime factorization of 54 is 2 ⋅ 3 ⋅ 3 ⋅ 3. 58. 70 2 35 2 5 7 · · The prime factorization of 70 is 2 ⋅ 5 ⋅ 7. 60. 79 is a prime number. 62. 315 9 35 3 3 5 · · · 7 The prime factorization of 315 is 3 ⋅ 3 ⋅ 5 ⋅ 7. 64. 10 = 2 · 5 14 = 2 · 7 2 · 5 · 7 The LCM is 2 ⋅ 5 ⋅ 7 = 70. 66. 42 = 2 · 18 = 2 · 3 · 3 2 · 3 · 3 · 7 3 · 7 The LCM of 42 and 18 is 2 ⋅ 3 ⋅ 3 ⋅ 7 = 126. 68. 55 = 5 · 11 6 = 2 · 3 2 · 3 · 5 · 11 The LCM of 55 and 6 is 2 ⋅ 3 ⋅ 5 ⋅ 11 = 330. 70. 8 = 2 · 2 · 2 60 = 2 · 2 · 2 · 3 · 5 2 · 2 · 3 · 5 The LCM of 8 and 60 is 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 = 120.

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    Chapter 1: Operations on Real Numbers and Algebraic Expressions ISM: Elementary & Intermediate Algebra 4 72. 15 = 3 · 5 20 = 2 · 2 · 5 2 · 2 · 3 · 3 · 5 9 = 3 · 3 The LCM of 9, 15, and 20 is 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 180. 74. 18 = 2 · 3 · 3 20 = 2 · 2 · 5 2 · 2 · 3 · 3 · 5 4 = 2 · 2 The LCM of 4, 18, and 20 is 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 180. 76. 4 4 3 12 5 5 3 15 ⋅ = = ⋅ 78. 5 5 2 10 14 14 2 28 ⋅ = = ⋅ 80. 4 10 40 4 1 10 10 ⋅ = = ⋅ 82. 4 = 2 · 2 12 = 2 · 2 · 3 LCD = 2 · 2 · 3 = 12 3 3 3 9 4 4 3 12 ⋅ = = ⋅ The equivalent fractions are 9 12 and 5 12 . 84. 4 = 2 · 2 9 = 3 · 3 LCD = 2 · 2 · 3 · 3 = 36 1 1 9 9 4 4 9 36 ⋅ = = ⋅ 2 2 4 8 9 9 4 36 ⋅ = = ⋅ The equivalent fractions are 9 36 and 8 36 . 86. 12 = 2 · 2 · 3 15 = 3 · 5 LCD = 2 · 2 · 3 · 5 = 60 5 5 5 25 12 12 5 60 ⋅ = = ⋅ 7 7 4 28 15 15 4 60 ⋅ = = ⋅ The equivalent fractions are 25 60 and 28 60 . 88. 4 = 2 · 2 6 = 2 · 3 LCD = 2 · 2 · 3 · 5 = 60 10 = 2 · 5 7 7 6 42 10 10 6 60 ⋅ = = ⋅ 1 1 15 15 4 4 15 60 ⋅ = = ⋅ 5 5 10 50 6 6 10 60 ⋅ = = ⋅ The equivalent fractions are 42 15 60 60 , , and 50 60 . 90. 9 3 3 3 3 3 15 3 5 3 5 5 ⋅ ⋅ = = = ⋅ ⋅ 92. 81 3 3 3 3 3 3 3 3 9 36 2 2 3 3 2 2 3 3 4 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = = = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 94. 9 3 3 3 3 1 27 3 3 3 3 3 3 3 ⋅ ⋅ = = = ⋅ ⋅ ⋅ ⋅ 96. 36 2 2 3 3 2 2 3 3 9 4 2 2 2 2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = = = ⋅ ⋅ 98. The 8 is three places to the left of the decimal; this is the hundreds place. 100. The 7 is one place to the right of the decimal; this is the tenths place. 102. The 9 is one place to the left of the decimal; this is the ones place. 104. The number 8 is in the hundredths place. The number to the right of 8 is 4. Since 4 is less than 5, round to 7298.08.

