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A P P E N D I X A Review of Fundamental Concepts of Algebra Appendix A.1 Review of Real Numbers and Their Properties . . . 1136 Appendix A.2 Exponents and Radicals . . . . . . . . . . . . . . . 1143 Appendix A.3 Polynomials and Factoring Appendix A.4 Rational Expressions Appendix A.5 Solving Equations . . . . . . . . . . . . . . . . . . 1172 Appendix A.6 Linear Inequalities in One Variable . . . . . . . . . 1194 Appendix A.7 Errors and the Algebra of Calculus . . . . . . . . . 1204 . . . . . . . . . . . . . 1150 . . . . . . . . . . . . . . . . 1164 A P P E N D I X A Review of Fundamental Concepts of Algebra Appendix A.1 ■ Review of Real Numbers and Their Properties You should know the following sets. (a) The set of real numbers includes the rational numbers and the irrational numbers. (b) The set of rational numbers includes all real numbers that can be written as the ratio pq of two integers, where q 0. (c) The set of irrational numbers includes all real numbers which are not rational. (d) The set of integers: . . . , 3, 2, 1, 0, 1, 2, 3, . . . (e) The set of whole numbers: 0, 1, 2, 3, 4, . . . (f) The set of natural numbers: 1, 2, 3, 4, . . . ■ The real number line is used to represent the real numbers. ■ Know the inequality symbols. ■ (a) a < b means a is less than b. (b) a ≤ b means a is less than or equal to b. (c) a > b means a is greater than b. (d) a ≥ b means a is greater than or equal to b. Interval Notation Inequality Notation a, b a ≤ x ≤ b a, b Graph a b a b a b a b a < x < b a, b a ≤ x < b a, b a < x ≤ b a, x ≥ a Type x Bounded and Closed x Bounded and Open x Bounded x Bounded x Unbounded x Unbounded x Unbounded x Unbounded x Unbounded a a, x > a a , b x ≤ b b , b x < b b , < x < a, if a ≥ 0 . a, if a < 0 ■ You should know that a ■ Know the properties of absolute value. (a) a ≥ 0 1136 (b) a a (c) ab a b (d) a a ,b0 b b —CONTINUED— Appendix A.1 Review of Real Numbers and Their Properties ■ The distance between a and b on the real line is da, b b a a b . ■ You should be able to identify the terms in an algebraic expression. ■ You should know and be able to use the basic rules of algebra. ■ Commutative Property (a) Addition: a b b a ■ (b) Multiplication: a Associative Property (a) Addition: a b c a b c ■ (b) Multiplication: abc abc Identity Property (a) Addition: 0 is the identity; a 0 0 a a. ■ bba (b) Multiplication: 1 is the identity; a 1 1 a a . Inverse Property (a) Addition: a is the additive inverse of a; a a a a 0. (b) Multiplication: 1a is the multiplicative inverse of a, a 0; a1a 1aa 1 . ■ Distributive Property (a) ab c ab ac ■ ■ ■ ■ (b) a bc ac bc Properties of Negation (a) 1a a (b) a a (c) ab ab ab (d) ab ab (e) a b a b a b Properties of Equality (a) If a b, then a ± c b ± c. (b) If a b, then ac bc. (c) If a ± c b ± c, then a b. (d) If ac bc and c 0, then a b. Properties of Zero (a) a ± 0 a (b) a 0 0 (d) a0 is undefined. (e) If ab 0, then a 0 or b 0. (c) 0 a 0a 0, a 0 Properties of Fractions b 0, d 0 (a) Equivalent Fractions: ab cd if and only if ad bc. (b) Rule of Signs: ab ab ab and ab ab (c) Equivalent Fractions: ab acbc, c 0 (d) Addition and Subtraction 1. Like Denominators: ab ± cb a ± cb 2. Unlike Denominators: ab ± cd ad ± bcbd (e) Multiplication: ab cd acbd (f) Division: ab cd ab dc adbc if c 0. Vocabulary Check 1. rational 2. irrational 3. absolute value 4. composite 5. prime 6. variables; constants 7. terms 8. coefficient 9. zero-factor property 1137 1138 Appendix A Review of Fundamental Concepts of Algebra 7 2 1. 9, 2, 5, 3, 2, 0, 1, 4, 2, 11 7 5 2. 5, 7, 3, 0, 3.12, 4, 3, 12, 5 (a) Natural numbers: 5, 1, 2 (a) Natural numbers: 12, 5 (b) Whole numbers: 0, 5, 1, 2 (b) Whole numbers: 0, 12, 5 (c) Integers: 9, 5, 0, 1, 4, 2, 11 (d) Rational numbers: 9, 72, 5, 2 3, (c) Integers: 7, 0, 3, 12, 5 7 5 (d) Rational numbers: 7, 3, 0, 3.12, 4, 3, 12, 5 0, 1, 4, 2, 11 (e) Irrational numbers: 2 (e) Irrational numbers: 5 4. 2.3030030003 . . . , 0.7575, 4.63, 10, 75, 4 3. 2.01, 0.666 . . . , 13, 0.010110111 . . . , 1, 6 (a) Natural numbers: 1 (a) Natural numbers: 4 (b) Whole numbers: 1 (b) Whole numbers: 4 (c) Integers: 13, 1, 6 (c) Integers: 75, 4 (d) Rational numbers: 2.01, 0.666 . . . , 13, 1, 6 (d) Rational numbers: 0.7575, 4.63, 75, 4 (e) Irrational numbers: 0.010110111 . . . (e) Irrational numbers: 2.3030030003 . . . , 10 1 6. 25, 17, 12 5 , 9, 3.12, 2 , 7, 11.1, 13 5. , 13, 63 , 12 2, 7.5, 1, 8, 22 (a) Natural numbers: 6 3 (a) Natural numbers: 25, 9, 7, 13 (since it equals 2), 8 (b) Whole numbers: 63, 8 (c) Integers: 6 3, (b) Whole numbers: 25, 9, 7, 13 (c) Integers: 25, 17, 9, 7, 13 1, 8, 22 (d) Rational numbers: 13, 63, 7.5, 1, 8, 22 (d) Rational numbers: 25, 17, 12 5 , 9, 3.12, 7, 11.1, 13 1 (e) Irrational numbers: , 2 2 1 (e) Irrational numbers: 2 7. 5 8 0.625 8. 1 3 0.3 9. 41 333 0.123 −8 14. 3.5 < 1 15. 3 2 < 7 − 3.5 −4 −3 −2 −1 0 5 6 > 16. 1 < −7 −6 −5 −4 16 3 3 2 1 1 17. 0.54 13. 4 > 8 12. 6 < 2.5 11. 1 < 2.5 6 11 10. 2 3 16 3 2 3 4 5 6 0 7 1 2 3 4 5 6 18. 87 < 37 − 87 2 5 3 6 0 −2 1 19. (a) The inequality x ≤ 5 denotes the set of all real numbers less than or equal to 5. (b) x 0 1 2 3 4 5 6 (c) The interval is unbounded. −1 − 37 0 20. (a) The inequality x ≥ 2 denotes the set of all real numbers greater than or equal to 2. (b) x −4 −3 −2 −1 0 (c) The interval is unbounded. Appendix A.1 21. (a) The inequality x < 0 denotes the set of all negative real numbers. (b) −1 0 1 (b) 3 4 5 6 2 4 5 6 24. (a) , 2 denotes the set of all real numbers less than 2. x 0 7 1 2 3 4 (c) The interval is unbounded. (c) The interval is unbounded. 25. (a) The inequality 2 < x < 2 denotes the set of all real numbers greater than 2 and less than 2. (b) 3 (b) x 2 1 (c) The interval is unbounded. 23. (a) The interval 4, denotes the set of all real numbers greater than or equal to 4. 1 x 0 2 (c) The interval is unbounded. 1139 22. (a) The inequality x > 3 denotes the set of all real numbers greater than 3. (b) x −2 Review of Real Numbers and Their Properties 26. (a) The inequality 0 ≤ x ≤ 5 denotes the set of all real numbers greater than or equal to zero and less than or equal to 5. x −2 −1 0 1 2 (b) x 0 1 2 3 4 5 (c) The interval is bounded. (c) The interval is bounded. 27. (a) The inequality 1 ≤ x < 0 denotes the set of all negative real numbers greater than or equal to 1. (b) (b) x −1 (b) 2 3 4 5 2 3 4 5 6 30. (a) The interval 1, 2 denotes the set of all real numbers greater than 1 and less than or equal to 2. (b) x 1 1 (c) The interval is bounded. 29. (a) The interval 2, 5 denotes the set of all real numbers greater than or equal to 2 and less than 5. 0 x 0 0 (c) The interval is bounded. −2 −1 28. (a) The inequality 0 < x ≤ 6 denotes the set of all real numbers greater than zero and less than or equal to 6. x −2 6 (c) The interval is bounded. −1 0 1 2 (c) The interval is bounded. 31. 2 < x ≤ 4 32. 6 ≤ y < 0 33. y ≥ 0 34. y ≤ 25 35. 10 ≤ t ≤ 22 36. 3 ≤ k < 5 37. W > 65 38. 2.5% ≤ r ≤ 5% 39. 10 10 10 42. 4 1 3 3 45. 5 5 5 1 5 5 5 41. 3 8 5 5 5 44. 3 3 3 3 6 40. 0 0 43. 1 2 1 2 1 46. 3 3 33 9 47. If x < 2, then x 2 is negative. Thus x 2 x 2 1. x2 x2 1140 Appendix A Review of Fundamental Concepts of Algebra 49. 3 > 3 since 3 > 3. 48. If x > 1, then x 1 is positive. x1 x1 Thus, 1. x1 x1 51. 5 5 since 5 5. 52. 6 < 6 since 6 6 and 6 6 6. 54. 2 > 2 since 2 2. 55. d126, 75 75 126 51 50. 4 4 since 4 4 and 4 4. 53. 2 2 since 2 2. 56. d126, 75 75 126 51 57. d 52, 0 0 52 52 58. d14, 11 4 11 4 112 59. d16 5 , 75 14 52 112 75 128 16 5 75 60. d9.34, 5.65 5.65 9.34 14.99 61. Budgeted Expense, b $112,700 Actual Expense, a a b 0.05b $113,356 $656 0.05112,700 $5635 Since $656 < $5635 but $656 > $500, the actual expense does not pass the “budget variance test.” 62. Budgeted Expense, b $9400 Actual Expense, a a b 0.05b $9772 $372 0.059400 $470 Since $372 < $470 and $372 < $500, the actual expense does pass the “budget variance test.” 63. Budgeted Expense, b $37,640 Actual Expense, a a b 0.05b $37,335 $305 0.0537,640 $1882 Since $305 < $500 and $305 < $1882, the actual expense passes the “budget variance test.” 64. Budgeted Expense, b $2575 Actual Expense, a a b 0.05b $2613 $38 0.052575 $128.75 $590.9 1990 $1253.2 2000 $1788.8 96 48 236.4 (s) 73.8 (d) 144 Year 2000 1980 192 1990 $195.6 240 1980 1970 92.5 92.2 $0.3 surplus 192.8 195.6 $2.8 deficit 517.1 590.9 $73.8 deficit 1032.0 1253.2 $221.2 deficit 2025.2 1788.8 $236.4 surplus (b) 0.3 (s) 2.8 (d) $92.2 Surplus or Deficit (in billions) 1970 1960 Expenditures (in billions) 1960 Year Surplus or deficit (in billions) 65. (a) 221.2 (d) Since $38 < $500, and $38 < $128.75, the actual expense passes the “budget variance test.” Appendix A.1 Review of Real Numbers and Their Properties 66. Total: 2213 3290 4666 5665 9784 25,618 Under 35 8.64% 2213 0.0863857 8.63857% 25,618 Under 35: 8.63857% of 360 31.1 3290 0.1284253 12.84253% 25,618 35–44: 35−44 12.84% 65 and older 38.19% 45−54 18.21% 55−64 22.11% 12.84253% of 360 46.2 4666 0.18213756 18.213756% 25,618 45–54: 18.213756% of 360 65.6 5665 0.221133578 22.1133578% 25,618 55–64: 22.1133578% of 360 79.6 65 and older: 9784 0.38191896 38.191896% 25,618 38.191896% of 360 137.5% 68. dx, 10 x 10 , and dx, 10 ≥ 6, thus, x 10 ≥ 6. 70. d y, a y a and d y, a ≤ 2, thus y a ≤ 2. 67. dx, 5 x 5 and dx, 5 ≤ 3, thus x 5 ≤ 3. 69. d y, 0 y 0 y and d y, 0 ≥ 6, thus y ≥ 6. 71. d326, 351 351 326 25 miles 73. 7x 4 72. d48, 82 82 48 34 75. 3x2 8x 11 74. 6x 3 5x Terms: 7x, 4 Terms: 6x 3, 5x Terms: 3x2, 8x, 11 Coefficient: 7 Coefficients: 6, 5 Coefficients: 3, 8 76. 3 3x2 1 77. 4x 3 Terms: 3 3x2 , 1 Coefficients: 3 3 x 5 2 x Terms: 4x 3, , 5 2 Coefficients: 4, 79. 4x 6 78. 3x 4 x2 4 Terms: 3x 4, 1 2 Coefficients: 3, 1 4 80. 9 7x (a) 41 6 4 6 10 (a) 9 73 9 21 30 (b) 40 6 0 6 6 (b) 9 73 9 21 12 81. x2 3x 4 x2 4 82. x2 5x 4 (a) 22 32 4 4 6 4 14 (a) 12 51 4 1 5 4 10 (b) 22 32 4 4 6 4 2 (b) 12 51 4 1 5 4 0 1141 1142 83. Appendix A Review of Fundamental Concepts of Algebra x1 x1 (a) 84. 11 2 11 0 Division by zero is undefined (b) 86. 2 x x2 85. x 9 9 x (a) 2 2 1 22 4 2 (b) 2 2 2 2 0 1 1 0 0 1 1 2 Division by 0 is undefined. 12 1 87. 89. 2x 3 2x 6 1 h 6 1, h 6 h 6 Associative Property of Addition Associative Property of Multiplication 712 Associative Property of Multiplication 12 Multiplicative Inverse Property 5 8 1 3 16 16 16 2 99. 5 8 98. 5 10 4 9 3 12 16 15 24 24 24 24 8 100. x 8x 3x 5x 2x 3 4 12 12 12 104. n 1 0.5 0.01 0.0001 0.000001 5n 5 10 500 50,000 5,000,000 (b) The value of 5n approaches infinity as n approaches 0. 107. False. If a < b, then 6 4 64 2 7 7 7 7 10 11 6 60 12 13 59 33 13 66 66 66 66 66 102. 