Solution Manual for College Algebra, 12th Edition
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S OLUTIONS M ANUAL B EVERLY F USFIELD C OLLEGE A LGEBRA T WELFTH E DITION Margaret L. Lial American River College John Hornsby University of New Orleans David I. Schneider University of Maryland Callie J. Daniels St. Charles Community College
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1 Chapter R REVIEW OF BASIC CONCEPTS Section R.1 Sets 1. The elements of the set of natural number are {1, 2, 3, 4, …}. 2. Set A is a subset of set B if every element of set A is also an element of set B . 3. The set of all elements of the universal set U that do not belong to set A is the complement of set A . 4. The intersection of sets A and B is made up of all the elements belonging to both set A and set B . 5. The union of sets A and B is made up of all the elements belonging to set A or set B (or both). 6. The set 1 1 1 3 9 27 1, , , , is infinite. 7. Using set notation, the set { x | x is a natural number less than 6} is {1, 2, 3, 4, 5}. 8. 4, 5 A 9. 16,18, 21, 50 15,16,17,18 16,18 10. 16,18, 21, 50 15,16,17,18 15,16,17,18, 21, 50 11. The set {4, 5, 6, …, 15} has a limited number of elements, so it is a finite set. Yes, 10 is an element of the set. 12. The set {1, 2, 3, 4, 5, …, 75} has a limited number of elements, so it is a finite set. Yes, 10 is an element of the set. 13. The set 1 1 1 2 4 8 1, , , , has an unlimited number of elements, so it is an infinite set. No, 10 is not an element of the set. 14. The set {4, 5, 6, …} has an unlimited number of elements, so it is an infinite set. Yes, 10 is an element of the set. 15. The set { x | x is a natural number larger than 11}, which can also be written as {11, 12, 13, 14, …}, has an unlimited number of elements, so it is an infinite set. No, 10 is not an element of the set. 16. The set { x | x is a natural number greater than or equal to 10}, which can also be written as {10, 11, 12, 13, …}, has an unlimited number of elements, so it is an infinite set. Yes, 10 is an element of the set. 17. There are infinitely many fractions between 1 and 2, so { x | x is a fraction between 1 and 2} is an infinite set. No, 10 is not an element of the set. 18. The set { x | x is an even natural number} has no largest element. Because it has an unlimited number of elements, this is an infinite set. Yes, 10 is an element of the set. 19. The elements of the set {12, 13, 14, …, 20} are all the natural numbers from 12 to 20 inclusive. There are 9 elements in the set, {12, 13, 14, 15, 16, 17, 18, 19, 20}. 20. The elements of the set {8, 9, 10, …, 17} are all the natural numbers from 8 to 17 inclusive. There are 10 elements in the set, {8, 9, 10, 11, 12, 13, 14, 15, 16, 17}. 21. Each element of the set 1 1 1 2 4 32 1, , , , after the first is found by multiplying the preceding number by 1 2 . There are 6 elements in the set, 1 1 1 1 1 2 4 8 16 32 1, , , , , . 22. Each element of the set {3, 9, 27, …, 729} after the first is found by multiplying the preceding number by 3. There are 6 elements in the set, {3, 9, 27, 81, 243, 729}. 23. To find the elements of the set {17, 22, 27, …, 47}, start with 17 and add 5 to find the next number. There are 7 elements in the set, {17, 22, 27, 32, 37, 42, 47}. 24. To find the elements of the set {74, 68, 62, …, 38}, start with 74 and subtract 6 (or add –6) to find the next number. There are 7 elements in the set, {74, 68, 62, 56, 50, 44, 38}. 25. When you list all elements in the set {all natural numbers greater than 8 and less than 15}, you obtain {9, 10, 11, 12, 13, 14}.