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    ISM: Elementary & Intermediate Algebra Chapter 1: Operations on Real Numbers and Algebraic Expressions 5 106. The number 3 is in the ones place. The number to the right of 3 is 5. Since 5 is 5 or more, round to 544. 108. The number 9 is in the hundredths place. The number to the right of 9 is 8. Since 8 is greater than 5, round to 683.10. 110. The number 9 is in the tenths place. The number to the right of 9 is 9. Since 9 is greater than 5, round to 38.0. 112. 0 75 4 3 00 28 20 20 0 . . 3 0 75 4 . = 114. 0 222 9 2 000 1 8 20 18 20 18 2 . . 2 0 222 or 0 2 9 . ... . = 116. 0 34375 32 11 00000 9 6 1 40 1 28 120 96 240 224 160 160 0 . . 11 0 34375 32 . = 118. 0 461538 13 6 000000 5 2 80 78 20 13 70 65 50 39 110 104 6 . . 6 0 461538 13 . = 120. 1 14 50 57 00 50 7 0 50 2 00 2 00 0 . . 57 1 14 50 . = 122. 25 1 25 0 25 100 . ⋅ = = 4 25 ⋅ 1 4 = 124. 4 2 2 2 0 4 10 2 5 5 . ⋅ = = = ⋅ 126. 358 2 179 179 0 358 1000 2 500 500 . ⋅ = = = ⋅ 128. 1 59 59 59 0 59 100 100 % % . % = ⋅ = = 130. 1 8 25 8 25 8 25 0 0825 100 100 . . % . % . % = ⋅ = = 132. 1 0 5 0 5 0 5 0 005 100 100 . . % . % . % = ⋅ = =

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    Chapter 1: Operations on Real Numbers and Algebraic Expressions ISM: Elementary & Intermediate Algebra 6 134. 100 0 5 0 5 50 1 % . . % = ⋅ = 136. 100 0 349 0 349 34 9 1 % . . . % = ⋅ = 138. 100 1 1 100 1 % % = ⋅ = 140. 1 87 8 15 00 8 7 0 6 4 60 56 4 . . 15 8 rounded to the nearest tenth is 1.9. 142. 1 285 7 9 000 7 2 0 1 4 60 56 40 35 5 . . 9 7 rounded to the nearest hundredth is 1.29. 144. 0 5806 31 18 0000 15 5 2 50 2 48 200 186 14 . . 18 31 rounded to the nearest thousandth is 0.581. 146. The number of laps is the LCM of 21, 18, and 15. 18 = 2 · 3 · 3 15 = 3 · 5 LCM = 2 · 3 · 3 · 5 · 7 = 630 21 = 3 · 7 The drivers would finish at the same time after 630 laps. 148. Find the LCM of 14 and 20. 14 = 2 · 7 20 = 2 · 2 · 5 LCM = 2 · 2 · 5 · 7 = 140 Pamela and Geoff are in Columbus every 140 days. 150. 450 of 750 450 2 3 3 5 5 2 3 3 5 5 3 750 2 3 5 5 5 2 3 5 5 5 5 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = = = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ The fraction of students that are enrolled in 15 or more semester hours is 3 5 . 152. (a) 640 32 20 1100 ⋅ = 55 20 ⋅ 32 55 = The fraction of adult women who stated that they believe in ghosts is 32 55 . (b) 0 5818 55 32 0000 27 5 4 50 4 40 100 55 450 440 10 . . J JJM 32 0 581 55 . = (c) 100 0 581 58 1818 58 18 1 % . . % . % ⋅ = ≈

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    ISM: Elementary & Intermediate Algebra Chapter 1: Operations on Real Numbers and Algebraic Expressions 7 154. 0 69565 115 80 00000 69 0 11 00 10 35 650 575 750 690 600 575 25 . . 80 100 0 6957 0 6957 69 57 115 1 % . . . % ≈ = ⋅ = 156. (a) 48 0 4752 101 100 0 4701 47 52 1 . % . . % = ⋅ = 47.52% were white ash. (b) 51 0 5050 101 100 0 5050 50 50 1 . % . . % = ⋅ = 50.50% were green ash. (c) 2 0 0198 101 100 0 0198 1 98 1 . % . . % = ⋅ = 1.98% were blue ash. 158. 0 45 11 5 00 4 4 60 55 5 . . 5 0 45 11 . = 100 0 4545 45 45 1 % . . % ⋅ = 45.45% is saturated fat. Section 1.3 Are You Prepared for This Section? P1. 0 625 8 5 000 4 8 20 16 40 40 0 . . 5 0 625 8 . = P2. 0 8181 11 9 0000 8 8 20 11 90 88 20 11 9 . . 9 0 8181 11 . ... = or 0 81 . 1.3 Quick Checks 1. The first four positive odd numbers are 1, 3, 5, and 7. If O represents this set, then O = {1, 3, 5, 7}. 2. The states whose names begin with the letter A are Alabama, Alaska, Arizona, and Arkansas. If A represents this set, then A = {Alabama, Alaska, Arizona, Arkansas}. 3. There are no states whose names begin with the letter Z. If Z represents this set, then Z = { } or ∅ . 4. Every integer is a rational number. True 5. Real numbers that can be represented with a terminating or repeating decimal are called rational numbers. 6. 12 is the only natural number. 7. 0 and 12 are the whole numbers.