6 1 4 101. 12 4 12 1 12 4 48 105. (a) Distributive Property Multiplicative Identity Property 97. 103. 95. 3t 4 3 t 3 4 Commutative Property of Multiplication 12 Multiplicative Identity Property 93. x y 10 x y 10 Right Distributive Property 1 91. 1 1 x 1 x Additive Identity Property 92. z 5x z x 5 x 96. 17 7 12 17 Additive Inverse Property 90. z 2 0 z 2 Distributive Property 3xy 88. x 3 x 3 0 Multiplicative Inverse Property Multiplicative Inverse Property 94. x3y x 3y Commutative Property of Addition 1 1 > , where a b 0. a b 5x 6 106. (a) 48 6 12 3 2 9 5x 3 1 5x 9 27 n 1 10 100 10,000 100,000 5n 5 0.5 0.05 0.0005 0.00005 (b) The value of 5n approaches 0 as n increases without bound. 108. False. The denominators cannot be added when adding fractions. Appendix A.2 (b) u v ≤ u v Exponents and Radicals 1143 109. (a) u v u v if u is positive and v is negative or vice versa. They are equal when u and v have the same sign. If they differ in sign, u v is less than u v . 110. Yes. y is nonnegative if y ≥ 0. y is positive if y > 0. 111. The only even prime number is 2, because its factors are itself and 1. 112. 113. (a) Since A > 0, A < 0. The expression is negative. Negative integers . . . , −3, −2, −1, Irrational numbers Integers . . . , −3, −2, −1, 0, 1, 2, 3, . . . Real numbers Rational numbers Noninteger fractions (positive and negative) Whole numbers 0, 1, 2, . . . (b) Since B < A, B A < 0. The expression is negative. Natural numbers 1, 2, 3, . . . Zero 114. (a) Since C < 0, C > 0. The expression is positive. (b) Since A > C, A C > 0. The expression is positive. Appendix A.2 ■ 115. Yes, if a is a negative number, then a is positive. Thus, a a if a is negative. Exponents and Radicals You should know the properties of exponents. (a) a1 a (b) a0 1, a 0 (c) a ma n a mn (d) aman amn, a 0 (e) an 1an 1an, a 0 (f) (a m)n a mn (g) (ab)n anbn (h) (ab)n a nb n, b 0 (i) (abn ba n, a 0, b 0 (j) a2 a 2 a2 ■ You should be able to write numbers in scientific notation, c 10n, where 1 ≤ c < 10 and n is an integer. ■ You should be able to use your calculator to evaluate expressions involving exponents. ■ You should know the properties of radicals. n am (a) n a , a > 0 m (c) n a n b ab, b 0 n n a (e) a n n a (b) (d) n b n ab m n a mn a n an a . (f) For n even, n an a. For n odd, n a (g) a1n ■ n a n a m, a ≥ 0 (h) a mn You should be able to simplify radicals. (a) All possible factors have been removed from the radical sign. (b) All fractions have radical-free denominators. (c) The index for the radical has been reduced as far as possible. ■ You should be able to use your calculator to evaluate radicals. m 1144 Appendix A Review of Fundamental Concepts of Algebra Vocabulary Check 1. exponent; base 2. scientific notation 3. square root 4. principal nth root 5. index; radicand 6. simplest form 7. conjugates 8. rationalizing 9. power; index 1. 85 8 8 8 8 8 2. 27 2222222 3. 4.94.94.94.94.94.9 4.96 4. 1010101010 105 5. (a) 32 3 33 27 6. (a) (b) 3 33 34 81 (b) 8. (a) 23 322 23 2 26 (b) 3 5 3 5 3 2 9. (a) 34 64 81 5184 33 13 3 5 3 5 3 5 4 32 4 22 22 31 52 1 32 (b) 32 9 3 44 31 4 44 1 35 34 41 332 523 32 1 4 4 31 16 3 32 32 1 25 32 11. (a) 21 31 1 1 3 2 5 2 3 6 6 6 (b) 212 212 22 4 12. (a) 31 22 1 1 4 3 7 3 4 12 12 12 (b) 322 34 13. 4352 6425 1600 1 1 34 81 36 729 2.125 73 343 14. 84103 0.244 15. 17. When x 2, 18. When x 4, 32 24. 3 20. When x 3, 3 7x2 742 2x3 7 7 . 42 16 4x2 4 12 414 1. 43 4334 5184 34 6x0 6100 61 6. 22. When x 2, 23 227 54. 3 527 135. 23. When x 12, 16. 19. When x 10, 21. When x 3, 5x 53 3 2 43 35 243 43 64 (b) 3225 (b) 20 1 3x3 7. (a) 330 1 1 1 32 32 2 34 3 9 32 2 1 10. (a) 55 53 125 52 3x 4 324 316 48. 1 24. When x 3, 25. (a) 5z3 53z3 125z3 1 5x3 5 13 5 27 275 . 3 (b) 5x 4x2 5x 42 5x6 Appendix A.2 26. (a) 3x2 32x2 9x2 (b) 4x30 40 27. (a) 6y 22y02 6y 22 x0 1, x0 (b) 28. (a) z33z4 13z33z4 1 3 29. (a) z34 3z7 (b) 25y 8 5 84 5 4 (b) y y 10y 4 2 2 30. (a) (b) 1 r4 r 46 r2 2 r6 r 3 4 y 3 y 4 43 3 y 32. (a) 2x 50 1, x 0 1 z 24 34. (a) 4y28y 4 48 y42 32y 2 36. (a) x3y4 5 x2 x3 xn xn 3 (b) 3 ab ab 3 5xy 3 3 4 a3 b3 7x 2 7 7x 23 7x1 x3 x 4 12x y3 4 x y31 x y2 9x y 3 3 1 1 2x22 4x4 (b) 2x22 a33 10 x 1 (b) 8x6 2 x 3 4x3 10 x 32n 3n2n 33n 2 125x9 y12 a3 (b) 35. (a) 3n x2n 1 x2n3n x1 x3n x 3 12 6y 24 24y 2 3x 5 3x 53 3x 2 x3 33. (a) 2x23 4x31 (b) z 23z 21 z 24 (b) 1145 31. (a) x 50 1, x 5 34 64 81 5184 34 7 y4 y y Exponents and Radicals ab ba ba ba ab 3 2 2 3 5 2 3 5 37. 57,300,000 5.73 107 square miles a0 b3 b33 b0 1 38. 9,460,000,000,000 9.460 1012 kilometers 39. 0.0000899 8.99 105 gram per cubic centimeter 40. 0.00003937 3.937 105 inch 41. 4.568 109 4,568,000,000 ounces 42. 1.5 107 15,000,000 degrees Celsius 43. 1.6022 1019 0.00000000000000000016022 coulomb 44. 9.0 105 0.00009 meter 45. (a) 25 108 5 104 50,000 3 8 1015 2 105 200,000 (b) 46. (a) 1.2 1075 103 6.0 104 60,000 (b) 6.0 108 2.0 1011 3.0 103 47. (a) 750 1 (b) 0.11 365 800 954.448 67,000,000 93,000,000 30,769,230,769.2 0.0052 3.077 1010 1146 Appendix A Review of Fundamental Concepts of Algebra 48. (a) 9.3 10636.1 104 4.907 1017 (b) 2.414 1046 1.479 1.68 1055 3 6.3 104 39.791 (b) 50. (a) 2.65 10413 0.064 51. (a) 9 3 (b) 9 104 0.03 53. (a) 3235 (b) 16 81 34 64 1 55. (a) (b) 49. (a) 4.5 109 67,082.039 1 1 5 3235 32 1 81 16 13 32 (b) 34 3 81 16 3 3 27 3 3 2 3 32 25 (b) 36 32 216 3 2 54. (a) 10032 100 3 27 8 (b) 3 64 4 6413 25 3 8 1 1 (2)3 8 4 278 3 27 3 52. (a) 2713 56. (a) 94 125 27 (b) 5 32 5 32 2 2 12 49 13 1 125 43 12 103 4 9 27 125 13 1 1000 2 3 3 27 3 3 5 5 3 125 12543 3 125 125134 4 54 625 57. (a) 57 7.550 59. (a) 12.41.8 0.011 3 452 12.651 58. (a) 5 27 3 27 35 (b) (b) 532.5 0.005 6 125 2.236 (b) 7.225 60. (a) (b) 7 4.13.2 3.495 2 13 3 32 3 2 133 3 4 61. (a) 433 41 4 3 5 96x5 5 32x5 (b) 5.906 3 2 (b) 24 3 8 1627 75 (b) 4 64. (a) 42 22 3 3 4 3x24 4 34x8 (b) 3x 2 3 3 23 x 3 2 2 2 3 3 3 3 65. (a) 72x3 36x2 22 2 32 x y22 3y 26x (b) 32a4 222 2 a22 b2 b2 4a2 b 2 3 16x5 3 8x3 67. (a) 2x2 (b) 3 182 z2 z 18 zz 4 3x4y2 3x4y 214 68. (a) 314 x y12 75x2 y4 25x2 3 y4 18z 2 3 2x2 2x (b) 75x2y4 2x 6x2x 3 53 52 3 3 3 8 3 2 3 66. (a) 54xy4 6 3 36 6 5 3 2x 2 315 63. (a) 8 4 62. (a) 12 5 x 3 y2 4 3y x (b) 5 160x8z4 160x8z415 25 5x5 x3 z415 2x5x3z415 5 5x3z4 2x Appendix A.2 69. (a) 250 128 225 2 124 2 252 1222 102 242 342 (b) 1032 618 1016 70. (a) 427 75 432 Exponents and Radicals 2 69 2 104 2 632 402 182 222 3 52 3 71. (a) 5x 3x 2x (b) 29y 10y 23y 10y 4 33 53 123 53 6y 10y 4y 73 3 16 3 3 54 3 2 (b) 3 2 23 3 33 3 2 3 3 2 2 3 3 2 9 3 2 2 3 2 11 72. (a) 849x 14100x 872 x 14102 x 8 7x 14 10x 56x 140x 84x (b) 348x2 775x2 33 42 x2 73 52 x2 12 x3 35x3 23x3 3 4 x 3 7 5 x 3 73. (a) 3x 1 10x 1 13x 1 (b) 780x 2125x 716 5x 225 5x 745x 255x 285x 105x 185x 74. (a) x3 7 5x3 7 4x3 7 (b) 11245x3 945x3 115 72 x x2 95 32 x x2 11 7x5x 9 3x5x 77x5x 27x5x 50x5x 75. 5 3 3.968 and 76. 5 3 8 2.828 113 3 77. 32 22 9 4 11 13 3.606 Thus, 5 3 > 5 3. Thus, 5 > 32 22. 78. 32 42 9 16 79. 25 5 1 3 1 3 3 3 3 3 80. 82. Thus, 5 32 42. 81. 5 10 5 10 10 10 510 10 10 2 2 5 3 2 5 3 25 3 2 5 3 5 3 2 5 3 5 3 5 3 5 2 32 25 3 22 11 3 5 6 3 5 6 5 6 5 6 35 6 35 6 35 6 36 5 56 1 1147 1148 83. 85. 86. Appendix A 8 2 4 2 5 3 7 3 4 2 22 3 Review of Fundamental Concepts of Algebra 2 5 3 3 7 3 4 2 1 2 2 5 3 5 3 7 3 7 3 2 84. 2 2 3 3 64 4, Given 88. 6413 4, Answer 5 32 2, 89. Answer 3215 2, Given 90. 144 12, Answer 14412 12, Given 3 216 6, 91. Given 21613 6, Answer 5 243 3, 92. Answer 24315 3, Given 4 813 27, 93. Given 8134 27, Answer 4 165 32, 94. Answer 1654 32, Given 2x232 232x232 212x4 212x4 x 121323 x3 x43y23 x43y23 13 13 x 33y13 xy13 xy13 x y 98. 512 5x52 512 5x52 5x32 532x32 1 , x > 0 x3 4 32 324 312 3 99. (a) 32 (3212)12 4 32 4 16 3214 x 512x52 51x , x > 0 532x32 5 6 x3 x36 x12 x 100. (a) 6 3 x 14 x 146 x 123 x 12 (b) (b) 96. 2 232x3 23212 x34 21x1 212x4 x x3 x 12 x 12 x1 x 32 x1 x 32 x 3 101. (a) 2 Rational Exponent Form 912 3, Answer 97. 3 2 2 32 53 2 35 3 35 3 87. 9 3, Given 2 2 1 79 47 3 47 3 27 3 Radical Form 95. 4 (b) 3x24 3x 2 102. (a) 243x 1 243x 1 1212 4 2 2 2 243x 114 4 2x 2x)1412 2x18 8 2x 4 243x 1 (b) 4 3 81x 1 3 3x 1 4 10a b 10a b 3 7 7 13 12 10a7b16 6 10a a6 b 6 a 10ab Appendix A.2 103. T 2 322 2 161 2 4 Exponents and Radicals 1149 3 104. Size 0.03v; For v : 4 34 Size 0.03 1 0.03 1.57 seconds 2 0.03 3 4 3 2 0.026 inch 105. t 0.031252 12 h52 , 0 ≤ h ≤ 12 (a) h (in centimeters) 106. Time (b) As h approaches 12, t approaches t (in seconds) 0 0 1 2.93 2 5.48 3 7.67 4 9.53 5 11.08 6 12.32 7 13.29 8 14.00 9 14.50 10 14.80 11 14.93 12 14.96 0.031252 8.643 14.96 seconds. Distance 93,000,000 miles Rate 11,180,000 miles per minute 107. True. When dividing variables, you subtract exponents. 8.32 minutes, or 8 minutes 19.1 seconds 109. 1 108. False. When a power is raised to a power, you multiply the exponents: a n k a nk. 110. (a) 3 is also raised to the negative one power so, 3x1 am amm a0, a 0 am 1 . 3x (b) When two powers have the same base, the exponents are added, y 3 y 2 y 5. (c) When a power is raised to a power, exponents are multiplied, a2b34 a8b12. (d) The square of a binomial contains a cross product term, a b2 a2 2ab b2. (e) If x < 0, then 4x2 > 0 but 2x < 0, 4x2 2 x . (f) Radicals can only be added together if they have the same radicand and index: 2 2 22. 1150 Appendix A Review of Fundamental Concepts of Algebra 25 54 0.8 2 111. When any positive integer is squared, the units digit is 0, 1, 4, 5, 6, or 9. Therefore, 5233 is not an integer. 112. 2 5 5 5 25 0.8944 5 Since 0.8 0.8944, 2 5 2 2 5 5 5 and squaring is not equivalent to rationalizing the denominator. Appendix A.3 ■ Polynomials and Factoring Given a polynomial in x, an x n a n1 x n1 . . . a1x a0, where an 0, and n is a nonnegative integer, you should be able to identify the following. (a) Degree: n (b) Terms: an x n, a n1 x n1, . . . , a1 x, a0 (c) Coefficients: a n, a n1, . . . , a1, a0 (d) Leading coefficient: an (e) Constant term: a0 ■ You should be able to add and subtract polynomials. ■ You should be able to multiply polynomials by the Distributive Properties. ■ You should be able to multiply two binomials by the FOIL Method. ■ You should know the special binomial products. (b) u ± v2 u2 ± 2uv v2 (a) (u v)(u v) u2 v2 (c) u ± v3 u3 ± 3u2v 3uv2 ± v3 ■ You should be able to factor out all common factors, the first step in factoring. ■ You should be able to factor the following special polynomial forms. (a) u2 v2 u vu v (b) u2 ± 2uv v2 u ± v2 (c) u3 ± v3 u ± vu2 uv v2 ■ You should be able to factor by grouping. ■ You should be able to factor some trinomials by grouping. Vocabulary Check 1. n; an; a0 2. descending 3. monomial; binomial; trinomial 4. like terms 5. First terms; Outer terms; Inner terms; Last terms 6. factoring 7. completely factored 1. (d) 12 is a polynomial of degree zero. 2. (e) 3x 5 2x 3 x is a polynomial of degree five. 3. (b) 1 2x3 2x3 1 is a binomial with leading coefficient 2. 