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2 Chapter R Review of Basic Concepts 26. When you list all elements in the set {all natural numbers not greater than 4} you obtain {1, 2, 3, 4}. 27. 6 is an element of the set {3, 4, 5, 6}, so we write 6 {3, 4, 5, 6}. 28. 9 is an element of the set {2, 3, 5, 9, 8}, so we write 9 {2, 3, 5, 9, 8}. 29. 5 is not an element of {4, 6, 8, 10}, so we write 5 {4, 6, 8, 10}. 30. 13 is not an element of the set {3, 5, 12, 14}, so we write 13 {3, 5, 12, 14}. 31. 0 is an element of {0, 2, 3, 4}, so we write 0 {0, 2, 3, 4}. 32. 0 is an element of the set {0, 5, 6, 7, 8, 10}, so we write 0 {5, 6, 7, 8, 10}. 33. {3} is a subset of {2, 3, 4, 5}, not an element of {2, 3, 4, 5}, so we write {3} {2, 3, 4, 5}. 34. {5} is a subset of {3, 4, 5, 6, 7}, not an element of {3, 4, 5, 6, 7}, so we write {5} {3, 4, 5, 6, 7}. 35. {0} is a subset of {0, 1, 2, 5}, not an element of {0, 1, 2, 5}, so we write {0} {0, 1, 2, 5}. 36. {2} is a subset of {2, 4, 6, 8}, not an element of {2, 4, 6, 8}, so we write {2} {2, 4, 6, 8}. 37. 0 is not an element of , because the empty set contains no elements. Thus, 0 . 38. is a subset of , not an element of . The empty set contains no elements. Thus we write, . 39. False. 3 is not one of the elements in {2, 5, 6, 8}. 40. False. 6 is not one of the elements of {2, 5, 8, 9}. 41. True. 1 is one of the elements of {11, 5, 4, 3, 1}. 42. True. 12 is one of the elements of {18, 17, 15, 13, 12}. 43. True. 9 is not one of the elements of {8, 5, 2, 1}. 44. True. 3 is not an element of {7, 6, 5, 4}. 45. True. Both sets contain exactly the same four elements. 46. True. Both sets contain exactly the same five elements. 47. False. These two sets are not equal because {5, 8, 9, 0} contains the element 0, which is not an element of {5, 8, 9}. 48. False. These two sets are not equal because {3, 7, 12, 14, 0} contains the element 0, which is not an element of {3, 7, 12, 14}. 49. True. 1 and 2 are the only natural numbers less than 3. 50. True. Both sets describe the same elements. For Exercises 51 − 62, A = {2, 4, 6, 8, 10, 12}, B = {2, 4, 8, 10}, C = {4, 10, 12}, D = {2, 10}, and U = {2, 4, 6, 8, 10, 12, 14}. 51. True. This statement says “ A is a subset of U .” Because every element of A is also an element of U , the statement is true. 52. True. This statement says “ C is a subset of U .” Because every element of C is also an element of U , the statement is true. 53. True. Because both elements of D , 2 and 10, are also elements of B , D is a subset of B . 54. True. Because both elements of D , 2, and 10, are also elements of A , D is a subset of A . 55. False. Set A contains two elements, 6 and 12, that are not elements of B . Thus, A is not a subset of B . 56. False. Set B contains two elements, 2 and 8, that are not elements of C . Thus, B is not a subset of C . 57. True. The empty set is a subset of every set. 58. True. The empty set is a subset of every set. 59. True. Because 4, 8, and 10 are all elements of B , {4, 8, 10} is a subset of B . 60. False. Because 0 is not an element of D , {0, 2} is not a subset of D . 61. False. Because B contains two elements, 4 and 8, that are not elements of D , B is not a subset of D . 62. False. There are three elements of A (2, 6, and 8) that are not elements of C , so A is not a subset of C . 63. Every element of {2, 4, 6} is also an element of {2, 3, 4, 5, 6}, so {2, 4, 6} is a subset of {2, 3, 4, 5, 6}. We write {2, 4, 6} {2, 3, 4, 5, 6}.
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Section R.1 Sets 3 64. Every element of {1, 5} is also an element of {0, 1 2, 3, 5}, so {1, 5} is a subset of the set {0, 1 2, 3, 5}. We write {1, 5} {0, 1 2, 3, 5}. 65. Because 0 is an element of {0, 1, 2}, but is not an element of {1, 2, 3, 4, 5}, {0, 1, 2} is not a subset of {1, 2, 3, 4, 5}. We write {0, 1, 2} {1, 2, 3, 4, 5}. 66. Because 8 is an element of {5, 6, 7, 8}, but is not an element of {1, 2, 3, 4, 5, 6, 7}, {5, 6, 7, 8} is not a subset of {1, 2, 3, 4, 5, 6, 7}. We write {5, 6, 7, 8} {1, 2, 3, 4, 5, 6, 7}. 67. The empty set is a subset of every set, so {1, 4, 6, 8}. 68. The empty set is a subset of every set, including itself, so . 69. True. 7 and 9 are the only elements belonging to both sets. 70. True. 8 and 11 are the only elements belonging to both sets. 71. False. {1, 2, 7} {1, 5, 9} {1, 2, 5, 7, 9}, while {1, 2, 7} {1, 5, 9} {1}. 72. False. {6, 12, 14, 16} {6, 14, 19} {6, 12, 14, 16, 19}, while {6, 12, 14, 16} {6, 14, 19} {6, 14}. 73. True. 2 is the only element belonging to both sets. 74. False. The sets {6, 8, 9} and {9, 8, 6} are equal because they contain exactly the same three elements. Their union contains the same elements, namely 8, 9, and 6. 