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    Chapter 1: Operations on Real Numbers and Algebraic Expressions ISM: Elementary & Intermediate Algebra 8 8. − 5, 12, and 0 are the integers. 9. 11 5 12 2 76 0 5 , , , . , , − and 18 4 are the rational numbers. 10. 2.737737773... is the only irrational number. 11. All the numbers listed are real numbers. 12. The point on the real number line whose coordinate is 0 is called the origin. 13. –2 –1 0 1 –3 4 3 2 –2 0 1 2 3 3.5 14. The symbols <, >, ≤ , ≥ are called inequality symbols. 15. 2 < 9 because 2 lies to the left of 9 on a real number line. 16. − 5 < − 3 because − 5 lies to the left of − 3 on a real number line. 17. 4 1 5 2 > because 4 8 5 10 = and 1 5 2 10 . = Having 8 parts out of 10 is more than having 5 parts out of 10. Also, 4 0 8 5 . = and 1 0 5 2 . = and 0.8 lies to the right of 0.5 on a real number line. 18. 4 0 5 7 . > because 4 0 571428 7 . = and 0 571428 . lies to the right of 0.5 on a real number line. 19. 4 20 3 15 = 20. 4 5 3 4 − < − 21. The distance from zero to a point on a real number line whose coordinate is a is called the absolute value of a . 22. | − 15| = 15 because the distance from 0 to − 15 on a real number line is 15. 23. 3 3 4 4 − = because the distance from 0 to 3 4 − on a real number line is 3 4 . 24. − | − 4| = − 4 (| − 4| = 4) because the distance from 0 to − 4 on a number line is 4. 1.3 Exercises 26. The set of natural numbers less than 25 is B = {1, 2, 3, 4, ..., 24}. 28. The set of integers between − 6 and 4 is C = { − 5, − 4, − 3, − 2, − 1, 0, 1, 2, 3}. 30. There are no whole numbers that are less than 0; F = { } or ∅ . 32. 3 and 0 are the whole numbers. 34. − 4, 3, 13 2 , − and 0 are the rational numbers. 36. All the numbers listed are real numbers. 38. − 4.2, 3 5 . , and 5 1 5 = are the rational numbers. 40. 5 1 5 , = so 5 5 is the only integer. 42. 5 1 5 , = so 5 5 is the only natural number. 44. 5 4 0 2 –0.5 1.5 0 1 2 –1 –2 – 3 4 46. 0 lies to the right of − 5 on a number line. The statement is false. 48. − 3 lies to the right of − 5 on a number line. The statement is true. 50. Since 4 7 4 777 . . ..., = 4 7 4 7 . . . < The statement is false. 52. Since 1 0 333 3 . ..., = 1 0 33 3 . . > The statement is false.

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    ISM: Elementary & Intermediate Algebra Chapter 1: Operations on Real Numbers and Algebraic Expressions 9 54. Since − 8 lies to the right of − 8.5 on a number line, − 8 > − 8.5. 56. Since 5 0 416 12 . = and 2 0 666 3 . , = 5 2 12 3 . < 58. Since 5 0 4545 11 . ..., = 5 0 45 11 . . = 60. Since 3 0 75 4 . = and 3 0 6 5 . , = 3 3 4 5 . > 62. | − 8| = 8 because the distance from 0 to − 8 on a real number line is 8. 64. |7| = 7 because the distance from 0 to 7 on a real number line is 7. 66. 13 13 9 9 − = because the distance from 0 to 13 9 − on a real number line is 13 9 . 68. − | − 3.2| = − 3.2 70. (a) 4 3 –1.5 –2 0 1 2 3 –1 –3 –4 –5 4 5 6 7 |–4| 0 – 15 3 – –2 8 8 ( b) From left to right on the number line, the order of the numbers is 15 3 , − − 2, − 1.5, 4 3 , − 0, | − 4|, 8. ( c) i. 15 5 3 , − = − − 2, 0, | − 4| = 4, 8 are integers. ii. All numbers listed are rational numbers. Natural Whole Integer Rational Irrational Real 72. 0 √ √ √ √ 74. 2 √ √ 76. 4 √ √ √ √ √ 78. 6 45 . √ √ 80. Only decimal numbers that are terminating or repeating are rational. The decimal number 4.010010001... is irrational. The statement is false. 82. 0 is neither positive nor negative. The statement is false. 84. The set of integers is a subset of the real numbers. The statement is true. 86. Consider the number in the form 8 4 2 , = which is an integer. The statement is true. 88. − 1 is a negative or nonpositive integer. The statement is true.