4. (a) 3x2 is a monomial of positive degree. 2 5. (f) 3x4 x2 10 is a trinomial 2 with leading coefficient 3. 6. (c) x 3 3x2 3x 1 is a third-degree polynomial with leading coefficient 1. Appendix A.3 7. 2x3; 2x3 5; 2x3 4x 2 3x 20, etc. (Answers will vary.) 10. 20x 3 5 (Answers will vary.) 8. 6x 5 3x 1 (Answers will vary.) 1 11. (a) Standard form: 2 x 5 14x 1 2 14. (a) Standard form: 7x (b) Degree: 1 (c) Monomial 16. (a) Standard form: 25y 2 y 1 (b) Degree: 0 (c) Monomial (c) Trinomial 19. (a) Standard form: 4x 5 6x 4 1 (b) Degree: 1 Leading coefficient: 4 Leading coefficient: 2 21. (a) Standard form: 4x3y (b) Degree: 4 (add the exponents on x and y) Leading coefficient: 4 (c) Binomial (c) Trinomial Leading coefficient: 1 (c) Binomial 20. (a) Standard form: 2x 3 (b) Degree: 5 18. (a) Standard form: t 2 9 (b) Degree: 2 Leading coefficient: 3 Leading coefficient: 25 Leading coefficient: 1 (c) Binomial 17. (a) Standard form: 3 (b) Degree: 2 15. (a) Standard form: x5 1 (b) Degree: 5 Leading coefficient: 7 (c) Trinomial Leading coefficient: 2 (c) Trinomial (c) Binomial Leading coefficient: 3 12. (a) Standard form: 2x 2 x 1 (b) Degree: 2 Leading coefficient: (b) Degree: 4 1151 9. 15x 4 1; 3x 4 7x2; 5x4 6x, etc. (Answers will vary.) (b) Degree: 5 13. (a) Standard form: 3x 4 2x 2 5 Polynomials and Factoring (c) Monomial Standard form: 3x 2x 8 24. 2x3 x 3x1 is not a polynomial because it includes a term with a negative exponent. x2 2x 3 is a polynomial. 2 27. y2 y 4 y3 is a polynomial. 23. 2x 3x3 8 is a polynomial. 22. (a) Standard form: x 5y 2x2y 2 xy 4 3 (b) Degree: 6 Leading coefficient: 1 (c) Trinomial 25. 3x 4 4 3 3 4x1 is not x x a polynomial because it includes a term with a negative exponent. 26. 1 3 Standard form: x2 x 2 2 28. y 2 y 4 is not a polynomial because of the square root. Standard form: y 4 y3 y2 29. 6x 5 8x 15 6x 5 8x 15 6x 8x 5 15 2x 10 30. 2x2 1 x2 2x 1 2x2 1 x2 2x 1 31. x3 2 4x3 2x x3 2 4x3 2x 2x2 x2 2x 1 1 4x3 x3 2x 2 x2 2x 3x3 2x 2 1152 Appendix A Review of Fundamental Concepts of Algebra 32. 5x2 1 3x2 5 5x2 1 3x2 5 5x2 3x2 1 5 2x2 4 33. 15x2 6 8.3x3 14.7x2 17 15x2 6 8.3x3 14.7x2 17 8.3x3 15x2 14.7x2 6 17 8.3x3 29.7x2 11 34. 15.2x 4 18x 19.1 13.9x 4 9.6x 15 15.2x 4 18x 19.1 13.9x 4 9.6x 15 15.2x 4 13.9x 4 18x 9.6x 19.1 15 1.3x 4 8.4x 34.1 35. 5z 3z 10z 8 5z 3z 10z 8 5z 3z 10z 8 5z 3z 10z 8 12z 8 36. y 3 1 y 2 1 3y 7 y 3 1 y 2 1 3y 7 y 3 1 y 2 1 3y 7 y 3 y 2 3y 1 1 7 y 3 y 2 3y 7 37. 3xx2 2x 1 3xx2 3x2x 3x 1 3x3 6x2 3x 39. 5z3z 1 5z3z 5z1 15z2 5z 41. 1 x34x 14x x34x 38. y 24y 2 2y 3 y 24y 2 y 22y y 23 4y 4 2y 3 3y 2 40. 3x5x 2 3x5x 3x2 15x 2 6x 42. 4x3 x 3 4x3 4xx 3 4x 4x 4 12x 4x 4 4x 4 4x 4x 4 12x 43. 2.5x2 33x 2.5x23x 33x 7.5x3 9x 45. 4x18 x 3 4x18 x 4x3 12 x2 12x 44. 2 3.5y2y 3 22y 3 3.5y2y 3 4y3 7y 4 7y 4 4y 3 46. 2y4 78 y 2y4 2y 78 y 8y 74 y 2 74 y2 8y 47. x 3x 4 x2 4x 3x 12 FOIL x2 7x 12 48. x 5x 10 x2 10x 5x 50 x2 5x 50 FOIL Appendix A.3 49. 3x 52x 1 6x2 3x 10x 5 FOIL Polynomials and Factoring 50. 7x 24x 3 28x2 21x 8x 6 6x 7x 5 2 51. Multiply: x2 0x3 1 0x3 FOIL 28x 29x 6 2 x2 1153 52. Multiply: x2 x2 x4 1 x4 0x3 x2 x4 0x3 x2 9x x4 0x3 x2 0x 1 x4 x4 0x3 x2 0x 1 x4 x2 1 0x3 3x 3x 3x3 3x3 2 2 2x2 9x2 6x 2x2 6x 4 13x2 0x 4 x 4 13x 2 4 53. x 10x 10 x2 102 x2 100 54. 2x 32x 3 2x2 32 4x2 9 55. x 2yx 2y x2 2y2 x2 4y2 56. 2x 3y2x 3y 2x2 3y2 4x2 9y2 57. 2x 32 2x2 22x3 32 58. 4x 52 4x2 24x5 52 4x2 12x 9 16x2 40x 25 59. 2x 5y2 2x2 22x5y 5y2 60. 5 8x2 52 258x 8x2 4x2 20xy 25y2 25 80x 64x2 61. x 13 x3 3x21 3x12 13 62. x 23 x3 3x22 3x22 23 x3 3x2 3x 1 x3 6x2 12x 8 63. 2x y3 2x3 32x2y 32xy2 y3 64. 3x 2y3 3x3 33x22y 33x2y2 2y3 8x3 12x2y 6xy2 y3 27x3 54x2y 36xy2 8y3 65. 4x3 32 4x32 24x33 32 66. 8x 32 8x2 28x3 32 16x6 24x3 9 64x2 48x 9 67. m 3 nm 3 n m 32 n2 68. x y 1x y 1 x y2 12 x 2 2xy y 2 1 m2 6m 9 n2 m2 n2 6m 9 69. x 3 y2 x 32 2yx 3 y2 70. x 1 y2 x 12 2x 1y y2 x2 6x 9 2xy 6y y2 x2 2x 1 2xy 2y y2 x2 2xy y2 6x 6y 9 x2 2xy y2 2x 2y 1 71. 2r2 52r2 5 2r 22 52 4r 4 25 72. 3a3 4b23a3 4b2 3a32 4b22 9a6 16b4 73. 12 x 32 12 x2 212 x3 32 1 4x2 3x 9 74. 23 t 52 23 t2 223 t5 52 49 t 2 20 3 t 25 75. 13x 213x 2 13x2 22 19x2 4 1154 Appendix A Review of Fundamental Concepts of Algebra 76. 2x 15 2x 15 2x2 15 2 77. 1.2x 32 1.2x2 21.2x3 32 1.44x2 7.2x 9 1 4x 2 25 78. 1.5y 32 1.5y2 21.5y3 32 79. 1.5x 41.5x 4 1.5x2 42 2.25y 2 9y 9 2.25x2 16 80. 2.5y 32.5y 3 2.5y2 32 81. 5xx 1 3xx 1 2xx 1 2x2 2x 6.25y 9 2 82. 2x 1x 3 3x 3 2x 2x 3 2x2 6x 2x 6 2x2 FOIL 8x 6 83. u 2u 2u2 4 u2 4u2 4 84. x yx yx2 y2 x2 y2x2 y2 x22 y 22 x4 y4 u4 16 85. x y x y x y 2 2 86. 5 x 5 x 52 x 2 xy 25 x 88. x 3 2 x2 2x3 3 2 87. x 5 x2 2x5 5 2 2 x2 23 x 3 x2 25x 5 89. 3x 6 3x 2 90. 5y 30 5 y 6 91. 2x3 6x 2xx2 3 92. 4x3 6x2 12x 2x2x2 3x 6 93. xx 1 6x 1 x 1x 6 94. 3xx 2 4x 2 x 23x 4 95. x 32 4x 3 x 3x 3 4 x 3x 1 96. 3x 12 3x 1 3x 1 13x 1 97. 12x 4 12x 82 12x 8 3x3x 1 1 1 15 1 98. 3 y 5 3 y 3 3 y 15 99. 12x3 2x2 5x 12x3 42x2 10 2x 12xx2 4x 10 6 2 100. 13 y 4 5y 2 2y 13 y 4 15 3 y 3y 2 2 12 101. 3xx 3 4x 3 3xx 3 3 x 3 3x 3x 6 2 13 y y3 15y 6 2 102. 45 y y 1 2 y 1 45 y y 1 10 5 y 1 5 y 12 y 5 103. x 2 81 x 2 92 x 9x 9 104. x2 49 x2 72 x 7x 7 105. 32y 2 18 216y 2 9 24y2 32 24y 34y 3 Appendix A.3 Polynomials and Factoring 107. 16x2 19 4x2 13 2 106. 4 36y 2 41 9y2 4 2 25 y 108. 64 25 y 82 2 5 y 825 y 8 4x 13 4x 13 412 3y2 1155 2 41 3y1 3y 109. x 12 4 x 12 22 110. 25 z 5 2 52 z 5 2 x 1 2x 1 2 5 z 55 z 5 x 1x 3 5 z 55 z 5 zz 10 111. 9u2 4v2 3u2 2v2 113. x2 4x 4 x2 22x 22 112. 25x2 16y 2 5x2 4y2 3u 2v3u 2v 114. x2 10x 25 x2 2(5x 52 x 52 x 22 5x 4y5x 4y 115. 4t2 4t 1 2t2 22t1 12 2t 12 116. 9x2 12x 4 3x2 23x2 22 3x 22 117. 25y2 10y 1 5y2 25y1 12 5y 12 118. 36y 2 108y 81 94y 2 12y 9 119. 9u2 24uv 16v2 3u2 23u4v 4v2 92y2 22y3 32 3u 4v2 92y 32 120. 4x2 4xy y 2 2x2 22xy y 2 2 x 23 2 2x y2 1 1 122. z2 z 4 z2 2z2 4 4 2 2 121. x2 3x 9 x2 2x3 3 12 2 z 2 1 2 123. x3 8 x3 23 x 2x2 2x 4) 124. x3 27 x3 33 x 3x2 3x 9 125. y3 64 y3 43 y 4 y2 4y 16 126. z3 125 z3 53 z 5z 2 5z 25 127. 8t3 1 2t3 13 2t 14t2 2t 1 128. 27x3 8 3x3 23 3x 29x2 6x 4 129. u3 27v3 u3 3v3 u 3vu2 3uv 9v2 130. 64x 3 y 3 4x3 y 3 4x y16x2 4xy y 2 131. x2 x 2 x 2x 1 132. x2 5x 6 x 2x 3 133. s2 5s 6 s 3s 2 134. t 2 t 6 t 2t 3 135. 20 y y2 y2 y 20 y 5 y 4 1156 Appendix A Review of Fundamental Concepts of Algebra 136. 24 5z z2 z 2 5z 24 z 8z 3 137. x2 30x 200 x 20x 10 138. x2 13x 42 x 6x 7 139. 3x2 5x 2 3x 2x 1 140. 2x2 x 1 2x 1x 1 141. 5x2 26x 5 5x 1x 5 142. 12x2 7x 1 3x 14x 1 143. 9z2 3z 2 9z2 3z 2 3z 23z 1 144. 5u2 13u 6 5u2 13u 6 145. x3 x2 2x 2 x2x 1 2x 1 5u 2u 3 146. x3 5x2 5x 25 x2x 5 5x 5 x 1x2 2 147. 2x3 x2 6x 3 x22x 1 32x 1 x 5x2 5 148. 5x3 10x2 3x 6 5x2x 2 3x 2 2x 1x2 3 149. 6 2x 3x3 x 4 23 x x33 x x 25x2 3 150. x5 2x3 x2 2 x3x2 2 x2 2 3 x2 x3 151. 6x3 2x 3x2 1 2x3x2 1 13x2 1 x2 2x3 1 3x2 12x 1 x 2 2x 1x 2 x 1 152. 8x 5 6x2 12x 3 9 2x24x 3 3 34x 3 3 4x 3 32x2 3 153. a c 38 24. Rewrite the middle term, 10x 6x 4x, since 64 24 and 6 4 10. 3x2 10x 8 3x2 6x 4x 8 3xx 2 4x 2 x 23x 4 154. a c 29 18. Rewrite the middle term, 9x 6x 3x, since 63 18 and 6 3 9. 155. a c 62 12. Rewrite the middle term, x 4x 3x, since 43 12 and 4 3 1. 2x2 9x 9 2x2 6x 3x 9 6x2 x 2 6x2 4x 3x 2 2xx 3 3x 3 2x3x 2 13x 2 x 32x 3 2x 13x 2 156. a c 615 90. Rewrite the middle term, x 10x 9x, since 109 90 and 10 9 1. 157. a c 152 30. Rewrite the middle term, 11x 6x 5x, since 65 30 and 6 5 11. 6x2 x 15 6x2 10x 9x 15 15x2 11x 2 15x2 6x 5x 2 2x3x 5 33x 5 3x5x 2 15x 2 2x 33x 5 3x 15x 2 158. a c 121 12. Rewrite the middle term, 13x 12x x, since 121 12 and 12 1 13. 12x2 13x 1 12x2 12x x 1 12xx 1 1x 1 x 112x 1 Appendix A.3 159. 6x2 54 6x2 9 160. 12x2 48 12x2 4 6x 3x 3 161. x3 4x2 x2x 4 164. 16 6x x 2 16 8x 2x x 2 8 x2 x xx 3x 3 165. 1 4x 4x2 1 2x2 166. 9x2 6x 1 9x2 3x 3x 1 3x 13x 1 167. 2x2 4x 2x3 2xx 2 x2 168. 2y3 7y2 15y y2y2 7y 15 2xx2 x 2 y2y2 10y 3y 15 2xx 1x 2 y2y 3 y 5 169. 9x2 10x 1 9x 1x 1 170. 13x 6 5x2 5x2 13x 6 5x2 10x 3x 6 5x 3x 2 171. 1 2 81 x 1157 12x 2x 2 163. x2 2x 1 x 12 162. x 3 9x xx2 9 Polynomials and Factoring 1 2 648 29x 8 81 x 18 81 x 81 1 2 81 x 18x 648 1 1 1 1 172. 8 x2 96 x 16 96 12x2 x 6 96 4x 33x 2 1 1 81 x 36x 18 173. 3x3 x2 15x 5 x23x 1 53x 1 3x 1x2 5 175. x 4 4x3 x2 4x xx3 4x2 x 4 174. 5 x 5x2 x3 15 x x25 x 5 x1 x2 176. 3u 2u2 6 u3 u3 2u2 3u 6 xx2x 4 x 4 u2u 2 3u 2 xx 4x2 1 u 2u2 3 u 23 u2 3 36 2 177. 14x3 3x2 34x 9 14x3 12 4 x 4x 4 1 1 178. 5x3 x2 x 5 5x3 5x2 5x 25 14x3 12x2 3x 36 15x2x 5 5x 5 14x2x 12 3x 12 1 5x2 5x 5 14x 12x2 3 179. t 12 49 t 12 72 180. x2 12 4x2 x2 1 2xx2 1 2x t 1 7t 1 7 x2 2x 1x2 2x 1 t 6t 8 x 12x 12 181. x2 82 36x2 x2 82 6x2 x2 8 6xx2 8 6x x2 6x 8x2 6x 8 x 4x 2x 4x 2 182. 2t 3 16 2t 3 8 2t 2t 2 2t 4 1158 Appendix A Review of Fundamental Concepts of Algebra 183. 5x3 40 5x3 8 184. 4x2x 1 2x 12 2x 14x 2x 1 5x3 23 5x 2 x2 2x 16x 1 2x 4 185. 53 4x2 83 4x5x 1 3 4x53 4x 85x 1 3 4x15 20x 40x 8 3 4x23 60x 186. 2x 1x 32 3x 12x 3 x 1x 32x 3 3x 1 x 1x 32x 6 3x 3 x 1x 3x 9 x 1x 3x 9 187. 73x 221 x2 3x 21 x3 3x 21 x273x 2 1 x 3x 21 x221x 14 1 x 3x 21 x220x 15 53x 21 x24x 3 188. 7x2x2 12x x2 127 7x2 14x2 x2 1 7x2 13x2 1 189. 3x 22x 14 x 234x 13 x 22x 133x 1 4x 2 x 22x 133x 3 4x 8 x 22x 137x 5 190. 2xx 54 x24x 53 2xx 53x 5 2x 2xx 53x 5 2xx 53x 5 191. 5x6 146x53x 23 33x 223x6 15 3x6 143x 2210x53x 2 3x6 1 3x6 143x 2230x6 20x5 3x6 3 3x6 143x 2233x6 20x5 3 3x23 143x 2233x6 20x5 3 3x2 1x 4 x2 143x 2233x6 20x5 3 3x2 14x 4 x2 143x 2233x6 20x5 3 192. x2 2 x2 x 14 x2 15 x2 14 x2 1 2 2 x2 14 x2 14 x2 1 2 x2 1 2 Appendix A.3 193. For x2 bx 15 to be factorable, b must equal m n where mn 15. Polynomials and Factoring 1159 194. For x2 bx 50 to be factorable, b must equal m n where mn 50. Factors of 15 Sum of factors Factors of 50 Sum of factors 151 15 1 14 150 1 50 51 151 15 1 14 150 1 50 51 35 3 5 2 510 5 10 15 35 3 5 2 510 5 10 15 225 2 25 27 225 2 25 27 The possible b-values are 14, 14, 2, or 2. The possible b-values are 51, 51, 15, 15, 27, 27. 195. For x2 bx 12 to be factorable, b must equal m n where mn 12. 196. For x2 bx 24 to be factorable, b must equal m n where mn 24. Factors of 12 Sum of factors 121 12 1 11 124 1 24 25 121 12 1 11 124 1 24 25 26 2 6 4 212 2 12 14 26 2 6 4 212 2 12 14 34 3 4 1 38 3 8 11 34 3 4 1 38 3 8 11 46 4 6 10 46 4 6 10 Factors of 24 The possible b-values are 11, 11, 4, 4, 1, 1. Sum of factors The possible b-values are 25, 25, 14, 14 11, 11, 10, 10. 