75. {3, 5, 9, 10} {3, 5, 9, 10} In order to belong to the intersection of two sets, an element must belong to both sets. Because the empty set contains no elements, {3, 5, 9, 10} , so the statement is false. 76. True. For any set A, . A A 77. True. Because the two sets are equal, their union contains the same elements, namely 1, 2, and 4. 78. False. {1, 2, 4} {1, 2, 4} {1, 2, 4} 79. True. 80. True. For Exercises 81–110, U = {0, 1, 2 , 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, M = {0, 2, 4, 6, 8}, N = {1, 3, 5, 7, 9, 11, 13}, Q = {0, 2, 4, 6, 8, 10, 12}, and R = {0, 1, 2, 3, 4}. 81. M R The only elements belonging to both M and R are 0, 2, and 4, so M R = {0, 2, 4}. 82. M U Because M U , the intersection of M and U will contain the same elements as M . M U = M or {0, 2, 4, 6, 8}. 83. M N The union of two sets contains all elements that belong to either set or to both sets. M N = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13} 84. M R The union M and R is made up of elements which belong to M or to R (or to both). M R = {0, 1, 2, 3, 4, 6, 8} 85. M N There are no elements which belong to both M and N , so M N . M and N are disjoint sets. 86. U N Because N U, the elements belonging to U and N are all the elements belonging to N, U N = N or {1, 3, 5, 7, 9, 11, 13} 87. N R = {0, 1, 2, 3, 4, 5, 7, 9, 11, 13} 88. M Q Because M Q , the elements belonging to M or Q are all the elements belonging to Q . M Q = Q or {0, 2, 4, 6, 8, 10, 12} 89. N The set N is the complement of set N , which means the set of all elements in the universal set U that do not belong to N . N Q or {0, 2, 4, 6, 8, 10, 12} 90. Q The set Q is the complement of set Q , which means the set of all elements in the universal set U that do not belong to Q . Q N or {1, 3, 5, 7, 9, 11, 13}
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4 Chapter R Review of Basic Concepts 91. M Q First form M , the complement of M . M contains all elements of U that are not elements of M . Thus, M {1, 3, 5, 7, 9, 10, 11, 12, 13}. Now form the intersection of M and Q . Thus, we have M Q {10, 12}. 92. Q R First form R , the complement of R . R contains all elements of U the are not elements of R . Thus, R {5, 6, 7, 8, 9, 10, 11, 12, 13}. Now form the intersection of Q and R . Thus, we have Q R {6, 8, 10, 12} 93. R Because the empty set contains no elements, there are no elements belonging to both and R . Thus, and R are disjoint sets, and . R 94. Q Because the empty set contains no elements, there are no elements belonging to both and Q . Thus, and Q are disjoint sets, and . Q 95. N Because contains no elements, the only elements belonging to N or are the elements of N . Thus, and N are disjoint sets, and N N or {1, 3, 5, 7, 9, 11, 13}. 96. R Because contains no elements, the only elements belonging to R or are the elements of R . Thus, and R are disjoint sets, and R R or {0, 1, 2, 3, 4}. 97. ( M N ) R First, form the intersection of M and N . Because M and N have no common elements (they are disjoint), M N . Thus, ( M N ) R . R Now, because contains no elements, the only elements belonging to R or are the elements of R . Thus, and R are disjoint sets, and R R or {0, 1, 2, 3, 4}. 98. ( N R ) M First form the union of N and R . We have N R = {0, 1, 2, 3, 4, 5, 7, 9, 11, 13}. Now form the intersection of this set with M . We have ( N R ) M = {0, 2, 4}. 99. ( Q M ) R First form the intersection of Q and M . We have Q M = {0, 2, 4, 6, 8} = M. Now form the union of this set with R . We have ( Q M ) R = M R = {0, 1, 2, 3, 4, 6, 8}. 100. ( R N ) M First form the union of R and N . We have R N = {0, 1, 2, 3, 4, 5, 7, 9, 11, 13}. Now find the complement of M . We have M = {1, 3, 5, 7, 9, 10, 11, 12, 13}. Now, find the intersection of these two sets. We have ( R N ) M N or {1, 3, 5, 7, 9, 11, 13}. 101. ( M Q ) R First, find M , the complement of M . We have M {1, 3, 5, 7, 9, 10, 11, 12, 13}. Next, form the union of M and Q . We have M Q = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} . U Thus, we have ( M Q ) R U R R or {0, 1, 2, 3, 4}. 102. Q ( M N ) First, form the union of M and N . We have M N = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13}. Now form the intersection of Q with this set. We have Q ( M N ) M or {0, 2, 4, 6, 8}. 103. Q N U First, find Q , the complement of Q . We have Q {1, 3, 5, 7, 9, 11, 13} . N Now find N , the complement of N . We have N {0, 2, 4, 6, 8, 10, 12} . Q Next, form the intersection of N and U . We have N U Q U Q Finally, we have Q N U Q Q Because the intersection of Q and N U is , Q and N U are disjoint sets. 104. U R Because , U and , U U U we have U R U U R U R U or {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}. 105. | , x x U x M This means all elements of U except those in set M . This gives the set {1, 3, 5, 7, 9, 10, 11, 12, 13}.