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    Chapter 1: Operations on Real Numbers and Algebraic Expressions ISM: Elementary & Intermediate Algebra 10 90. Nonnegative integers are 0, 1, 2, ..., or the whole numbers. 92. Terminating or repeating decimals are rational numbers. 94. There is no number that is both negative and positive. This is the empty set or ∅ . 96. The element f in Z is not an element of Y . The statement is false. 98. The element f in Z is not an element of X . The statement is false. 100. A ∩ B is the set of all elements common to both A and B , so A ∩ B = {10, 11, 12}. 102. B ∩ C is the set of all elements common to both B and C , so B ∩ C = {11, 12, 13, 14, 15}. 104. A ∪ C is the set of elements in either A or C , so A ∪ C = {7, 8, 9, 10, 11, 12, 13, 14, 15}. 106. Since Y is a subset of X , then X ∩ Y = Y , so X ∩ Y = {60, 62, 64, ..., 80}. 108. (a) Answers may vary. One possibility: There are subsets with no elements, subsets with one element, subsets with two elements, and subsets with three elements. 0: ∅ 1: { a }, { b }, { c } 2: { a , b }, { a , c }, { b , c } 3: { a , b , c } (b) There are a total of 8 subsets. 110. An irrational number is any number that cannot be written as the quotient of two integers. Non- repeating, non-terminating decimals are irrational numbers. Numbers such as π and 2 and 8.98987987698765... are irrational numbers. Section 1.4 Are You Prepared for This Section? P1. 16 2 2 2 2 2 2 2 2 2 2 4 36 2 2 3 3 2 2 3 3 3 3 9 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = = = = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ P2. | − 5| = 5 because the distance from 0 to − 5 on a number line is 5. 1.4 Quick Checks 1. The answer to an addition problem is called the sum. 2. 11 12 13 14 10 9 8 6 steps right Start End 8 + 6 = 14 3. –6 –5 –4 –3 –7 –8 –9 5 steps left Start End − 3 + ( − 5) = − 8 4. 1 2 3 2 0 –1 –2 4 steps right Start End − 1 + 4 = 3 5. 1 2 3 4 0 –1 –2 4 steps left Start End 3 + ( − 4) = − 1 6. –6 –5 –4 –3 –7 –8 –9 4 steps right Start End − 8 + 4 = − 4 7. 16 17 18 15 14 13 3 steps left Start End 17 + ( − 3) = 14 8. The sum of two negative integers will be negative. 9. | − 11| = 11 |7| = 7 The signs of − 11 and 7 are different, so we subtract the absolute values: 11 − 7 = 4. The larger absolute value, 11, corresponds to a negative number in the original problem, so the sum is negative. − 11 + 7 = − 4