197. For 2x2 5x c to be factorable, the factors of 2c must add up to 5. Possible c-values 2c Factors of 2c that add up to 5 2 4 14 4 and 1 4 5 3 6 23 6 and 2 3 5 3 6 61 6 and 6 1 5 7 14 72 14 and 7 2 5 12 24 83 24 and 8 3 5 These are a few possible c-values. There are many correct answers. If c 2: 2x 2 5x 2 2x 1x 2 If c 3: 2x 2 5x 3 2x 3x 1 If c 3: 2x 2 5x 3 2x 1x 3 If c 7: 2x 2 5x 7 2x 7x 1 If c 12: 2x 2 5x 12 2x 3x 4 1160 Appendix A Review of Fundamental Concepts of Algebra 198. For 3x2 10x c to be factorable, the factors of 3c must add up to 10. Possible c-values 3c Factors of 3c that add up to 10 3 9 19 9 and 1 9 10 8 24 122 24 and 12 2 10 8 24 64 24 and 6 4 10 These are a few possible c-values. There are many correct answers. If c 3: 3x2 10x 3 3x 1x 3 If c 8: 3x2 10x 8 3x 2x 4 If c 8: 3x2 10x 8 3x 4x 2 199. For 3x2 x c to be factorable, the factors of 3c must add up to 1. Possible c-values 3c Factors of 3c must add up to 1 2 6 23 6 and 2 3 1 4 12 34 12 and 3 4 1 10 30 56 30 and 5 6 1 These are a few possible c-values. There are many correct answers. If c 2: 3x 2 x 2 3x 2x 1 If c 4: 3x 2 x 4 3x 4x 1 If c 10: 3x 2 x 10 3x 5x 2 200. For 2x2 9x c to be factorable, the factors of 2c must add up to 9. There are many possibilities. Possible c-values 2c Factors of 2c that add up to 9 4 8 18 8 and 1 8 9 7 14 27 14 and 2 7 9 9 18 36 18 and 3 6 9 10 20 45 20 and 4 5 9 11 22 211 22 and 2 11 9 18 36 312 36 and 3 12 9 These are a few possible c-values. 2x2 9x 4 2x 1x 4 2x2 9x 7 2x 7x 1 2x2 9x 9 2x 3x 3 2x2 9x 10 2x 5x 2 2x2 9x 11 2x 11x 1 2x2 9x 18 2x 3x 6 Appendix A.3 201. (a) Profit Revenue Cost Polynomials and Factoring 202. (a) Profit Revenue Cost Profit 95x 73x 25,000 P 36x 460 12x 95x 73x 25,000 22x 25,000 36x 460 12x (b) For x 5000: 24x 460 Profit 225000 25,000 (b) When x 42, P 2442 460 $548. 110,000 25,000 $85,000 203. (a) 5001 r2 500r 12 500r 2 2r 1 500r 2 1000r 500 (b) r 212% 3% 4% 412% 5% 5001 r2 $525.31 $530.45 $540.80 $546.01 $551.25 (c) As r increases, the amount increases. 204. (a) 12001 r3 12001 3r 3r 2 r 3 1200r 3 3r 2 3r 1 1200r 3 3600r 2 3600r 1200 (b) r 2% 3% 312% 4% 412% 12001 r3 $1273.45 $1311.27 $1330.46 $1349.84 $1369.40 (c) Amount increases with increasing r. 205. (a) V l w h 26 2x18 2xx (b) x cm 1 2 3 384 616 720 x (cm) 3 5 7 Volume cm3 486 375 84 213 x29 xx V cm3 4x1x 131x 9 4xx 13x 9 4x3 88x 2 468x 206. (a) Volume length width height 1245 3x15 2xx 1 2 45 1 2 675x 3x15x 2x2 90x2 45x2 6x3 126x3 135x2 675x (b) When x 3: V 12633 13532 6753 126 27 135 9 2025 1 2 162 1215 2025 12972 486 cubic centimeters When x 5: V 1 3 2 65 1355 6755 126 2 125 135 25 3375 1 2 750 3375 3375 12750 375 cubic centimeters When x 7: V 1 3 2 67 1357 6757 126 2 343 135 49 4725 1 2 2058 6615 4725 12168 84 cubic centimeters 1161 1162 Appendix A Review of Fundamental Concepts of Algebra 208. A 18 2x14 x 252 18x 28x 2x2 207. Area length width 2x 1422 2x2 46x 252 2x22 1422 44x 308 209. (a) Area of shaded region Area of outer rectangle Area of inner rectangle A 2x2x 6 xx 4 4x2 12x x2 4x 3x2 8x (b) Area of shaded region Area of outer triangle Area of inner triangle A 12 9x12x 12 6x8x 54x2 24x2 30x2 210. (a) T R B 1.1x 0.0475x2 0.001x 0.23 (b) 0.0475x2 1.099x 0.23 x mi hr 30 40 55 T feet 75.95 120.19 204.36 (c) Stopping distance increases at an accelerating rate as speed increases. 212. x2 4x 3 x 3x 1 211. 3x2 7x 2 3x 1x 2 x x x x x 1 x 1 1 x x x x 1 x 1 x x 1 x x 1 x x 1 1 x x x 1 1 1 x x x 1 x 1 x 1 1 1 1 1 1 213. 2x2 7x 3 2x 1x 3 x 214. x2 3x 2 x 2x 1 x x x x 1 1 x x x 1 x 1 x 1 x 1 x x 1 x 1 1 x x x 1 x 1 1 x 1 x 1 x 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 215. A r 22 r 2 216. Area 2r2 r2 217. A 818 4x2 r 22 r 2 4r2 r2 436 x2 r 2 4r 4 r 2 r24 46 x6 x 4r 4 4 r 1 Appendix A.3 1 5 1 218. Area 2x 34 x 3 254 Polynomials and Factoring 219. (a) V R 2h r 2h 58x2 6x 9 5816 hR 2 r 2 58x2 6x 9 16 hR rR r 5 2 8 x 5 8 x 1163 (b) The average radius is R r 2. The thickness of the tank is R r. 6x 7 7x 1 V hR rR r 2 R 2 rR rh 2 average radiusthicknessh 220. kQx kx2 kxQ x 221. False. 4x2 13x 1 12x 3 4x2 3x 1 222. False. 223. True. a2 b2 a ba b (4x 3) (4x 6) 4x 3 4x 6 369 224. False. A perfect square trinomial can be factored as the binomial sum squared. 225. Since x mx n x mn, the degree of the product is m n. 226. If the degree of one polynomial is m and the degree of the second polynomial is n (and n > m), the degree of the sum of the polynomials is n. 227. The unknown polynomial may be found by adding x3 3x2 2x 1 and 5x2 8: x 3 3x2 2x 1 5x2 8 x 3 3x2 5x2 2x 1 8 x 3 8x2 2x 7 228. x y2 x2 y 2 Let x 3 and y 4. 3 42 72 49 32 42 9 16 25 Not Equal If either x or y is zero, then x y2 would equal x2 y2. 229. x2n y 2n x n2 y n2 x n y nx n y n This is not completely factored unless n 1. For n 2: x 2 y 2x 2 y 2 x 2 y 2x yx y For n 3: x3 y3x3 y3 x yx 2 xy y 2x yx 2 xy y 2 For n 4: x 4 y 4x 4 y 4 x 4 y 4x2 y 2x yx y 230. x 3n y 3n x n 3 ( y n3 x n y nx 2n x n y n y 2n Depending on the value of n, this may factor further. 232. Answers will vary. A possible answer: A polynomial is in factored form when written as a product of polynomials of lesser degree than the given polynomial. 231. x 3n y 2n x n3 y n2 x 3n y 2n is completely factored. For integer values of n greater than 4, the factorizations become more complicated. 233. Answers will vary. Some examples: x 2 3; x 2 x 1; x 2 16 1164 Appendix A Appendix A.4 Review of Fundamental Concepts of Algebra Rational Expressions ■ You should be able to find the domain of a rational expression. ■ You should know that a rational expression is the quotient of two polynomials. ■ You should be able to simplify rational expressions by reducing them to lowest terms. This may involve factoring both the numerator and the denominator. ■ You should be able to add, subtract, multiply, and divide rational expressions. ■ You should be able to simplify complex fractions. ■ You should be able to simplify expressions with negative or fraction exponents. Vocabulary Check 1. domain 2. rational expression 3. complex 4. smaller 5. equivalent 6. difference quotient 1. The domain of the polynomial 3x2 4x 7 is the set of all real numbers. 2. The domain of the polynomial 2x2 5x 2 is the set of all real numbers. 3. The domain of the polynomial 4x3 3, x ≥ 0 is the set of non-negative real numbers, since the polynomial is restricted to that set. 4. The domain of the polynomial 6x2 9, x > 0 is the set of all positive real numbers because the polynomial is restricted to that set. 5. The domain of 1x 2 is the set of all real numbers x such that x 2. 6. The domain of x 12x 1 is the set of all real numbers such that x 12. 7. The domain of x 1 is the set of all real numbers x such that x ≥ 1. 8. The domain of 6 x is the set of all real numbers x such that x ≤ 6. 9. 5 53x 53x , 2x 2x3x 6x2 x0 10. The missing factor is 3x, x 0. 3 3x 1 4 4x 1 11. 15x2 5x3x 3x , 10x 5x2 2 x0 The missing factor is x 1, where x 1. 12. 6y2 3 3 18y 2 2 5 3 60y 6y 10y 10y3 13. 3xy x3y 3y , x0 xy x xy 1 y 1 14. 2x2y 2x2y 2x2 xy y yx 1 x 1 15. 4y 8y2 4y2y 1 10y 5 52y 1 16. 9x2 9x 9xx 1 2x 2 2x 1 17. x5 x5 10 2x 2x 5 18. 4y 1 , y 5 2 12 4x 43 x 4, x 3 x3 x3 19. 1 , x5 2 9x , x 1 2 y2 16 y 4 y 4 y4 y4 y 4, y 4 20. x2 25 x 5x 5 5x 1x 5 x 5, x 5 Appendix A.4 21. x3 5x2 6x xx 2x 3 xx 3 , x2 4 x 2x 2 x2 23. y2 7y 12 y 3 y 4 y 4 , y2 3y 18 y 6 y 3 y 6 25. 2 x 2x2 x3 2 x x22 x x2 4 x 2x 2 z2 y3 22. 24. 26. x2 8x 20 x 10x 2 x 2 , x2 11x 10 x 10x 1 x 1 x 10 x2 7x 6 x 6x 1 11x 10 x 10x 1 x3 x2 9 x2 9 2 2 x 9x 9 x x 1 9x 1 2 x1 x2 x 2x 2 x2 9 x2 9x 1 x 2x2 1 x 2x 2 1 , x ±3 x1 x2 1 , x2 x2 z3 8 z 2z2 2z 4 z2 2z 4 z2 2z 4 28. y3 2y2 3y y y 3 y 1 y3 1 y 1 y2 y 1 29. 1165 x2 27. x 2 Rational Expressions x 0 1 2 3 4 5 6 x2 2x 3 x3 1 2 3 Undef. 5 6 7 x1 1 2 3 4 5 6 7 30. The expressions are equivalent except at x 3. y y 3 , y 1 y2 y 1 x 0 1 2 3 4 5 6 x3 x2 x 6 1 2 1 3 1 4 Undef. 1 6 1 7 1 8 1 x2 1 2 1 3 1 4 1 5 1 6 1 7 1 8 The expressions are equivalent except at x 3. 31. 5x3 5x3 3 4 2x 2 32. 2x3 There are no common factors so this expression cannot be simplified. In this case factors of terms were incorrectly cancelled. 33. r2 r2 , r0 2 2r 4r2 4 x2 xx2 25 x3 25x 2x 15 x 5x 3 The expression cannot be simplified. 34. Area of shaded portion: x5 2 2 x 5 2 4 Area of total figure: 2x 3x 5 x 52 x 5 x5 4 4 Ratio: 2x 3x 5 2x 3 42x 3 35. 5 x1 x1 1 25x 2 5x 2, x1 36. x 13 x33 x xx 3 x 13 3 5 x x 31 xx 3 5 x 13 x 13 , x3 5x2 5x2 1166 37. Appendix A r r1 Review of Fundamental Concepts of Algebra r 2 1 rr 1r 1 r 1 , r 1, r 0 r2 r 2r 1 r 38. 4y 16 5y 15 2y 6 4 y 4 4y 5 y 3 t 3t 2t 3 39. t2 t 6 t2 6t 9 40. x 2 xy 2y 2 x3 x2y 41. x2 36 x3 6x2 x2 36 2 x x x x 43. x x 2 3xy 2y 2 t3 t 2 x2 x x3 6x2 x 6x 6 xx 1 x2x 6 x x 6x 1 , x6 x2 5 5 6x 3 x3 x 3 x3 42. x2 14x 49 3x 21 x 7x 7 x2 49 x7 x 7x 7 1 , x ±7 3 44. 2x 1 1 x 2x 1 1 x x x3 x3 x3 x3 46. 3 3 5x 1 5 x1 x1 x1 6x 3 5 x3 3 5x 1 x1 6x 18 5 x3 3 5x 5 x1 6x 13 x3 8 5x x1 5 3 5 2 3 x2 2x x2 x2 x2 48. 2x 5 2x 51 x 5 5 x x 5 15 x 49. 50. 8 8 , y 3, 4 5 5 x 2yx y x xy x 2yx y xx y)2, x 2y x 2x y 5 x 5x x5 x1 x1 x1 x1 45. 6 47. t3 t2 4 t 32t 2t 2 t 3t 2, 2 y 3 1 y 4 5 2x 5 2x x5 x5 x5 1 x 1 x x2 x 2 x2 5x 6 x 2x 1 x 2x 3 x 3 xx 1 x 3 x2 x x 1x 2x 3 x 1x 2x 3 x2 3 x2 3 x 1x 2x 3 x 1x 2x 3 2 10 2 10 x2 x 2 x2 2x 8 x 2x 1 x 4x 2 2x 4 10x 1 x 2x 1x 4 x 2x 1x 4 12x 18 62x 3 2x 8 10x 10 x 2x 1x 4 x 2x 1x 4 x 2x 1x 4 x7 3x 7 Appendix A.4 Rational Expressions 1167 2 1 x2 1 1 2x 1 51. 2 x x 1 x3 x xx2 1 xx2 1 xx2 1 x2 1 2x 1 x2 2x xx 2 xx2 1 xx2 1 xx2 1 52. 53. 2x x2 , x2 1 x2 1 x0 2 1 2 2 1 2 x 1 x 1 x2 1 x 1 x 1 x 1x 1 2x 1 2x 1 1 x 1x 1 x 1x 1 x 1x 1 2x 2 2x 2 1 4x 1 x 1x 1 x 1x 1 x 4 3x 8 x 4 3x 8 x2 x2 x2 x 4 3x 8 2x 12 2x 6 x2 x2 x2 The error was incorrect subtraction in the numerator. 54. 6x x2 8 x6 x x 22 8 2 2 2 2 xx 2 x x x 2 x x 2 x x 2 x2x 2 6x x2 x2 4x 4 8 10x 12 25x 6 2 2 x2x 2 x x 2 x x 2 The error was an incorrect expansion of x 22 in the numerator. x4 x4 x 4 1 1 2 2 56. x 4 x 16 x 16 4 x 4x 4x 4x 2x 1 2x 22 55. x 2 x 1 2 x2 2 1 , 2 1 x2 2 2 x 1 x x4 1 x2 16 x4 1 x 4x 4 x 4 , 2 3 2 3 xx 1, x 1, 0 x 21 x x 21 x x 4x 4x 4x 2 2 x x 1 x 58. x 1 x x2 59. x2 x 1 x 57. x x 1 x 1 x 2x 2x 1 , x>0 2x 2x 1 x x x 12 x 1x 1 x x 1x 1 x x1 , x0 x1 x 0, 4 1168 Appendix A Review of Fundamental Concepts of Algebra t2 t2 t 2 1 t 2 1 2 t 1 t 1 60. t2 t2 2 t 2 1 t 2 1 t 2 t 2 1 1 2 2 t 2t 2 1 t t 1 61. x5 2x2 x2x7 2 x7 2 x2 x8 5 x3 62. x 5 5x3 x3x8 5 63. x2x2 15 x 2 14 x 2 15 x 2 x 2 1 1 x 2 15 64. 2xx 53 4x2x 54 2xx 54 x 5 2x 2xx 54x 5 2x 2xx 5 x 54 65. 2x2x 112 5x 112 x 112 2x2x 11 5 2x3 2x 2 5 x 112 66. 4x 32x 132 2x2x 112 2x 1124x 32x 12 2x 67. 68. 