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Section R.2 Real Numbers and Their Properties 5 106. | , x x U x R This means all elements of U except those in set R . This gives the set {5, 6, 7, 8, 9, 10, 11, 12, 13}. 107. | and x x M x Q This means all elements that are members of both set M and set Q , or . M Q Note that set M is a subset of set Q because all members of set M are included in set Q . Thus, the answer is set M or {0, 2, 4, 6, 8}. 108. | and x x Q x R This means all elements that are members of both set Q and set R , or . Q R Thus, the answer is {0, 2, 4}. 109. | or x x M x Q This means all elements that are either members of set M or set Q , or . M Q Note that set M is a subset of set Q because all members of set M are included in set Q . Thus, the answer is set Q or {0, 2, 4, 6, 8, 10, 12}. 110. | or x x Q x R This means all elements that are either members of set Q or set R , or . Q R Thus, the answer is {0, 1, 2, 3, 4, 6, 8, 10, 12}. Section R.2 Real Numbers and Their Properties 1. {0, 1, 2, 3, …} describes the set of whole numbers. 2. {…, –3, –2, –1, 0, 1, 2, 3, …} describes the set of integers. 3. In the expression 3 6 , 6 is the base and 3 is the exponent. 4. If the real number a is to the left of the real number b on a number line, then a < (is less than) b . 5. The distance on a number line from a number to 0 is the absolute value of that number. 6. (a) 0 is a whole number. Therefore, it is also an integer, a rational number, and a real number. 0 belongs to B, C, D, F. (b) 34 is a natural number. Therefore, it is also a whole number, an integer, a rational number, and a real number. 34 belongs to A, B, C, D, F. (c) 9 4 is a rational number and a real number. 9 4 belongs to D, F. (d) 36 6 is a natural number. Therefore, it is also a whole number, an integer, a rational number, and a real number. 36 belongs to A, B, C, D, F. (e) 13 is an irrational number and a real number. 13 belongs to E, F . (f) 216 54 100 25 2.16 is a rational number and a real number. 2.16 belongs to D, F. 7. 3 10 1000 8. 2 5 10 2 10 5 5 9. 4 4 10. 7 4 7 28 x y x y For Exercises 11 − 16, 5 12 1 4 8 4 6, , , 3, 0, , 1, 2 , 3, 12 . A 11. 1 and 3 are natural numbers. 12. 0, 1, and 3 are whole numbers. 13. –6, 12 4 (or –3), 0, 1, and 3 are integers. 14. 5 12 1 4 8 4 6, (or 3), , 0, , 1, and 3 are rational numbers. 15. 3, 2 and 12 are irrational numbers. 16. All are real numbers. 17. 4 2 2 2 2 2 16 18. 5 3 3 3 3 3 3 243 19. 4 2 2 2 2 2 16 20. 6 2 2 2 2 2 2 2 64 21. 5 3 3 3 3 3 3 243 22. 5 ( 2) ( 2)( 2)( 2)( 2)( 2) 32 23. 4 2 3 2 3 3 3 3 2 81 162 24. 3 4 5 4 5 5 5 4 125 500
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6 Chapter R Review of Basic Concepts 25. 2 5 12 3 10 12 3 10 4 6 26. 9 3 16 4 27 16 4 27 4 23 27. 3 3 4 9 8 7 2 4 1 7 2 4 1 7 8 4 7 8 4 56 60 28. 4 6 5 3 2 6 5 3 16 30 3 16 30 48 30 48 18 29. 3 (4 2 )( 2 25) (4 8)( 2 5) ( 4)(3) 12 30. 2 3 3 5 3 16 2 5 9 4 2 4 4 8 4 4 16 31. 2 1 5 1 9 4 18 2 8 9 5 9 36 36 18 18 17 4 17 8 25 36 18 36 36 36 32. 5 2 3 11 8 5 2 10 25 16 15 11 9 4 40 40 10 10 40 10 9 16 25 5 40 40 40 8 33. 8 4 6 12 8 24 12 4 3 4 3 8 2 7 6 6 7 7 34. 15 5 4 6 8 3 4 6 8 6 5 8 2 6 5 4 12 6 8 2 8 6 5 4 1 4 6 6 5 5 For Exercises 35 − 48, p = − 4, q = 8, and r = –10. 35. 2 2 2 2 2 7 4 7 8 10 4 7 8 100 16 7 8 100 16 56 100 72 100 28 p q r 36. 2 2 2 4 2 8 10 16 2 8 10 16 16 10 32 10 42 p q r 37. 8 ( 10) 2 1 8 ( 4) 4 2 q r q p 38. 4 ( 10) 14 7 4 8 4 2 p r p q 39. 3 5 3 8 5 24 5 12 5 10 4 10 4 5 4 12 5 48 25 23 5 4 20 20 20 q r p 40. 3 2 3( 10) 2 30 2 8 10 8 10 15 1 75 4 71 4 5 20 20 20 r q r 41. 5 5( 10) 50 2 3 2( 4) 3( 10) 8 30 50 25 22 11 r p r 42. 3 3 8 24 3 2 3 4 2 10 12 2 10 24 24 24 3 12 20 12 20 8 q p r 43. 8 10 10 22 4 11 2 3 2 3 3 3 3 3( 4) 8 3 1 2 3 4 8 4 8 q r p q 44. 8 10 10 2 4 5 4 5 5 4 8 8 2 2 2 2 2 2 2 2 2 2 4 2 2 4 2 2 q r p q