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    ISM: Elementary & Intermediate Algebra Chapter 1: Operations on Real Numbers and Algebraic Expressions 11 10. |5| = 5 | − 8| = 8 The signs of 5 and − 8 are different, so we subtract the absolute values: 8 − 5 = 3. The larger absolute value, 8, corresponds to a negative number in the original problem, so the sum is negative. 5 + ( − 8) = − 3 11. | − 8| = 8 | − 16| = 16 The signs of − 8 and − 16 are the same, so we add the absolute values: 8 + 16 = 24. Both numbers in the original problem are negative, so the sum is negative. − 8 + ( − 16) = − 24 12. |94| = 94 | − 38| = 38 The signs of 94 and − 38 are different, so we subtract the absolute values: 94 − 38 = 56. The larger absolute value, 94, corresponds to a positive number in the original problem, so the sum is positive. 94 + ( − 38) = 56 13. For any real number a other than 0, there is a real number − a , called the additive inverse, or opposite, of a such that a + ( − a ) = − a + a = 0. 14. The additive inverse of 7 is − 7 because 7 + ( − 7) = 0. 15. The additive inverse of − 21 is − ( − 21) = 21 because − 21 + 21 = 0. 16. The additive inverse of 8 5 − is 8 8 5 5 ⎛ ⎞ − − = ⎜ ⎟ ⎝ ⎠ because 8 8 0 5 5 . − + = 17. The additive inverse of 5.75 is − 5.75 because 5.75 + ( − 5.75) = 0. 18. The answer to a subtraction problem is called the difference. 19. The subtraction problem − 3 − 10 is equivalent to − 3 + ( − 10). 20. 59 − ( − 21) = 59 + 21 = 80 21. − 32 − 146 = − 32 + ( − 146) = − 178 22. − 19 − ( − 40) = − 19 + 40 = 21 23. 8 13 5 8 13 5 5 5 0 ( ) − + = + − + = − + = 24. 27 49 18 27 49 18 76 18 58 ( ) − − + = − + − + = − + = − 25. 3 14 8 3 3 14 8 3 17 8 3 9 3 12 ( ) ( ) ( ) − − − + = + + − + = + − + = + = 26. − 825 + 375 − ( − 735) + 265 = − 825 + 375 + 735 + 265 = − 450 + 735 + 265 = 285 + 265 = 550 27. The product of two integers with the same sign is positive. 28. − 3(7) = − 21 29. 13( − 4) = − 52 30. 5 ⋅ 16 = 80 31. − 9( − 12) = 108 32. ( − 13)( − 25) = 325 33. The product of thirteen negative factors is negative. True 34. − 3 ⋅ 9( − 4) = − 27( − 4) = 108 35. 3 4 5 6 12 5 6 60 6 360 ( )( )( ) ( )( ) ( ) − − − = − − − = − = − 36. The reciprocal of 6 is 1 6 . 37. The reciprocal of − 2 is 1 2 . − 38. In the division problem 18 6 3 , = 18 is called the dividend, 3 is called the divisor, and 6 is called the quotient.

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    Chapter 1: Operations on Real Numbers and Algebraic Expressions ISM: Elementary & Intermediate Algebra 12 39. The quotient of two negative numbers is positive. True 40. 20 5 4 5 4 5 5 4 1 4 1 4 1 ⋅ ⋅ = = = = − − − ⋅ − ⋅ − 41. 72 1 4 18 1 4 18 18 20 4 5 4 5 5 − − ⋅ ⋅ − ⋅ ⋅ = = = − ⋅ ⋅ 42. 9 7 63 9 7 63 7 7 1 7 ( ) ( ) ( ) ( ) − − − − ÷ − = = = − − 1 7 ( ) − 9 = 1.4 Exercises 44. 6 + 4 = 10 46. − 4 + 12 = 8 48. 7 + ( − 13) = − 6 50. − 13 + ( − 5) = − 18 52. − 32 + 49 = 17 54. − 145 + ( − 68) = − 213 56. ( − 13) + 37 + ( − 22) = 24 + ( − 22) = 2 58. 34 46 12 72 12 12 72 0 72 72 ( ) ( ) − + + − + = + − + = + = 60. The additive inverse of − 34 is − ( − 34) = 34 because − 34 + 34 = 0. 62. The additive inverse of 7 is − 7 because 7 + ( − 7) = 0. 64. 35 − 23 = 35 + ( − 23) = 12 66. 12 − 19 + 12 + ( − 19) = − 7 68. − 15 − 9 = − 15 + ( − 9) = − 24 70. 14 − ( − 18) = 14 + 18 = 32 72. − 15 − ( − 15) = − 15 + 15 = 0 74. 0 − 18 = 0 + ( − 18) = − 18 76. 46 − ( − 25) = 46 + 25 = 71 78. 49 − ( − 49) = 49 + 49 = 98 80. 7 ⋅ 9 = 63 82. 9( − 7) = − 63 84. − 21 ⋅ 0 = 0 86. ( − 22)( − 5) = 110 88. ( − 128)7 = − 896 90. − 6 ⋅ 4 ⋅ 8 = − 24 ⋅ 8 = − 192 92. − 8(2)( − 9) = − 16( − 9) = 144 94. 3 4 6 1 12 6 1 72 1 72 ( )( )( )( ) ( )( ) ( ) − − − = − = − = − 96. The reciprocal of 10 is 1 10 . 98. The reciprocal of − 3 is 1 3 . − 100. The reciprocal of 2 is 1 2 . 102. 36 4 9 4 9 36 9 4 9 1 9 1 9 ⋅ ⋅ ÷ = = = = ⋅ ⋅ 104. 9 7 63 9 7 7 1 7 ( ) ( ) ( ) − − − = = − − 1 7 ( ) − 9 = 106. 144 24 6 24 6 24 6 1 6 1 6 − − ⋅ − ⋅ = = = − ⋅ ⋅ 108. 20 5 4 5 4 5 16 4 4 4 4 4 ⋅ ⋅ = = = ⋅ ⋅ 110. 120 20 6 20 6 20 20 66 11 6 11 6 11 11 ⋅ ⋅ = = = = − − − ⋅ − ⋅ − 112. 20 4 80 20 4 12 3 4 ( ) ( ) ( ) − − − = = − − 3 4 ( ) − 20 3 = 114. 7( − 15) = − 105 116. 87 − 19 = 87 + ( − 19) = 68 118. 7 ⋅ 209 = 1463