3x13 x23 3x13 x23 3x23 3x23 1 x3 2x1 x4 x2 12 x23 x23 x2 12 4x32x 12 2x 2x 112 3x1 x0 3x 1 , x0 3x0 3 x3 2x1 x212 1 x212 x4 x3 2x1 x2121 x212 x3 2x1 x2 2 12 2 12 1 x 1 x 1 x212 4 x x4 69. x 1 h 1x x 1 h 1x h h x3 2x 2x3 1 x212 x2 2 xx2 2 3 2 12 x 1 x x 1 x212 1 x3 2x 1 x4 1 x212 x4 4 xx h xx h x 1 h 2 70. h 1 x2 x 1 h 2 1 x2 h x2x h2 x2x h2 x h2 hx2x h2 x x h hxx h h hxx h x2 x2 2xh h2 hx2x h2 h2x h hx2x h2 1 , xx h h0 x2 2x h , h0 x2x h2 Appendix A.4 71. x h1 4 x 1 4 x h1 4 x 1 4 h h x 4 x h 4 hx 4x h 4 h hx 4x h 4 xh 72. 73. 1 , h0 x 4x h 4 x hx 1 x h xx h 1 x 2 x 2 h1 x hx 1 x h 1x 1 x h 1x 1 h h 1 1 , h0 x h 1x 1 h x h 1x 1 x2 x hx h x2 xh x x h 1x 1 x 2 x 2 h 1 x 2 x 74. x 2 x z 3 z 3 z 3 z 3 z 3 z z 3 z x 2 x 2x 2 x z 3 z 3z 3 z 2 2x 2 x 3 3z 3 z 1 x 2 x h x h 2 x 2 x h 1 x 1 h x h 1 x 1 x h 1 x 1 hx h 1 x 1 h hx h 1 x 1 1 x h 1 x 1 h0 , x h 2 x 2 xh2x2 hx h 2 x 2 h hx h 2 x 2 x h 2 x 2 Shaded area xx2 x2 Total area x2x 1 2x 1 1 z 3 z x h 1 x 1 77. Probability 1 x h 1 x 1 h xx h 1 76. x 4x h 4 x 4x h 4 x h 1 x 1 x h 1x 1 x h 1x 1 75. Rational Expressions 2 1 x h 2 x 2 x 2 22x 1 , h0 1169 1170 Appendix A 78. Probability 79. (a) 1 2 4 x x 2x x 4 Shaded area Total area 1 4 x 4 x 2 x 2 2 x 4x 22x 4 x 4 x 4x 2 1 x 8x 22 x 4 2x 22 x x 4x 2 1 4 x 1 x 4x 2 1 4 x 8x 22 8x 2 x 4x 2x 4 x 42 1 minute 16 (b) x (c) Review of Fundamental Concepts of Algebra 80. t t 5t 3t 8t 3 5 15 15 161 16x minutes 60 15 minutes 16 4 24[48(400) 16,000] 48 81. (a) r 0.0909 9.09% 48400 16,000 12 24(NM P) N 24NM P (b) r NM N P 12 r 288NM P 12 12P NM N12P NM 28848400 16,000 0.0909 9.09% 481216,000 48400 24NM P 2460 525 28,000 231,500 28,000 N 60 5 82. (a) r NM 60 525 31,500 P 28,000 28,000 12 12 12 23500 5 2700 1400 4.57% 28,000 2625 30,625 30,625 (b) r 24NMN P 12P NM 12 24NM P N 12 288NM P 12P NM N12P NM 28831,500 28,000 2883500 1,008,000 28860 525 28,000 4.57% 6012 28,000 60 525 60336,000 31,500 60367,500 22,050,000 Appendix A.4 83. T 10 (a) 4tt 2 2 16t 75 4t 10 Rational Expressions 1171 t 0 2 4 6 8 10 12 14 16 18 20 22 T 75 55.9 48.3 45 43.3 42.3° 41.7 41.3 41.1 40.9 40.7 40.6 (b) T is approaching 40. 84. (a) Year 2002 2003 2004 2005 2006 2007 Banking 21.9 27.0 31.4 35.6 40.2 45.6 Paying Bills 13.9 17.8 21.5 25.0 28.1 31.0 (b) The estimates and actual values are quite close for banking. For paying bills, they are close for 2002–2004, but the model generally tends to overestimate the number. 4.39t 5.5 0.002t 2 0.01t 1.0 4.39t 5.5 (c) 0.728t 2 23.81t 0.3 0.002t 2 0.01t 1.0 0.049t 2 0.61t 1.0 (d) 0.049t 2 0.61t 1.0 0.728t 2 23.81t 0.3 0.21511t 3 2.6779t 2 4.39t 0.2695t2 3.355t 5.5 0.001456t 4 0.04762t 3 0.0006t 2 72.8t 3 0.2381t 2 0.003t 0.728t 2 23.81t 0.3 0.21511t3 2.4084t 2 7.745t 5.5 0.001456t 4 72.75238t 3 0.4905t2 23.807t 0.3 Year 2002 2003 2004 2005 2006 2007 Paying BillsBanking 0.63 0.66 0.69 0.70 0.70 0.68 The ratio is approximately 23, and it appears to peak in 2005–2006. 85. False. In order for the simplified expression to be equivalent to the original expression, the domain of the simplified expression needs to be restricted. If n is even, x ± 1. If n is odd, x 1. 86. False. The two expressions are equivalent for all values of x such that x 1. 87. Completely factor the numerator and the denominator. A rational expression is in simplest form if there are no common factors in the numerator and the denominator other than ± 1. 1172 Appendix A Appendix A.5 Review of Fundamental Concepts of Algebra Solving Equations ■ You should know how to solve linear equations. ax b 0 ■ An identity is an equation whose solution consists of every real number in its domain. ■ To solve an equation you can: (a) Add or subtract the same quantity from both sides. (b) Multiply or divide both sides by the same nonzero quantity. ■ To solve an equation that can be simplified to a linear equation: (a) Remove all symbols of grouping and all fractions. (b) Combine like terms. (c) Solve by algebra. (d) Check the answer. ■ A “solution” that does not satisfy the original equation is called an extraneous solution. ■ You should be able to solve a quadratic equation by factoring, if possible. ■ You should be able to solve a quadratic equation of the form u2 d by extracting square roots. ■ You should be able to solve a quadratic equation by completing the square. ■ You should know and be able to use the Quadratic Formula: For ax2 bx c 0, a 0, x b ± b2 4ac . 2a ■ You should be able to solve polynomials of higher degree by factoring. ■ For equations involving radicals or fractional powers, raise both sides to the same power. ■ For equations with fractions, multiply both sides by the least common denominator to clear the fractions. ■ For equations involving absolute value, remember that the expression inside the absolute value can be positive or negative. Vocabulary Check 1. equation 2. solve 3. identities; conditional 4. ax b 0 5. extraneous 6. quadratic equation 7. factoring; extracting square roots; completing the square; Quadratic Formula 1. 2x 1 2x 2 is an identity by the Distributive Property. It is true for all real values of x. 2. 3x 2 5x 4 is conditional. There are real values of x for which the equation is not true (for example, x 0). 3. 6x 3 5 2x 10 is conditional. There are real values of x for which the equation is not true. 4. 3x 2 5 3x 1 is an identity by simplification. It is true for all real values of x. 3x 2 5 3x 6 5 3x 1 5. 4x 1 2x 4x 4 2x 2x 4 2x 2 This is an identity by simplification. It is true for all real values of x. 6. 7x 3 4x 37 x is an identity by simplification. It is true for all real values of x. 7x 3 4x 7x 21 4x 21 3x 37 x Appendix A.5 7. x 42 11 x2 8x 16 11 x2 8x 5 2 1 4x is conditional. There are real values x1 x1 of x for which the equation is not true. 9. 3 11. x 11 15 x 11 11 15 11 10. 5 3 24 is conditional. There are real values of x for x x which the equation is not true (for example, x 1). 7 x 19 12. 7 2x 25 13. 7 x x 19 x x4 1173 8. x2 23x 2 x2 6x 4 is an identity by simplification. It is true for all real values of x. Thus, 8x 5 x 4 11 is an identity by simplification. It is true for all real values of x. x2 Solving Equations 7 7 2x 25 7 7 19 x 2x 18 7 19 19 x 19 18 2x 2 2 12 x x 9 14. 7x 2 23 8x 5 3x 20 15. 7x 3 3 3x 3x 17 3 3x 8x 3x 5 3x 3x 20 7x 2 2 23 2 7x 3 3x 17 16. 7x 21 5x 5 20 4x 20 7x 21 7 7 5x 5 5 20 5 x 5 5x 25 x3 5x 25 5 5 x5 17. 2x 5 7 3x 2 3x 3 51 x 1 18. 2x 10 7 3x 6 19. x 32x 3 8 5x 3x 9 5 5x 1 x 6x 9 8 5x 2x 3 3x 6 3x 9 4 5x 5x 9 8 5x 2x 3x 3 3x 3x 6 3x 9 5x 9 4 5x 5x 9 5x 5x 9 8 5x 5x 8x 5 9 8 x 3 6 x 3 3 6 3 x x 9 58 No solution x9 20. 9x 10 5x 22x 5 1 5x 1 x 4 2 2 21. 9x 10 5x 4x 10 4 9x 10 9x 10 The solution is the set of all real numbers. 4 5x4 21 4x 21 5x4 412 4x 412 5x 2 4x 2 x 4 x x 3x 3 5 2 10 22. 10 5x 2x 103 103x 2x 5x 30 3x 6x 30 x 5 1174 Appendix A Review of Fundamental Concepts of Algebra 3 1 z 5 z 24 0 2 4 23. 3x 1 x 2 10 2 4 24. 32 z 5 4z 24 40 1 4 4 2 4 x 2 410 4 32z 5 414z 24 40 4 3x 1 2 44 x 2 410 3x 1 6z 5 z 24 0 6x x 2 40 6z 30 z 24 0 7x 2 40 5z 6 7x 42 6 z 5 x6 25. 0.25x 0.7510 x 3 26. 0.60x 0.40100 x 50 0.25x 7.5 0.75x 3 0.60x 40 0.40x 50 0.50x 7.5 3 0.20x 10 0.50x 4.5 x 50 x9 27. x 8 2x 2 x 28. 8x 2 32x 1 2x 5 x 8 2x 4 x 8x 16 6x 3 2x 10 x8x4 2x 13 2x 10 8 4 13 10 Contradiction; no solution 100 4x 5x 6 6 3 4 29. 12 31. Contradiction; no solution 17 y 32 y 100 y y 30. 100 3 4x 125x 4 6 126 y 17 y 32 y y 100 y y y 4100 4x 35x 6 72 17 y 32 y 100y 400 16x 15x 18 72 49 2y 100y 31x 310 49 98y x 10 1 y 2 5x 4 2 5x 4 3 32. 10x 3 1 5x 6 2 35x 4 25x 4 210x 3 15x 6 15x 12 10x 8 20x 6 5x 6 5x 20 15x 0 x4 x0 Appendix A.5 33. 10 5 13 4 x x 34. 10x 13 4x 5 x x 15 6 4 3 x x x3 3z 6 2z 4 2 z0 9 7x 1 2 0 x x5 Multiply both sides by xx 5. 37. x 4 20 x4 x4 1x 5 2x 0 x4 20 x4 3x 5 0 120 3x 5 x 38. 30 5 3 Contradiction; no solution 7 8x 4 2x 1 2x 1 Multiply both sides by 2x 12x 1. 72x 1 8x2x 1 42x 12x 1 14x 7 16x2 8x 16x2 4 6x 11 x 39. 11 6 2 1 2 x 4x 2 x 4 x 2 Multiply both sides by x 4x 2. 2 1x 2 2x 4 2 x 2 2x 8 2 3x 10 12 3x 4x A check reveals that x 4 is an extraneous solution—it makes the denominator zero. There is no real solution. 4 6 15 x 1 3x 1 3x 1 40. x 13x 1 Multiply both sides by x 13x 1. 4 6 15 x 13x 1 x 13x 1 x1 3x 1 3x 1 43x 1 6x 1 15x 1 12 x 4 6x 6 15x 15 18x 2 15x 15 3x 13 x 13 3 2 z 2 z2 9 x 7 36. 1175 2 z2 3z 2 2 9 7 x 6x 18 32 35. 15 6 7 x x 10x 13 4x 5 Solving Equations 1176 Appendix A Review of Fundamental Concepts of Algebra 41. 1 1 10 x 3 x 3 x2 9 1 1 10 x 3 x 3 x 3x 3 Multiply both sides by x 3x 3. 1x 3 1x 3 10 2x 10 x5 42. 3 4 1 2 x2 x3 x x6 3 4 1 x 2 x 3 x 3x 2 Multiply both sides by x 3x 2. x 3 3x 2 4 x 3 3x 6 4 4x 3 4 4x 7 x 43. 7 4 3 4 1 x2 3x x x3 3 4 1 xx 3 x x3 Multiply both sides by xx 3. 3 4x 3 x 3 4x 12 x 3x 9 x3 A check reveals that x 3 is an extraneous solution since it makes the denominator zero, so there is no solution. 44. 6 2 3x 5 x x3 xx 3 Multiply both sides by xx 3. Check: 6x 3 2x 3x 5 2 6x 18 2x 3x 15 x 3 x 22 5 x 32 x2 4x 4 5 x2 6x 9 4x 9 6x 9 2x 0 x0 2 6 0 30 Division by zero is undefined. Thus, x 3 is not a solution, and the original equation has no solution. 4x 18 3x 15 45. 2 33 5 6 3 3 3 33 3 46. x 12 2x 2 x 1x 2 x2 2x 1 2x 4 x2 x 2 5x 1 1 x5 Appendix A.5 47. x 22 x2 4x 1 Solving Equations 1177 2x 12 4x2 x 1 48. x2 4x 4 x2 4x 4 4x2 4x 1 4x2 4x 4 44 14 The equation is an identity; every real number is a solution. 49. 2x2 3 8x This is a contradiction. Thus, the equation has no solution. 50. x2 16x General form: 2x2 8x 3 0 51. x 32 3 x2 6x 9 3 General form: x2 16x 0 General form: x2 6x 6 0 13 3x 72 0 1 53. 53x2 10 18x 13 3x2 14x 49 0 3x2 10 90x 52. 13 3x2 42x 147 0 55. 4x2 2x 1 0 General form: 14x2 2x 1 10 3x2 42x 134 0 General form: 4x2 2x 1 0 6x2 3x 0 3x 0 x0 x 12 3x 1 0 ⇒ x 59. x2 10x 25 0 x 9x 1 0 x 52 0 x90 ⇒ x9 x50 x4 60. x 2 or 4x2 12x 9 0 2x 3 0 ⇒ x 23 or 1 2x 0 x 1 2 x2 8x 12 2x2 19x 33 0 x2 4x 12 0 2x 3x 11 0 x 6x 2 0 x60 2x 3 0 ⇒ x 23 65. x2 4x 12 63. x 6 x 11 0 ⇒ x 11 64. or x 2 0 2x 32x 3 0 2x2 19x 33 62. 3 x1 2x 0 or x40 1 3 x 5 3 5x 2x2 0 x3 x2 2x 8 0 x 4x 2 0 3x 1 0 ⇒ x 31 x2 10x 9 0 3x0 57. 3x 13x 1 0 or 2x 1 0 or 9x2 1 0 56. x10 ⇒ x1 61. x2 2x 5x2 1 General form: 3x2 90x 10 0 3x2x 1 0 58. xx 2 5x2 1 54. 3 2 4x 8x 20 0 66. 1 2 8x or x 2 0 or x2 x 16 0 x2 8x 12 0 434x2 8x 20 40 x2 8x 128 0 1x2 8x 12 10 3x2 32x 80 0 x 16x 8 0 3x 20x 4 0 x 16 0 ⇒ x 16 x2 8x 12 0 x 6x 2 0 x60 ⇒ x6 x20 ⇒ x2 3x 20 0 or x 4 0 x 20 3 or x 4 x 8 0 ⇒ x 8 1178 Appendix A Review of Fundamental Concepts of Algebra x a2 b2 0 69. x2 49 x a2 0 x a bx a b 0 x ±7 xa0 x a bx a b 0 67. x2 2ax a2 0 68. x a x a b 0 ⇒ x a b x a b 0 ⇒ x a b 70. x2 169 71. x2 11 72. x2 32 x ± 11 x ± 169 ± 13 73. 3x2 81 x ± 32 ± 42 75. x 122 16 74. 