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Section R.2 Real Numbers and Their Properties 7 45. 2 2 2 2 3 4 2 3 10 2 2 8 2 3 10 6 4 3 10 4 30 6 6 4 30 26 13 6 6 3 p r q 46. 2 2 2 6 2 8 6 2 4 4 4 4 2 8 4 8 4 4 8 4 1 8 2 q p p 47. 3 3 3 3 4 3 4 4 3 3 4 8 10 8 12 3 0 12 6 2 2 p p r 48. 3 3 3 5 8 2 1 4 5 2 1 3 10 3 5 8 2 27 5 8 2 3 7 7 40 2 27 40 54 7 7 14 2 7 q p r 49. distributive 50. distributive 51. inverse 52. inverse 53. identity 54. identity 55. commutative 56. commutative 57. associative 58. associative 59. closure 60. closure 61. No; in general . a b b a Examples will vary, i.e. if a = 15 and b = 0, then a − b = 15 − 0 = 15, but b − a = 0 − 15 = − 15. 62. No; in general a b c a b c . Examples will vary, i.e. if a = 15, b = 0, and c = 3, then 15 0 3 15 3 12, a b c but 15 0 3 15 3 18. a b c 63. 10 10 22 22 20 11 11 z z z 64. 3 3 3 12 12 12 9 4 4 4 r r r r 65. 5 6 5 6 11 m m m 66. 8 7 8 7 8 7 15 (or 15) a a a a a 67. 3 16 32 40 8 9 27 9 3 16 3 32 3 40 8 9 8 27 8 9 3 16 3 32 5 8 9 8 27 3 2 4 5 + 3 9 3 y z y z y z y z 68. 1 20 8 32 4 1 1 1 20 8 32 4 4 4 1 1 1 20 8 32 4 4 4 5 + 2 8 = 5 2 8 m y z m y z m y z m y z m y z 69. 8 14 (8 14) 6 p p p p 70. 15 10 15 10 5 x x x x 71. 4 4 4 4 4 z y z y z y 72. 3 3 3 3 3 m n m n m n 73. The process in your head should be the following: 72 17 28 17 (72 28) 17 (100) 17 1700 74. The process in your head should be the following: 32 80 32 20 32 80 20 32 100 3200 75. The process in your head should be the following: 5 1 5 1 5 5 1 123 1 23 1 123 23 1 8 2 8 2 8 8 2 1 (100) 1 150 2
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8 Chapter R Review of Basic Concepts 76. The process in your head should be the following: 2 3 2 3 2 3 3 17 14 17 4 17 14 4 5 4 5 4 5 4 4 2 17 10 17.4 10 5 174 77. This statement is false because 6 8 2 2 and 6 8 6 8 2. A corrected statement would be 6 8 6 8 or 6 8 8 6 78. This statement is false because 3 3 27 27 and 3 3 27 27 . A corrected statement would be 3 3 3 3 or 3 3 3 3 . 79. This statement is true because 5 6 5 6 30 and 5 6 30 30 . 80. This statement is true because 14 14 7 2 2 and 14 7 7 2 . 81. This statement is false. For example if you let 2 a and 6 b then 2 6 4 4 and 2 6 2 6 4 a b . A corrected statement is a b b a , if b > a > 0. 82. This statement is true by the algebraic definition of absolute value stated in the text. 83. 10 10 84. 15 15 85. 4 4 7 7 86. 7 7 2 2 87. 8 8 88. 12 12 89. 120 116 120 4 4 d P P The d P value for a woman whose actual systolic blood pressure is 116 and whose normal value should be 120 is 4. 90. Consider the relation, 120 17. d P P Because 103 and 137 both differ from 120 by 17, these are the two possible values for the patient’s systolic blood pressure. For Exercises 91 − 98, x = − 4 and y = 2. 91. 3 2 3( 4) 2(2) 12 4 16 16 x y 92. 2 5 2 4 5 2 8 10 18 18 x y 93. 3 4 3( 4) 4(2) 12 8 20 20 x y 94. 5 5 2 4 10 4 14 14 y x 95. 2 3 2 2 3 4 4 2 2 2 3 4 4 12 8 1 8 8 8 y x xy 96. 4 4 4 4 4 2 4 4 4 4 2 16 8 24 6 4 4 4 x y x 97. 8 8(2) ( 4) 16 ( 4) 4 4 20 20 5 4 4 y x x 98. 2 4 2 2 4 2 2 8 2 4 4 4 x y x 99. True because 25 25 and 25 25. 100. True because the absolute value of any number is always greater than or equal to 0. 101. False. 5 13 8 8 while 5 13 5 13 18. 102. False. 8 12 4 4 while 8 12 8 12 4. 103. True because 11 6 11 6 66 and 66 66. 104. True because 10 5 5 2 and 10 10 5. 2 2