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    ISM: Elementary & Intermediate Algebra Chapter 1: Operations on Real Numbers and Algebraic Expressions 13 120. 6 26 156 6 26 26 1 26 ( ) ( ) ( ) − − − = = − − 1 26 ( ) − 6 = 122. 103 + ( − 66) = 37 124. 42 42 7 6 7 6 7 12 12 2 6 2 6 2 − ⋅ ⋅ = = = = ⋅ ⋅ 126. | − 10| + ( − 62) = 10 + ( − 62) = − 52 128. | − 193| − ( − 20) = 193 − ( − 20) = 193 + 20 = 213 130. The sum of 32 and − 64 is written as 32 + ( − 64) = − 32. 132. − 85 subtracted from − 16 is written as − 16 − ( − 85) = − 16 + 85 = 69. 134. 32 multiplied by − 8 is written as 32 ⋅ ( − 8) = − 256. 136. − 40 divided by 100 is written as 40 2 20 2 20 40 100 100 5 20 − − ⋅ − ⋅ − ÷ = = = ⋅ 5 20 ⋅ 2 5 . = − 138. The product of − 11 and − 5 is written as − 11( − 5) = 55. 140. The difference of − 16 and − 42 is written as − 16 − ( − 42) = − 16 + 42 = 26 142. − 14 ° ; the number is negative because the temperature is below zero. 144. +$125,000 or $125,000; the number is positive because there was a profit. 146. +12,368 or 12,368; the number is positive because the population grew. 148. 15 + ( − 7) + 14 + 26 = 48 The ball is on the 48-yard marker. 150. 399 + 839 + ( − 69) + ( − 78) + ( − 739) = 352 She has $352 left for spending money. 152. 35,000 + 4290 + ( − 10,400) + ( − 2605) = 26,285 The plane is flying at an elevation of 26,285 feet. 154. 14,495 − ( − 280) = 14,495 + 280 = 14,775 The difference is 14,775 feet. 156. 6 + ( − 4) = 2 6( − 4) = − 24 The integers are 6 and − 4. 158. − 15 + ( − 3) = − 18 − 15( − 3) = 45 The integers are − 15 and − 3. 160. Consider 5( − 3). The statement 5( − 3) means ( − 3) + ( − 3) + ( − 3) + ( − 3) + ( − 3) = − 15. We conclude that the product of a positive number and a negative number is negative. Section 1.5 Are You Prepared for This Section? P1. 12 = 2 ⋅ 2 ⋅ 3 16 = 2 ⋅ 2 ⋅ 2 ⋅ 2 LCD = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 = 48 The LCD of 5 12 and 3 16 is 48. P2. 30 = 5 ⋅ 6 4 4 6 4 6 24 5 5 6 5 6 30 ⋅ = ⋅ = = ⋅ P3. 2. 18 3 6 3 6 3 30 5 6 5 6 5 ⋅ ⋅ − = − = − = − ⋅ ⋅ 3. 24 6 4 6 4 6 6 4 1 4 1 4 1 ⋅ ⋅ − = − = − = − = − ⋅ ⋅ 1.5 Quick Checks 1. 3 9 3 9 27 4 8 4 8 32 ⋅ ⋅ = = ⋅ 2. 5 56 5 56 7 15 7 15 1 5 7 8 7 3 5 1 5 7 8 7 3 5 8 3 − − ⋅ ⋅ = ⋅ − ⋅ ⋅ ⋅ = ⋅ ⋅ − ⋅ ⋅ ⋅ = ⋅ ⋅ = −

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