9x2 36 x2 4 x2 27 x 12 ± 4 x ± 4 ± 2 x ± 33 x 12 ± 4 x 16 77. x 22 14 76. x 132 25 78. x 52 30 x 2 ± 14 x 13 ± 25 x 5 ± 30 x 5 ± 30 x 2 ± 14 x 13 ± 5 or x 8 x 13 ± 5 8, 18 80. 4x 72 44 79. 2x 12 18 4x 7 ± 44 2x 1 ± 18 2x 1 ± 32 x 4x 7 ± 211 1 ± 32 2 x 81. x 72 x 32 82. x 52 x 42 x 5 ± x 4 x 7 ± x 3 x7x3 7 3 7 11 7 ± 211 ± 4 4 2 x 5 x 4 or or x 7 x 3 54 2x 4 or x 5 x 4 or x 5 x 4 2x 9 x2 x 92 The only solution to the equation is x 2. The only solution to the equation is x 92. 83. x2 4x 32 0 84. 85. x 2 12x 25 0 x2 2x 3 0 x2 4x 32 x2 2x 3 x 2 12x 25 x2 4x 22 32 22 x2 2x 12 3 12 x 2 12x 62 25 6 2 x 22 36 x 12 4 x 62 11 x 2 ±6 x 1 ± 4 x1±2 x 2 ± 6 x4 or x 8 x3 or x 1 x 6 ± 11 x 6 ± 11 Appendix A.5 86. x2 8x 14 0 87. 9x2 18x 3 x2 8x 14 x2 8x 42 x2 2x 14 16 x 4 2 2 x 4 ± 2 x 12 14 4 x2 x 3 9 1 3 32 2 x1 ± 3 8 4x x2 0 90. x 2 x 1 0 x2 4x 8 0 x2 x 1 0 x2 4x 8 0 x2 x2 x 4x 8 x 22 12 2 x 3 2 x 3 2 2 x 6 14 4 9 9 18 9 2 2 ± 2 3 x 3 91. 2 ± 2 3 2x 2 5x 8 0 2x 2 5x 8 1 1 1 4 4 x 21 x2 4x 22 8 22 2 2 x1± 89. 4 2 x2 x 3 3 2 3 x1 ± 3 4 5 x2 x 4 2 2 x 45 2 5 5 x2 x 2 4 No real solution x 2 ± 12 x 2 ± 23 x 4 93. 2x2 x 1 0 b ± b2 4ac 2a 2 89 5 ± 4 4 x 4x2 4x 99 0 54 89 16 x 92. 1179 9x2 12x 14 88. 1 x2 2x 12 12 3 x 4 ± 2 Solving Equations 5 89 ± 4 4 5 ± 89 4 94. 2x2 x 1 0 b ± b2 4ac 2a x2 x 99 4 2 2 99 1 4 4 1 ± 12 421 22 1 ± 12 421 22 x 2 2 100 4 1 ± 3 1 , 1 4 2 1 ± 1 8 4 x 2 2 1±3 1 1, 4 2 x2 x 1 1 1 x 25 1 ± 25 2 x 11 9 1 ± 5 , 2 2 2 x x 1180 Appendix A Review of Fundamental Concepts of Algebra 95. 16x2 8x 3 0 x 96. 25x2 20x 3 0 b ± b2 4ac 2a 8 ± x 4163 216 82 8 ± 16 1 3 , 32 4 4 x b ± b2 4ac 2a 10 ± 10 21 b ± b2 4ac 2a x2 2x 2 0 x 122 24 10 ± 100 88 2 x 20 ± 20 4253 225 2 20 ± 400 300 50 20 ± 10 3 1 , 50 5 5 99. x2 14x 44 0 98. x2 10x 22 0 2 2x x2 0 97. 2 ± 22 412 21 2 ± 23 1 ± 3 2 6x 4 x2 100. x2 6x 4 0 b ± b2 4ac 2a 14 ± b ± b2 4ac 2a x 14 4144) 2 21 14 ± 25 7 ± 5 2 10 ± 23 5 ± 3 2 b ± b2 4ac 2a 6 ± 62 414 21 6 ± 36 16 2 6 ± 213 2 3 ± 13 101. x2 8x 4 0 x b ± b2 4ac 2a 8 ± 82 414 21 8 ± 45 2 4 ± 25 16x2 22 40x 104. 8x2 20x 11 0 x b ± b2 4ac 2a 102. 4x2 4x 4 0 x2 x 1 0 x b ± b2 4ac 2a 12x 9x2 3 103. 9x2 12x 3 0 x b ± b2 4ac 2a 1 ± 12 411 21 12 ± 122 493 29 1 ± 1 4 2 12 ± 67 2 7 ± 18 3 3 1 5 ± 2 2 105. 9x2 24x 16 0 x b ± b2 4ac 2a 106. 36x2 24x 7 0 x b ± b2 4ac 2a 20 ± 202 4811 28 24 ± 242 4916 29 24 ± 242 4367 236 20 ± 400 352 16 24 ± 0 18 24 ± 576 1008 72 24 ± 14411 72 5 3 ± 4 4 4 3 1 11 ± 3 6 Appendix A.5 108. 16x2 40x 5 0 4x2 4x 7 107. 4x2 4x 7 0 x x b ± b2 4ac 2a 447 24 4 ± 42 4 ± 82 1 ± 2 8 2 110. 3x x2 1 0 b ± b2 4ac 2a 8t 5 2t2 t 3 13 3 ± 13 ± 2 2 2 b ± 4ac 2a 12 ± 122 4125 21 12 ± 211 6 ± 11 2 57 x 14 28 ± 282 4494 249 28 ± 0 2 98 7 h 8 ± 82 425 22 6 8 ± 26 2 ± 4 2 14z 36 0 b ± b2 4ac z 2a 2 b ± b2 4ac 2a 80 ± 802 42561 225 80 ± 6400 6100 50 8 103 ± 5 50 8 3 ± 5 5 1 2 3 x x2 2 8 115. 4x 2 3x 16 4x 2 3x 16 0 x b ± b 2 4ac 2a 14 ± 142 4136 21 3 ± 32 4416 24 14 ± 52 2 3 ± 265 8 7 ± 13 116. z2 12z 36 2z z2 b ± b2 4ac 2a 112. 25h2 80h 61 0 b ± b2 4ac 2a z 62 2z 114. y2 12y 25 0 y 4165 40 ± 1600 320 32 5 5 ± 4 2 3 ± 32 411 21 b2 x 2t2 8t 5 0 y 52 2y 113. 40 ± 40 216 111. x2 3x 1 0 x 49x2 28x 4 0 2 40 ± 165 32 3 265 ± 8 8 8x 25 2 x 20x 196 8x 49 25 2 x 28x 196 0 49 25x2 1372x 9604 0 x 1181 28x 49x2 4 109. b ± b2 4ac 2a Solving Equations b ± b2 4ac 1372 ± 13722 4259604 1372 ± 921,984 686 ± 1966 2a 225 25 50 1182 Appendix A Review of Fundamental Concepts of Algebra 117. 5.1x2 1.7x 3.2 0 x 1.7 ± 1.72 45.13.2 25.1 x 0.976, 0.643 118. 2x2 2.50x 0.42 0 x b ± b2 4ac 2a 2.50 ± 2.502 420.42 22 2.50 ± 9.61 4 1.400, 0.150 119. 0.067x2 0.852x 1.277 0 x 0.852 ± 0.8522 40.0671.277 20.067 x 14.071, 1.355 120. 0.005x2 0.101x 0.193 0 x b ± b2 4ac 2a 0.101 ± 0.1012 40.0050.193 20.005 0.101 ± 0.006341 0.01 2.137, 18.063 121. 422x2 506x 347 0 x 506 ± 5062 4422347 2422 x 1.687, 0.488 123. 12.67x2 31.55x 8.09 0 x 31.55 ± 31.552 412.678.09 212.67 x 2.200, 0.290 125. x2 2x 1 0 Complete the square. x 2x 1 122. 1100x2 326x 715 0 x b ± b2 4ac 2a 326 ± 3262 41100715 21100 326 ± 3,252,276 0.672, 0.968 2200 124. 3.22x2 0.08x 28.651 0 x b ± b2 4ac 2a 0.08 ± 0.082 43.2228.651 23.22 0.08 ± 369.031 2.995, 2.971 6.44 126. 11x2 33x 0 11 2 x2 x2 2x 12 1 12 x 12 2 Factor. 3x 0 xx 3 0 x0 or x 3 0 x 3 x 1 ± 2 x 1 ± 2 127. x 32 81 Extract square roots. x 3 ±9 x39 x6 x 7 0 2 or x 3 9 or 128. x2 14x 49 0 x 12 x70 x7 Extract square roots. Appendix A.5 129. x2 x 11 4 0 x2 x Complete the square. 130. x2 3x 34 0 Solving Equations Complete the square. x2 3x 2 34 94 3 2 11 4 1 x2 x 12 11 4 2 2 x 32 2 3 2 x 12 2 124 x 32 ± 3 x 12 ± 12 4 x 32 ± 3 x 12 ± 3 131. x 12 x2 Extract square roots. 132. Factor. ax bax b 0 x2 x 12 x ± x 1 ax b 0 ⇒ x For x x 1: 01 a2 x 2 b2 0 ax b 0 ⇒ x No solution For x x 1: b a b a 2x 1 x 12 133. 3x 4 2x2 7 0 x 2x2 Quadratic Formula 134. 4x2 2x 4 2x 8 3x 11 4x2 3 ± 32 4211 22 135. 40 4x2 1 0 x 1x 1 0 3 ± 97 4 x10 or x 1 0 x 1 3 97 ± 4 4 4x4 18x2 0 2x22x2 9 0 2x2 0 ⇒ x 0 32 2x2 9 0 ⇒ x ± 2 Factor. 136. x1 20x3 125x 0 5x4x2 25 0 5x2x 52x 5 0 5x 0 ⇒ x 0 2x 5 0 ⇒ x 2x 5 0 ⇒ x 137. x4 81 0 x2 9x 3x 3 0 x2 9 0 ⇒ No real solution x 3 0 ⇒ x 3 2 x30 ⇒ x3 2 5 2 5 2 1183 1184 Appendix A Review of Fundamental Concepts of Algebra x6 64 0 138. x3 8x3 8 0 x 2x2 2x 4x 2x2 2x 4 0 x20 ⇒ x2 x2 2x 4 0 ⇒ No real solution (by the Quadratic Formula) x 2 0 ⇒ x 2 x2 139. 2x 4 0 ⇒ No real solution (by the Quadratic Formula) x3 216 0 x3 63 0 x 6x2 6x 36 0 x 6 0 ⇒ x 6 x2 6x 36 0 ⇒ No real solution (by the Quadratic Formula) 140. 27x 3 512 0 3x 89x2 24x 64 0 3x 8 0 ⇒ x 83 9x2 24x 64 0 ⇒ No real solution (by the Quadratic Formula) 141. 5x3 30x2 45x 0 142. 9x4 24x3 16x2 0 5xx2 6x 9 0 x29x2 24x 16 0 5xx 32 0 x23x 42 0 5x 0 ⇒ x 0 x2 0 ⇒ x 0 x 3 0 ⇒ x 3 143. x3 3x2 x 3 0 3x 4 0 ⇒ x 43 144. x3 2x2 3x 6 0 x2x 3 x 3 0 x2x 2 3x 2 0 x 3x2 1 0 x 2x2 3 0 x 3x 1x 1 0 x30 ⇒ x3 x 1 0 ⇒ x 1 x10 ⇒ x1 145. x4 x3 x 1 0 x3x 1 x 1 0 x 1x3 1 0 x 1x 1x2 x 1 0 x10 ⇒ x1 x 1 0 ⇒ x 1 x2 x 1 0 ⇒ No real solution (by the Quadratic Formula) x 2 0 ⇒ x 2 x2 3 0 ⇒ No real solution Appendix A.5 146. Solving Equations 1185 x4 2x3 8x 16 0 x3x 2 8x 2 0 x3 8x 2 0 x 2x2 2x 4x 2 0 x20 ⇒ x2 x2 2x 4 0 ⇒ No real solution (by the Quadratic Formula) x 2 0 ⇒ x 2 147. x4 4x2 3 0 x4 5x2 36 0 148. x2 3x2 1 0 x2 9x2 4 0 x 3 x 3 x 1x 1 0 x2 9x 2x 2 0 x 3 0 ⇒ x 3 x2 9 0 ⇒ No real solution x 3 0 ⇒ x 3 x 2 0 ⇒ x 2 x 1 0 ⇒ x 1 x20 ⇒ x2 x10 ⇒ x1 149. 4x4 65x2 16 0 36t4 29t2 7 0 150. 4x2 1x2 16 0 36t 2 7t 2 1 0 2x 12x 1x 4x 4 0 6t 7 6t 7 t 2 1 0 2x 1 0 ⇒ x 21 2x 1 0 ⇒ x 6t 7 0 ⇒ t 1 2 1x 4 0 ⇒ x 4 6t 7 0 ⇒ t 1x 4 0 ⇒ x 4 151. x3 8x3 1 0 x 2x2 2x 4x 1x2 x 1 0 x 2 0 ⇒ x 2 x 2x 4 0 ⇒ No real solution (by the Quadratic Formula) 2 x10 ⇒ x1 152. x 1 0 ⇒ No real solution (by the Quadratic Formula) x6 3x3 2 0 x3 2x3 1 0 3 2 x2 3 2x x 3 2 2x 1x2 x 1 0 3 2 0 ⇒ x 3 2 x 3 2x x2 3 2 0 ⇒ No real solution (by the Quadratic Formula) 2 x 1 0 ⇒ x 1 x x 1 0 ⇒ No real solution (by the Quadratic Formula) 2 6 7 6 t 1 0 ⇒ No real solution 2 x6 7x3 8 0 x2 7 1186 Appendix A Review of Fundamental Concepts of Algebra 153. 2x 10 0 2x 10 2x 100 x 50 156. 5 x 3 0 5 x 3 5x9 x 4 154. 4x 3 0 155. x 10 4 0 4x 3 x 10 4 16x 9 x 10 16 9 16 x 26 x 3 2x 5 3 0 157. 3 3x 1 5 0 158. 3 3 2x 5 3 3 3x 1 5 2x 5 27 3x 1 125 2x 32 3x 124 x 16 159. 26 11x 4 x x 160. x 31 9x 5 4 x 26 11x 31 9x 5 x 16 8x x2 26 11x 31 9x 25 10x x2 x2 3x 10 0 0 x2 x 6 x 5x 2 0 0 x 3x 2 x 5 0 ⇒ x 5 0x3 ⇒ x3 x20 ⇒ x2 0 x 2 ⇒ x 2 161. x 1 3x 1 162. x 5 x 5 x 1 3x 1 x5x5 2x 0 5 5 x0 163. x 532 8 x 53 82 3 x 5 64 x549 No solution 164. x 332 8 165. x 323 8 x 33 82 x 32 83 x 33 64 x 3 ± 83 3 64 x3 x 3 ± 512 x 3 4 1 166. x 223 9 x 22 93 x 2 ± 729 x 2 ± 27 29, 25 124 3 x 3 ± 162 167. x2 532 27 x2 53 272 3 272 x2 5 x2 5 9 x2 14 x ± 14 Appendix A.5 Solving Equations 168. x2 x 2232 27 x2 x 22 2723 x2 x 22 9 x2 x 31 0 x 1 ± 12 4131 1 ± 125 1 ± 55 21 2 2 169. 3xx 112 2x 132 0 170. 4x2x 113 6xx 143 0 x 1123x 2x 1 0 2x2xx 113 3x 143 0 x 1125x 2 0 2xx 1132x 3x 1 0 2xx 1135x 3 0 x 112 0 ⇒ x 1 0 ⇒ x 1 2x 0 ⇒ x 0 5x 2 0 ⇒ x 25, extraneous x10 ⇒ x1 3 5x 3 0 ⇒ x 5 x 171. 3 1 x 2 2xx 2x 172. x 2x2 3 1 4 5 x x 3 6 4 5 x 6x 6x 6x x 3 6 24 10x x2 2x2 6 x x2 10x 24 0 2x2 x 6 0 x 12x 2 0 2x 3x 2 0 x 12 0 ⇒ x 12 3 2x 3 0 ⇒ x 2 x20 ⇒ x2 x20 ⇒ x2 1 1 3 x x1 173. 174. 1 1 xx 1 xx 1 xx 13 x x1 x 1 x 3xx 1 1 3x2 3x 0 3x2 3x 1 a 3, b 3, c 1 x 3 ± 3 431 3 ± 21 23 6 2 4 3 1 x1 x2 4x 2 3x 1 x 1x 2 4x 8 3x 3 x2 3x 2 x2 2x 3 0 x 1x 3 0 x10 ⇒ x1 x 3 0 ⇒ x 3 1187 1188 Appendix A Review of Fundamental Concepts of Algebra 20 x x x 175. 4x 1 176. 20 x x2 3 x x4x x1 x 0 x2 x 20 3 x 4x2 x 3 0 x 5x 4 4x2 x 3 0 x 5 0 ⇒ x 5 4x 3x 1 0 x40 ⇒ x4 4x 3 0 ⇒ x 3 4 x 1 0 ⇒ x 1 x 1 3 x2 4 x 2 177. x 2x 2 x1 x1 0 3 x2 178. x 1 x 2x 2 3x 2x 2 x2 4 x2 3x 2 x 2x 1) 3x 1 0 x x 2 3x2 12 3x2 x1 x1 3x 2 0 3 x2 x2 3x 2 3x 3 0 2x 10 0 a 3, b 2, c 10 x x2 1 0 2 ± 22 4310 23 x 1x 1 0 x 1 0 ⇒ x 1 2 ± 124 2 ± 231 1 ± 31 6 6 3 179. x10 ⇒ x1 2x 1 5 180. 3x 2 7 3x 2 7 ⇒ x 53 2x 1 5 ⇒ x 3 3x 2 7 2x 1 5 ⇒ x 2 3x 2 7 ⇒ x 3 181. x x2 x 3 First equation: x x2 x 3 x2 3 0 x ± 3 Second equation: x x2 x 3 x2 2x 3 0 x 1x 3 0 x10 ⇒ x1 x 3 0 ⇒ x 3 Only x 3 and x 3 are solutions to the original equation. x 3 and x 1 are extraneous. Appendix A.5 Solving Equations 1189 182. x2 6x 3x 18 Second equation: First equation: x2 6x 3x 18 x2 6x 3x 18 x2 3x 18 0 0 x2 9x 18 x 3x 6 0 0 x 3x 6 x30 ⇒ x3 0 x 3 ⇒ x 3 x 6 0 ⇒ x 6 0 x 6 ⇒ x 6 The solutions to the original equation are x ± 3 and x 6. 183. x 1 x2 5 Second equation: First equation: x 1 x2 5 x 1 x2 5 x2 x 6 0 x2 x 4 0 x 3x 2 0 x30 ⇒ x3 x 2 0 ⇒ x 2 Only x 3 and x x 1 x2 5 x 1 ± 17 2 1 17 1 17 are solutions to the original equation. x 2 and x are extraneous. 2 2 184. x 10 x2 10x First equation: Second equation: x 10 x 10 x2 10x x2 10x 0 x2 11x 10 0 x2 9x 10 0 x 1x 10 0 x 10x 1 0x1 ⇒ x1 0 x 10 ⇒ x 10 0 x 10 ⇒ x 10 0 x 1 ⇒ x 1 The solutions to the original equation are x 10 and x 1. x 1 is extraneous. 185. (a) Female: y 0.432x 10.44 For y 16: 16 0.432x 10.44 26.44 0.432x 26.44 x 0.432 x 61.2 inches —CONTINUED— (b) Male: y 0.449x 12.15 For y 19: 19 0.449x 12.15 31.15 0.449x 69.4 x Yes, it is likely that both bones came from the same person because the estimated height of a male with a 19-inch thigh bone is 69.4 inches. 1190 Appendix A Review of Fundamental Concepts of Algebra 185. —CONTINUED— (c) Height x Female Femur Length Male Femur Length 60 15.48 14.79 70 19.80 19.28 80 24.12 23.77 90 28.44 28.26 100 32.76 32.75 110 37.08 37.24 The lengths of the male and female femurs are approximately equal when the lengths are 100 inches. 186. 10,000 0.32m 2500 y 0.25t 8 187. 1 0.25t 8 7500 0.32m 0.25t 7 7500 m 0.32 t 28 hours m 23,437.5 miles 189. 