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Section R.2 Real Numbers and Their Properties 9 105. , 1 4 1 4 3 3 d P Q or , 4 1 4 1 3 3 d P Q 106. , 8 4 8 4 12 12 d P R or , 4 8 12 12 d P R 107. , 8 1 8 1 9 9 d Q R or , 1 8 9 9 d Q R 108. , 12 1 13 13 d Q S or , 1 12 13 13 d Q S 109. 0 xy if x and y have the same sign. 110. 2 0 x y if y is positive, because 2 x is positive for any nonzero x . 111. 0 x y if x and y have different signs. 112. 2 0 y x if x is negative, because 2 y is positive for any nonzero y . 113. Because 3 x has the same sign as x , 3 0 x y if x and y have the same sign. 114. 0 x y if x and y have different signs. 115. Because 18 ( 1) 19 19 and 1 ( 18) 19 19, the number of strokes between their scores is 19. 116. 18, 355 3224 15 21, 564 yards. No, it is not the same, because the sum of the absolute values is 18, 355 3224 15 18, 355 3224 15 21, 594 The fact that | –15 | = 15 changes the two answers. 117. BAC 48 3.2 0.075 190 2 0.015 0.031 118. BAC 36 4.0 0.075 135 3 0.015 0.035 119. BAC 3 20 3.8 0.075 200 4 0.015 0.026 If the man’s weight was greater, the BAC would be lower. For example, if the man weighed 250 pounds, his BAC would be 3 20 3.8 0.075 250 4 0.015 0.0084. 120. BAC 2 6 14 0.075 150 2 0.015 0.054 If the woman consumed the same two glasses of wine over a longer period of time, her BAC would be lower. For example, if she drank the wine over 3 hours, her BAC would be 2 6 14 0.075 150 3 0.015 0.039.
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10 Chapter R Review of Basic Concepts For exercises 121 − 132, we use the formula 250 1000 12.5 6.25 1250 , 3 C T Y I A A A A R where 0 0.775, 0 0.11875, 0 12.5, C T Y A A A and 0 0.095. I A 121. First, find the value of the numerator. 250 1000 12.5 6.25 1250 304 34 3705 9 250 1000 12.5 6.25 1250 339.73 435 435 435 435 C T Y I A A A A 339.73 113.2 3 R 122. First, find the value of the numerator. 250 1000 12.5 6.25 1250 341 38 4381 5 250 1000 12.5 6.25 1250 336.56 520 520 520 520 C T Y I A A A A 336.56 112.2 3 R 123. First, find the value of the numerator. 250 1000 12.5 6.25 1250 408 32 4952 9 250 1000 12.5 6 6.25 1250 309.95 608 608 408 608 C T Y I A A A A 309.95 103.3 3 R 124. First, find the value of the numerator. 250 1000 12.5 6.25 1250 395 39 4727 15 250 1000 12.5 6.25 1250 304.55 597 597 597 597 C T Y I A A A A 304.55 101.5 3 R 125. First, find the value of the numerator. 250 1000 12.5 6.25 1250 373 33 4109 9 250 1000 12.5 6.25 1250 292.10 582 582 582 582 C T Y I A A A A 292.10 97.4 3 R
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Section R.2 Real Numbers and Their Properties 11 126. First, find the value of the numerator. 250 1000 12.5 6.25 1250 456 33 4952 17 250 1000 12.5 6.25 1250 291.00 659 659 659 659 C T Y I A A A A 291.00 97.0 3 R 127. First, find the value of the numerator. 250 1000 12.5 6.25 1250 380 40 4761 16 250 1000 12.5 6.25 1250 289.55 616 616 616 616 C T Y I A A A A 289.55 96.5 3 R 128. First, find the value of the numerator. 250 1000 12.5 6.25 1250 141 11 1626 3 250 1000 12.5 6.25 1250 286.72 224 224 224 224 C T Y I A A A A 286.72 95.6 3 R 129. First, find the value of the numerator. 250 1000 12.5 6.25 1250 197 17 2483 8 250 1000 12.5 6.25 1250 286.02 312 312 312 312 C T Y I A A A A 286.02 95.3 3 R 130. First, find the value of the numerator. 250 1000 12.5 6.25 1250 285 20 3475 7 250 1000 12.5 6.25 1250 284.87 452 452 452 452 C T Y I A A A A 284.87 95.0 3 R 131. First, find the value of the numerator. 250 1000 12.5 6.25 1250 415 28 4694 14 250 1000 12.5 6.25 1250 281.61 628 628 628 628 C T Y I A A A A 281.61 93.9 3 R
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12 Chapter R Review of Basic Concepts 132. First, find the value of the numerator. 250 1000 12.5 6.25 1250 379 31 4286 18 250 1000 12.5 6.25 1250 281.38 570 570 570 570 C T Y I A A A A 281.38 93.8 3 R 133. First, find the value of the numerator. 250 1000 12.5 6.25 1250 343 45 4643 6 250 1000 12.5 6.25 1250 367.38 502 502 502 502 C T Y I A A A A 367.38 122.5 3 R 134. First, find the value of the numerator. 250 1000 12.5 6.25 1250 336 49 4557 10 250 1000 12.5 6.25 1250 363.32 497 497 497 497 C T Y I A A A A 363.32 121.1 3 R 135. First, find the value of the numerator. 250 1000 12.5 6.25 1250 324 36 3900 4 250 1000 12.5 6.25 1250 332.98 492 492 492 492 C T Y I A A A A 332.98 110.0 3 R 136. We use the maximum values for , , and , C T Y A A A but the minimum value for I A (because we want the fewest interceptions possible.) 250 1000 12.5 6.25 1250 250 0.775 1000 0.11875 12.5 12.5 6.25 1250 0 475 C T Y I A A A A 475 158.3 3 R Section R.3 Polynomials 1. The polynomial 5 2 4 x x is a trinomial of degree 5. 2. A polynomial containing exactly one term is a monomial. 3. A polynomial containing exactly two terms is a binomial. 4. In the term 2 6 , x y –6 is the coefficient (or numerical coefficient. 5. A convenient way to find the product of two binomials is to use the FOIL method. 6. True 7. True
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Section R.3 Polynomials 13 8. False. 3 2 6 a a 9. False. 2 2 2 2 x y x xy y 10. False. 2 2 2 2 x x x 11. 5 2 5 2 5 2 7 4 4 4 4 16 16 x x x x x x 12. 4 3 4 3 4 3 7 3 6 3 6 18 18 y y y y y y 13. 6 4 6 4 1 11 n n n n n 14. 8 5 8 5 1 14 a a a a a 15. 3 5 3 5 8 9 9 9 9 16. 2 8 2 8 10 4 4 4 4 17. 4 2 5 4 2 5 4 2 5 11 3 6 4 3 6 4 72 72 m m m m m m m m 18. 3 6 4 3 6 4 3 6 4 13 8 2 5 8 2 5 80 80 t t t t t t t t 19. 2 3 4 2 3 4 2 3 1 4 5 5 5 3 5 3 15 15 x y x y x x yy x y x y 20. 3 2 2 3 1 2 3 1 3 4 4 7 4 7 28 28 xy x y xx y y x y x y 21. 2 2 2 2 1 2 1 2 3 3 1 1 8 8 2 2 4 4 mn m n mm nn m n m n 22. 4 2 4 2 4 1 1 2 5 3 2 2 35 35 7 7 10 10 m n mn m m nn m n m n 23. 2 5 2 5 10 (2 ) 2 2 24. 4 3 4 3 12 (6 ) 6 6 25. 3 3 3 3 2 2 6 6 6 6 6 216 x x x x 26. 5 5 5 5 5 5 25 25 2 2 2 32 x x x x 27. 3 0 2 2 3 2 0 2 2 3 2 0 2 2 6 0 2 6 2 6 6 (4 ) 4 ( ) ( ) 4 4 4 1 4 16 m n m n m n m n m m m 28. 0 4 3 3 0 3 4 3 3 0 3 4 3 3 0 12 12 12 (2 ) 2 ( ) ( ) 2 2 8 1 8 x y x y x y x y y y 29. 3 8 8 3 8 3 24 2 2 3 2 3 6 ( ) ( ) r r r r s s s s 30. 2 4 4 2 4 2 8 2 2 2 ( ) p p p p q q q q 31. 4 2 4 2 4 4 8 8 2 4 4 8 4 8 4 2 4 ( 4) ( ) ( 4) 256 m m m m tp t p t p t p 32. 3 4 3 4 3 3 12 12 2 2 3 6 6 5 ( 5) ( ) ( 5) 125 ( ) n n n n r r r r 33. 0 3 5 1 x y z 34. 0 2 3 3 1 p q r 35. (a) B; 0 6 1 (b) C; 0 6 1 (c) B; 0 6 1 (d) C; 0 6 1 36. (a) D; 0 3 3 1 3 p (b) E; 0 3 3 1 3 p (c) B; 0 3 1 p (d) B; 0 3 1 p 37. 11 5 x is a polynomial of degree 11. It is a monomial because it has one term. 38. 5 4 y is a polynomial of degree 5 and is a monomial. 39. 4 6 3 x x is a polynomial. It has degree 4 because the term 4 3 x has the greatest degree. It is a binomial because there are two terms.