188. (a) S x2 4xh 84 x2 4x2 w 0 x2 8x 84 w + 14 0 x 14x 6 (b) ww 14 1632 x 14 (c) w2 14w 1632 0 Since x must be positive, we have x 6 inches. The dimensions of the box are 6 inches 6 inches 2 inches. w 48w 34 0 w 48 or w 34 Since w must be greater than zero, we have w 34 feet and the length is w 14 48 feet. 190. x2 x2 52 Pythagorean Theorem 2x2 25 x2 x 25 2 25 2 5 x 5 2 or x 6 x 52 3.54 centimeters 2 Each leg in the right triangle is approximately 3.54 centimeters. Appendix A.5 191. Model: Labels: Solving Equations 1191 height2 half of side2 side2 height 10 inches, side s, half of side Equation: 102 2s 2 s 2 s 10 s2 1s 2 s2 100 s2 4 3 2 s 100 4 s2 s 400 3 4003 203 3 11.55 inches Each side of the equilateral triangle is approximately 11.55 inches long. dN 3 hoursr 50 mph 192. dE 3 hoursr mph dN dE2 24402 2 2440 miles 3(r + 50) 9r 502 9r 2 24402 18r 2 900r 5,931,100 0 r 900 ± 9002 4185,931,100 900 ± 60118,847 218 36 3r Using the positive value for r, we have one plane moving northbound at r 50 600 miles per hour and one plane moving eastbound at r 550 miles per hour. 193. (a) P 200 million when: (b) For P 230: 182.45 3.189t 200 1.00 0.026t 182.45 3.189t 230 1.00 0.026t 182.45 3.189t 2001.00 0.026t 182.45 3.189t 2301.00 0.026t 182.45 3.189t 200 5.2t 182.45 3.189t 230 5.98t 2.011t 17.55 t 8.7 So the total voting-age population reached 200 million during 1998. 194. When C 2.5 we have: 2.5 0.2x 1 so 2.791t 47.55 t 17 The model predicts that the total voting-age population will reach 230 million during 2007. This value is reasonable but the model is reaching its limit since it soon begins to rise very fast due to its asymptotic behavior. 195. 1x 37.55 40 0.01x 1 0.01x 1 2.45 6.25 0.2x 1 0.01x 1 6.0025 5.25 0.2x 1 0.01x 5.0025 x 26.25 26,250 passengers 1 0.01x 500.25 Rounding x to the nearest whole unit yields x 500 units. 1192 Appendix A Review of Fundamental Concepts of Algebra 197. False. x3 x 10 ⇒ 3x x2 10 196. When p $750, we have: This is a quadratic equation. The equation cannot be written in the form ax b 0. 750 800 0.01x 1 50 0.01x 1 2500 0.01x 1 0.01x 2499 x 249,900 So the demand is 249,900 units when the price is $750. 198. False. The product must equal zero for the Zero-Factor property to be used. 200. False. x 0 has only one solution to check, 0. 199. False—See Example 14 on page A55. 201. Equivalent equations are derived from the substitution principle and simplification techniques. They have the same solution(s). 2x 3 8 and 2x 5 are equivalent equations. 202. To transform an equation into an equivalent equation, you should first remove symbols of grouping, combine like terms, and reduce fractions. Then, as needed, you may add (or subtract) the same quantity to (from) both sides of the equation, multiply (divide) both sides of the equation by the same nonzero quantity, or interchange the two sides of the equation. 203. The student should have subtracted 15x from both sides so that the equation is equal to zero. By factoring out an x, there are two solutions, x 0 and x 6. 204. 3x 42 x 4 2 0 (b) 3x2 8x 16 x 4 2 0 (a) Let u x 4 3u2 u 2 0 3x2 24x 48 x 4 2 0 3u 2u 1 0 3x2 25x 50 0 3u 2 0 u 2 3 x4 2 3 x 10 3 or u 1 x 4 1 or 3x 10x 5 0 u10 x 5 205. 3 and 6 3x 10 0 x 10 3 or x50 x 5 (c) The method of part (a) reduces the number of algebraic steps. 206. x 4x 11 0 One possible equation is: x 4x 11 0 x 3x 6 0 x2 15x 44 0 x 3x 6 0 x2 3x 18 0 Any non-zero multiple of this equation would also have these solutions. Appendix A.5 Solving Equations 1 208. x 6 ⇒ 6x 1 ⇒ 6x 1 is a factor. 207. 8 and 14 One possible equation is: x 25 ⇒ 5x 2 ⇒ 5x 2 is a factor. x 8x 14 0 6x 15x 2 0 22x 112 0 30x2 7x 2 0 x2 1193 Any non-zero multiple of this equation would also have these solutions. 209. 1 2 and 1 2 210. x 3 5, x 3 5, so: One possible equation is: x 1 2 x 1 2 0 x 1 2x 1 2 0 2 x 12 2 0 x 3 5 x 3 5 0 x 3 5 x 3 5 0 x 2 6x 4 0 x 2 2x 1 2 0 x 2 2x 1 0 Any non-zero multiple of this equation would also have these solutions. 9 a b 9 211. 9 9 a b 9ab9 OR a b 18 a 18 b 9 a b 9 9 a b 9 212. Isolate the absolute value by subtracting x from both sides of the equation. The expression inside the absolute value signs can be positive or negative, so two separate equations must be solved. Each solution must be checked since extraneous solutions may be included. a b ab Thus, a 18 b or a b. From the original equation we know that b ≥ 9. Some possibilities are: b 9, a 9 b 10, a 8 or a 10 b 11, a 7 or a 11 b 12, a 6 or a 12 b 13, a 5 or a 13 b 14, a 4 or a 14 213. (a) ax2 bx 0 (b) ax2 ax 0 xax b 0 axx 1 0 x0 ax b 0 ⇒ x ax 0 ⇒ x 0 b a x10 ⇒ x1 1194 Appendix A Review of Fundamental Concepts of Algebra Appendix A.6 ■ Linear Inequalities in One Variable You should know the properties of inequalities. (a) Transitive: a < b and b < c implies a < c. (b) Addition: a < b and c < d implies a c < b d. (c) Adding or Subtracting a Constant: a ± c < b ± c if a < b. (d) Multiplying or Dividing a Constant: For a < b, ■ 1. If c > 0, then ac < bc and a b < . c c 2. If c < 0, then ac > bc and a b > . c c You should be able to solve absolute value inequalities. (b) x > a if and only if x < a (a) x < a if and only if a < x < a. or x > a. Vocabulary Check 1. solution set 2. graph 3. negative 4. equivalent 5. double 6. union 1. Interval: 1, 5 2. Interval: 2, 10 3. Interval: 11, (a) Inequality: 1 ≤ x ≤ 5 (a) Inequality: 2 < x ≤ 10 (a) Inequality: x > 11 (b) The interval is bounded. (b) The interval is bounded. (b) The interval is unbounded. 5. Interval: , 2 4. Interval: 5, (a) Inequality: 5 ≤ x < or x 6. Interval: , 7 (a) Inequality: x < 2 ≥ 5 (a) Inequality: < x ≤ 7 or x ≤ 7 (b) The interval is unbounded. (b) The interval is unbounded. (b) The interval is unbounded. 9. 3 < x ≤ 4 8. x ≥ 5 7. x < 3 Matches (b). Matches (f). 10. 0 ≤ x ≤ Matches (d). 9 2 Matches (c). 11. x < 3 ⇒ 3 < x < 3 12. x > 4 ⇒ x > 4 or x < 4 Matches (e). Matches (a). 13. 5x 12 > 0 (b) x 3 (a) x 3 ? 53 12 > 0 ? 53 12 > 0 3 > 0 27 > 0 Yes, x 3 is a solution. No, x 3 is not a solution. (d) x 32 (c) x 52 ? 552 12 > 0 1 2 > 0 Yes, x 52 is a solution. ? 532 12 > 0 92 > 0 No, x 32 is not a solution. Appendix A.6 Linear Inequalities in One Variable 1195 14. 2x 1 < 3 (c) x 4 (b) x 14 (a) x 0 ? 20 1 < 3 ? 2 14 1 < 3 1 < 3 No, x 0 is not a solution. 1 2 No, x a solution. ? 2 32 1 < 3 7 < 3 2 < 3 < 3 14 is not (d) x 32 ? 24 1 < 3 Yes, x 4 is a solution. No, x 32 is not a solution. x2 < 2 4 15. 0 < (a) x 4 (b) x 10 ? 42 ? 0 < < 2 4 Yes, x 4 is a solution. 16. 1 < No, x 10 is not a solution. (d) x ? 02 ? 0 < < 2 4 0 < 1 < 2 2 No, x 0 is not a solution. 7 2 ? 72 2 ? 0 < < 2 4 0 < 3 < 2 8 Yes, x 27 is a solution. 3x ≤ 1 2 (a) x 0 1 < (c) x 1 (b) x 5 ? 30 ? 1 < ≤ 1 2 3 ≤ 1 2 No, x 0 is not a solution. ? 10 2 ? 0 < < 2 4 0 < 2 < 2 1 < 2 2 0 < (c) x 0 (d) x 5 ? 35 ? 1 < ≤ 1 2 ? 31 ? 1 < ≤ 1 2 1 < 4 ≤ 1 1 < 1 ≤ 1 No, x 5 is not a solution. Yes, x 1 is a solution. ? 35 ? 1 < ≤ 1 2 ? 1 < 1 ≤ 1 No, x 5 is not a solution. 17. x 10 ≥ 3 (a) x 13 (b) x 1 ? 13 10 ≥ 3 3 ≥ 3 Yes, x 13 is a solution. (c) x 14 ? 1 10 ≥ 3 9 10 ≥ 3 4 ≥ 3 1 ≥ 3 11 ≥ 3 Yes, x 1 is a solution. (d) x 9 ? 14 10 ≥ 3 Yes, x 14 is a solution. ? No, x 9 is not a solution. 18. 2x 3 < 15 (a) x 6 (b) x 0 ? 26 3 < 15 15 < 15 No, x 6 is not a solution. (c) x 12 ? 20 3 < 15 3 < 15 Yes, x 0 is a solution. (d) x 7 ? 212 3 < 15 27 3 < 15 21 < 15 11 < 15 No, x 12 is not a solution. ? Yes, x 7 is a solution. 1196 19. Appendix A Review of Fundamental Concepts of Algebra 20. 10x < 40 4x < 12 1 4 4x 1 4 12 < 2x > 3 21. 122x x < 4 x < 3 −5 −4 −3 1 3 2 x < x −6 < 2 3 −2 3 2 x 1 2 3 4 5 x −2 22. 6x > 15 x < or x < 52 −3 −2 11 12 x ≥ 3 4 28. 6x 4 ≤ 2 8x 6 7 7x ≥ 2 1 2 x ≥ 2 7 2 7 0 1 x −2 2 −1 0 1 2 30. 4x 1 < 2x 3 4 2x < 9 3x 4x 4 < 2x 3 x < 5 2x < 1 x ≥ 3 x < 12 x x −2 5 27. 2x 1 ≥ 1 5x 29. 4 2x < 33 x 2x ≤ 6 −3 4 x −1 −4 3 1 2 x −5 x 14 2x ≥ 1 x > 2 2 13 26. 3x 1 ≥ 2 x 2x < 4 1 3 −1 25. 2x 7 < 3 4x 0 2 x 10 x −4 1 x ≤ 5 x ≥ 12 − 52 0 24. x 7 ≤ 12 23. x 5 ≥ 7 15 6 −1 −1 3 4 5 6 7 − 21 x −2 3 31. 4x 6 ≤ x 7 32. 3 27 x > x 2 −1 0 1 33. 128x 1 ≥ 3x 52 14x ≤ 1 21 2x > 7x 14 4x 12 ≥ 3x 52 x ≥ 4 5x > 35 x ≥ 2 x < 7 x 2 3 4 5 x 6 0 1 2 3 4 x 5 34. 9x 1 < 34 16x 2 6 1 6 x 1 36. 15.6 1.3x < 5.2 1.3x < 20.8 x ≥ 4 x > 16 x −6 1 6 0 9 3.6x ≥ 14.4 12x < 2 −1 8 35. 3.6x 11 ≥ 3.4 36x 4 < 48x 6 x > 7 2 3 −5 −4 −3 −2 x 14 15 16 17 18 Appendix A.6 38. 8 ≤ 3x 5 < 13 37. 1 < 2x 3 < 9 2 < 2x < 6 8 ≤ 3x 5 < 13 1 < x < 3 3 ≤ 3x < 18 0 1 2 39. 14 < 2x 3 < 4 3 19 < 2x < 15 1 92 < x < 3 x − 6 − 5 − 4 − 3 − 2 −1 1197 12 < 2x 3 < 12 6 < x ≤ 1 x −1 Linear Inequalities in One Variable 0 15 2 15 2 −9 2 1 x − 6 −4 − 2 x3 < 5 2 0 ≤ 40. 0 ≤ x 3 < 10 3 ≤ x < 7 −3 x 0 2 4 6 − 3 4 4 6 8 x < 1 3 3 < 6 x < 3 1 3 > x > 4 4 8 2 42. 1 < 2 9 < x < 3 3 1 < x < 4 4 7 −4 −2 3 1 > x1 > 4 4 41. 0 3 < x < 9 x −1 4 1 3 5 7 9 11 4 6 x −1 3.2 ≤ 0.4x 1 ≤ 4.4 43. 44. 4.5 > 4.2 ≤ 0.4x ≤ 5.4 13.5 11 12 1.5x 6 > 10.5 2 45. x < 6 6 < x < 6 x 3 > 1.5x > 15 13 −6 −4 −2 0 2 2 > x > 10 x 10 1 9 > 1.5x 6 > 21 10.5 ≤ x ≤ 13.5 10.5 0 14 There is no solution. 47. 46. x > 4 x < 4 or x > 4 x − 6 − 4 −2 0 2 4 x > 1 2 48. x > 3 5 x x < 3 or > 3 5 5 x x < 1 or > 1 2 2 6 x < 2 x < 15 x > 2 x > 15 x − 3 − 2 −1 49. x 5 < 1 No solution. The absolute value of a number cannot be less than a negative number. 0 1 2 52. x 8 ≥ 0 6 ≤ x 20 ≤ 6 x 8 ≥ 0 or x 8 ≥ 0 14 ≤ x ≤ 26 x 8 ≥ 0 x ≥ 8 x ≥ 8 26 14 x 10 15 20 − 10 25 0 10 20 50. There is no solution because the absolute value of a number cannot be less than a negative number. 51. x 20 ≤ 6 x − 20 3 x ≤ 8 30 All real numbers x x −3 −2 −1 0 1 2 3 1198 Appendix A Review of Fundamental Concepts of Algebra 53. 3 4x ≥ 9 54. 1 2x < 5 or 3 4x ≥ 9 3 4x ≤ 9 5 < 1 2x < 5 4x ≥ 6 4x ≤ 12 x ≥ 3 x ≤ 6 < 2x < 4 32 3 > x > 2 2 < x < 3 −3 2 x −2 −1 55. 0 1 2 3 x 4 −2 −1 56. 1 x3 ≥ 4 2 x 3 ≤ 8 x3 ≥ 8 x ≤ 5 2 < x ≥ 11 5 3 2x < 1 3 1 < 1 2x < 0 3 3 > x > 0 x 0 2 2x < 1 3 11 − 15 − 10 − 5 1 x3 ≥ 4 2 x3 ≤ 4 or 2 0 10 0 < x < 3 15 x −1 57. 9 2x 2 < 1 9 2x < 1 0 1 2 3 4 58. x 14 3 > 17 1 < 9 2x < 1 x 14 > 14 x 14 < 14 or x 14 > 14 10 < 2x < 8 x < 28 x > 0 5 > x > 4 x − 35 − 28 − 21 −14 −7 0 7 4 < x < 5 x 3 4 5 6 4 5x ≤ 3 x 10 ≥ 29 60. 3 4 5x ≤ 9 59. 2 x 10 ≥ 9 x 10 ≤ 29 x ≤ − x 10 ≥ or 29 2 29 2 x ≥ 3 ≤ 4 5x ≤ 3 11 2 7 ≤ 5x ≤ 1 − 11 2 −12 −8 −4 62. 3x 1 ≤ 5 61. 6x > 12 ≥ x ≥ 1 5 1 5 ≤ x ≤ 7 5 5 2x ≤ 2 10 10 −10 2 52x ≥ 4 x ≤ 2 10 7 5 1 63. 5 2x ≥ 1 3x ≤ 6 6x > 2 7 5 x 0 9 2 x −16 1 5 10 − 10 10 − 10 10 − 10 −10 − 10 Appendix A.6 64. 3x 1 < x 7 Linear Inequalities in One Variable 65. x 8 ≤ 14 2x 9 < 13 or 2x 9 > 13 6 ≤ x ≤ 22 2x < 4 x < 2 66. 2x 9 > 13 14 ≤ x 8 ≤ 14 3x 3 < x 7 1199 10 2x < 22 2x > 4 x < 11 x > 2 10 10 −10 −10 24 − 15 10 9 −10 −10 −10 x 7 ≥ 132 67. 