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14 Chapter R Review of Basic Concepts 40. 3 9 5 y y is a polynomial. It has degree 3 because the term 3 5 y has the greatest degree. It is a binomial because there are two terms. 41. 5 3 7 2 1 z z is a polynomial. It has degree 5 because the term 5 7 z has the greatest degree. It is a trinomial because there are three terms. 42. 4 3 9 8 7 t t is a polynomial. It has degree 4 because the term 4 9 t has the greatest degree. It is a trinomial because there are three terms. 43. 2 3 3 8 5 6 15 12 13 12 a b a b b b is a polynomial. It has degree 11 because the term 3 8 12 a b has the greatest degree. It is not a monomial, binomial, or trinomial. 44. 5 7 3 8 9 10 16 12 4 18 x y x y xy x is a polynomial. It has degree 12 because the term 5 7 16 x y has the greatest degree. It is not a monomial, binomial, or trinomial. 45. 5 2 3 1 9 8 x x is not a polynomial because one term has a variable in the denominator. 46. 6 5 2 3 1 3 t t is not a polynomial because one term has a variable in the denominator. 47. 5 is a polynomial of degree 0. It is a monomial. 48. 9 is a polynomial of degree 0. It is a monomial. 49. 2 2 2 2 2 5 4 7 4 3 5 5 4 4 3 7 5 1 1 2 2 x x x x x x x x x x 50. 3 2 3 2 3 2 3 2 3 2 3 3 4 2 6 3 2 3 1 4 6 1 4 10 4 10 m m m m m m m m m m 51. 2 2 2 2 2 2 2 2 12 8 6 4 3 4 2 2 12 2 8 2 6 4 3 4 4 4 2 24 16 12 12 16 8 12 4 y y y y y y y y y y y y y 52. 2 2 2 2 2 2 2 3 8 5 5 3 2 4 3 8 3 5 5 3 5 2 5 4 24 15 15 10 20 9 5 20 p p p p p p p p p p p p p p 53. 4 2 3 2 2 4 2 3 2 2 4 3 2 4 3 2 4 3 2 6 3 2 5 4 6 3 2 5 4 6 2 3 5 1 1 4 1 6 2 7 4 6 2 7 4 m m m m m m m m m m m m m m m m m m m m m m m m m m m m 54. 3 3 2 2 3 3 2 2 3 2 3 2 3 2 8 3 2 4 3 1 8 3 2 4 3 1 8 2 1 4 1 3 3 1 6 3 4 3 1 6 3 4 4 x x x x x x x x x x x x x x x x x x x x x 55. 2 2 (4 1)(7 2) 4 (7 ) 4 (2) 1(7 ) 1(2) 28 8 7 2 28 2 r r r r r r r r r r r 56. 2 2 5 6 3 4 5 3 5 4 6 3 6 4 15 20 18 24 15 2 24 m m m m m m m m m m m 57. 2 2 2 2 1 3 5 3 3 2 1 3 5 3 3 1 2 2 1 3 5 3 5 3 3 3 3 x x x x x x x x x x x 2 2 2 2 2 2 4 3 2 10 2 15 3 9 3 10 2 15 3 3 9 7 2 7 2 15 15 3 9 3 9 x x x x x x x x x x x x x x 58. 3 3 3 2 3 2 5 4 3 1 1 2 3 4 2 1 1 1 1 2 3 2 3 2 4 4 2 3 1 6 4 8 1 1 1 1 6 6 4 8 4 8 m m m m m m m m m m m m m m m m m m
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