2 x 7 ≥ 13 68. x 7 ≤ 13 2 x7 ≥ or x ≤ 27 2 1 2 x 1 ≤ 3 x 1 ≤ 6 6 ≤ x 1 ≤ 6 13 2 7 ≤ x ≤ 5 x ≥ 21 10 10 −15 − 10 1 10 − 10 − 10 69. y 2x 3 2 70. y 3 x 1 3 (a) 2 3y ≥ 1 −4 8 y ≤ 5 (a) 2 3x 2x 3 ≥ 1 32x ≥ 4 1 ≤ 5 2 3x −5 2 3x ≥ 2 2 3x −2 1 ≥ 0 2 3x 32x ≤ 3 ≥ 1 3 x ≥ 2 3 2 71. y 12x 2 72. y 3x 8 6 0 ≤ 12 x 2 ≤ 3 2 ≤ 12 x ≤ 1 −6 1 ≤ 3x 8 ≤ 3 6 9 ≤ 3x ≤ 5 −2 3 ≥ x ≥ 5 3 y ≥ 0 12 x 2 ≥ 0 12 x ≥ 2 x ≤ 4 9 (a) 1 ≤ y ≤ 3 0 ≤ y ≤ 3 4 ≥ x ≥ 2 (b) ≤ 4 7 y ≥ 0 (b) 2x 3 ≤ 0 (a) −5 x ≤ 6 (b) 2 3y ≤ 0 2 3x ≤ 6 (b) 5 3 ≤ x ≤ 3 y ≤ 0 3x 8 ≤ 0 3x ≤ 8 x ≥ 8 3 −9 9 −3 1200 Appendix A Review of Fundamental Concepts of Algebra 73. y x 3 74. y 8 1 2 x 3 ≤ 2 −5 x 1 1 2 10 2 ≤ x 3 ≤ 2 −2 or or x 1 1 2 x3 ≥ 4 1 2x x ≥ 7 ≥ 1 1 ≤ 1 or 76. x 10 5, x ≥ 10 x ≤ 3x ≥ 0 x ≥ 3 3, 3 ≥ x , 6x ≥ 15 82. x 8 > 4 81. x 10 < 8 6x 15 ≥ 0 All real numbers within 8 units of 10. All real numbers more than 4 units from 8 x ≥ 52 7 2 52, 83. The midpoint of the interval 3, 3 is 0. The interval represents all real numbers x no more than 3 units from 0. 84. The graph shows all real numbers more than 3 units from 0. 86. The graph shows all real numbers no more than 4 units from 1. 87. All real numbers within 10 units of 12 x 1 ≤ 4 89. All real numbers more than 4 units from 3 85. The graph shows all real numbers at least 3 units from 7. x 7 ≥ 3 x 0 > 3 x > 3 x 0 ≤ 3 x ≤ 3 x 3 > 4 x 3 > 4 , 3 4 6x 15 80. 7 2 ≥ 0 78. 3 x 10, 2x ≥ 7 1 ≥ 1 x ≥ 0 77. x 3 ≥ 0 x 10 ≥ 0 x ≥ 5 1 2x 1 2x ≤ 2 x ≤ 4 79. 7 2x ≥ 0 −1 (b) y ≥ 1 1 2x 75. x 5 ≥ 0 1 10 ≤ x ≤ 6 x 3 ≥ 4 x ≤ 1 −5 1 5 ≤ 2 x ≤ 3 y ≥ 4 x 3 ≤ 4 ≤ 4 4 ≤ 12 x 1 ≤ 4 1 ≤ x ≤ 5 (b) 3 (a) y ≤ 4 y ≤ 2 (a) x 1 x 12 < 10 88. All real numbers at least 5 units from 8 x 8 ≥ 5 90. All real numbers no more than 7 units from 6 x 6 ≤ 7 Appendix A.6 91. Let x the number of checks written in a month. Type A account charges: 6.00 0.25x Type B account charges: 4.50 0.50x 6.00 0.25x < 4.50 0.50x 1.50 < 0.25x Linear Inequalities in One Variable 1201 92. 3000 0.03x ≤ 0.1x 3000 < 0.07x 42,857 < x You must make more than 42,857 copies to justify buying the copier. 6 < x If you write more than six checks a month, then the charges for the type A account are less than the charges for the type B account. 93. 10001 r 2 > 1062.50 1 2r > 1.0625 2r > 0.0625 r > 0.03125 r > 3.125% 94. 825 < 7501 r 2 825 < 7501 2r 825 < 750 1500r 75 < 1500r 0.05 < r The rate must be more than 5%. 96. 24.55x > 15.4x 150,000 95. 115.95R > C 115.95x > 95x 750 9.15 > 150,000 120.95x > 750 x > 16,393.44262 120.95x > 35.7995 Because the number of units x must be an integer, the product will return a profit when at least 16,394 units are sold. 120.95x ≥ 36 units 97. Let x daily sales level (in dozens) of doughnuts. Revenue: R 2.95x Cost: C 150 1.45x Profit: PRC 2.95x 150 1.45x 98. The goal is to lose 164 128 36 pounds. At 112 pounds per week, it will take 24 weeks. 36 112 36 23 12 2 24 1.50x 150 50 ≤ P ≤ 200 50 ≤ 1.50x 150 ≤ 200 200 ≤ 1.50x ≤ 350 13313 ≤ x ≤ 23313 In whole dozens, 134 ≤ x ≤ 234. 99. (a) y 0.067x 5.638 (b) From the graph we see that y ≥ 3 when x ≥ 129. Algebraically we have: 5 3 ≤ 0.067x 5.638 8.638 ≤ 0.067x 75 150 0 x ≥ 129 IQ scores are not a good predictor of GPAs. Other factors include study habits, class attendance, and attitude. 1202 Appendix A Review of Fundamental Concepts of Algebra 100. (a) and (b) 300 x 165 184 150 210 196 240 y 170 185 200 255 205 295 1.3x 36 179 203 159 237 219 276 x 202 170 185 190 230 160 y 190 175 195 185 250 155 1.3x 36 227 185 205 211 263 172 (c) One estimate is x ≥ 181 pounds. (d) 1.3x 36 ≥ 200 1.3x ≥ 236 x ≥ 181.5385 181.54 pounds 101. S 1.05t 31.0, 0 ≤ t ≤ 12 (a) 32 ≤ 1.05t 31 ≤ 42 140 140 260 (e) An athlete’s weight is not a particularly good indicator of the athlete’s maximum bench press weight. Other factors, such as muscle tone and exercise habits, influence maximum bench press weight. 102. (a) 70 ≤ 1.64t 67.2 ≤ 80 2.8 ≤ 1.64t ≤ 12.8 1.7 ≤ t ≤ 7.8 1 ≤ 1.05t ≤ 11 0.95 ≤ t ≤ 10.48 Rounding to the nearest year, 1 ≤ t ≤ 10. The average salary was at least $32,000 but not more than $42,000 between 1991 and 2000. (b) 1.05t 31 > 48 The number of eggs produced was between 70 and 80 billion from late 1991 until late 1997. (b) E 1.64t 67.2 ≥ 95 when t ≥ 16.95. So the annual egg production will exceed 95 billion in 2007. 1.05t > 17 t > 16 According to the model, the average salary will exceed $48,000 in 2006. 103. s 10.4 ≤ 1 16 1 16 104. 24.2 0.25 ≤ s ≤ 24.2 0.25 ≤ s 10.4 ≤ 1 16 0.0625 ≤ s 10.4 ≤ 0.0625 10.3375 ≤ s ≤ 10.4625 Since A s2, we have 23.95 ≤ s ≤ 24.45 The interval containing the possible side lengths s in centimeters of the square is 23.95, 24.45, so the interval containing the possible areas in square centimeters is 23.95 2, 24.45 2, or 573.6025, 597.8025. 10.33752 ≤ area ≤ 10.46252 106.864 ≤ area ≤ 109.464. 105. x 15 ≤ 1 10 1 10 ≤ x 15 ≤ 1 1 106. 1 oz 16 lb, so 21 oz 32 lb 1 10 14.9 ≤ x ≤ 15.1 gallons 1 10 $1.89 $0.19 You might have been undercharged or overcharged by $0.19. 1 14.99 32 0.4684375, so you could have been underor overcharged as much as $0.47. Appendix A.6 107. 1 < t 15.6 1.9 < 1 t 15.6 < 1 1.9 1.9 < t 15.6 < 1.9 1203 h 68.5 ≤ 1 2.7 1 ≤ 17.5 t 12 13 14 15 16 17 18 19 h 68.5 ≤ 1 2.7 2.7 ≤ h 68.5 ≤ 2.7 65.8 inches ≤ h ≤ 71.2 inches 13.7 < t < 17.5 Two-thirds of the workers could perform the task in the time interval between 13.7 minutes and 17.5 minutes. h 50 ≤ 30 109. 108. 13.7 Linear Inequalities in One Variable 65.8 71.2 h 65 66 67 68 69 70 71 72 110. (a) Estimate from the graph: when the plate thickness is 2 millimeters, the frequency is approximately 330 vibrations per second. 30 ≤ h 50 ≤ 30 20 ≤ h ≤ 80 The minimum relative humidity is 20 and the maximum is 80. (b) Estimate from the graph: when the frequency is 600, the plate thickness is approximately 3.6 millimeters. (c) Estimate from the graph: when the frequency is between 200 and 400 vibrations per second, the plate thickness is between 1.2 and 2.4 millimeter. (d) Estimate from the graph: when the plate thickness is less than 3 millimeters, the frequency is less than 500 vibrations per second. 111. False. If c is negative, then ac ≥ bc. 112. False. If 10 ≤ x ≤ 8, then 10 ≥ x and x ≥ 8. 113. x a ≥ 2 Matches (b). x a ≤ 2 x ≤ a 2 or xa ≥ 2 x ≥ a2 114. ax b ≤ c ⇒ c must be greater than or equal to zero. c ≤ ax b ≤ c b c ≤ ax ≤ b c Let a 1, then b c 0 and b c 10. This is true when b c 5. One set of values is a 1, b 5,c 5. (Note: This solution is not unique. Any positive multiple of these values will also work, such as a 2, b c 10 or a 3, b c 15. 1204 Appendix A Appendix A.7 Review of Fundamental Concepts of Algebra Errors and the Algebra of Calculus ■ You should be able to recognize and avoid the common algebraic errors involving parentheses, fractions, exponents, radicals, and cancellation. ■ You should be able to “unsimplify” algebraic expressions by the following methods. (a) Unusual Factoring (b) Rewriting with Negative Exponents (c) Writing a Fraction as a Sum of Terms (d) Inserting Factors or Terms Vocabulary Check 1. numerator 1. 2x 3y 4 2x 3y 4 3. 2. reciprocal 2. 5z 3x 2 5z 3x 2 Change all signs when distributing the minus sign. The 3 is distributed to both terms. 2x 3y 4 2x 3y 4 5z 3x 2 5z 3x 6 4 4 16x 2x 1 14x 1 4. Change all signs when distributing the minus sign. The expression on the right should be negative. 4 4 4 16x 2x 1 16x 2x 1 14x 1 1x x1 5 xx xx 5 5. 5z6z 30z 6. x yz xyxz z occurs twice as a factor. yz is one term, not two. 5z6z xyz xyz 7. a 30z2 xy ayax 8. 4x2 4x2 The exponent applies to the coefficient also. The fraction as a whole is multiplied by a, not the numerator and denominator separately. a 4x2 16x2 xy 1a xy axy 9. x 9 x 3 Do not apply the radical to the terms. 10. 25 x2 5 x Do not apply radicals term-by-term. x 9 does not simplify. 11. 1x x1 5 xx xx 5 2x2 1 2x 1 5x 5 Divide out common factors not common terms. 2x 1 cannot be simplified. 5x 2 25 x2 5 x5 x 12. 6x y does not simplify. 6x y Reduce common factors of the numerator and denominator, not common factors of terms. Appendix A.7 13. 1 1 a1 b1 ab 1 14. ab ab ab b a 1 y x y1 x 1 1 1 x y1 x 1y y y y xy 1 16. x2x 12 2x2 x2 15. x2 5x12 xx 512 17. Factor within grouping symbols before applying the exponent to each factor. Factor within grouping symbols before applying the exponent to each factor. x2 5x12 xx 512 x12x 512 x2x 12 x4x2 4x 1 3 4 3 x y x 4 y 3y 4x xy x yyx 1205 The negative exponent is on a term of the denominator, not a factor. To get rid of negative exponents: 1 1 a1 b1 a1 b1 Errors and the Algebra of Calculus 18. To add fractions, they must have a common denominator. 1 1 y 2y 2 Be careful when using a slash to denote division. 12y 21 y 2 y 19. 3x 2 1 3x 2 5 5 20. The required factor is 3x 2. 7x2 7 x2 10 10 The required factor is x2. 2 1 2 1 15 1 21. 3x2 3x 5 3x2 3x 3 32x2 x 15 3 1 3 2 1 22. 4x 2 4x 4 43x 2 The required factor is 3x 2. The required factor is 2x2 x 15. 1 23. x2x3 14 x3 143x2 3 24. x1 2x23 4x 1 1 2x23 4x1 2x23 4 4 411 2x 4x 1 The required factor is 3. 2 3 1 The required factor is 4. 25. 4x 6 22x 3 2 2x 3 1 1 x2 3x 73 2x2 3x 73 2x 3 x2 3x 73 x2 3x 73 1 1 The required factor is 2. 26. x1 1 x 2x 32 2 2 2x 1 x 2 2x 32 12x The required factor is 12. 2 27. 1 2x 2 2x 32 5 6x 3 3 5 3x 3 2 x 2 2 2 x 2x 2 2x 2x 2x 2x1 6x 5 3x The required factor is 3 2 1 . 2x2 1206 28. Appendix A Review of Fundamental Concepts of Algebra x 12 x 1x 12 y 52 y 52 169 x 1169 29. 9x2 16y2 9 25 49 25 x 13 y 52 169x 1 The required factor is x 1. x2 16 1 49 3x2 9y 2 4 16 13 3x2 19 9y 2 x2 y 2 13 4 19 16 43 169 31. 4 16 The required factors are 3 and 9 . 32. 1 x2 1 4916 1 259 x2 y2 259 4916 x2 y2 9 8 9x2 8y 2 y2 x2 49 78 4 7 4 7 34. 32x 1x12 4x32 x1232x 1 4x 2x 2x 35. 1 3x43 4x1 3x13 1 3x131 3x1 4x 1 3x131 7x The required factor is 1 7x. 5x322x 10x522x 2x 2x 1 1 10x32x 20x52x 2x 1 1 10x2 20x3 2x The required factor is 1 10x2 20x3. 37. 1 10 2x 3 152 162x 132 30 2x 1 322x 11 305 2x 132 1 30 2x 13232x 1 5 1 30 2x 1326x 2 1 30 2x 13223x 1 1 15 2x 1323x 1 The required factor is 3x 1. 38. 3 3 12 21 t 173 t 143 t 143t 133 t 143 7 4 28 28 3t 143 4t 1 7 28 3t 143 4t 3 28 The required factor is 4t 3. The required factor is 1 5x. The required factor is 10x 3. 1 49 and 16. 33. x13 5x43 x131 5x33 x131 5x x1210x 3 5x32 10x52 y2 1 x2 12x2 3y2 y2 12 3 y2 x2 112 23 1 2 1 2 x126x 3 4x 1 25 9 The required factors are 1 and 2. The required factors are 4 and 7. 36. y2 1 The required factors are 30. Appendix A.7 Errors and the Algebra of Calculus 1207 39. 3x2 3x22x 13 2x 13 40. x1 x 1x16 x12 x6 x12 41. 4 4 7x 3 43x1 4x4 7x2x13 3x x4 2x 42. x 8 8 1 xx 21 x2 9x3 x 2 x2 39x3 3 43. 16 5x x2 16 5x x2 16 5x x x x x x 44. x3 5x2 4 x3 5x2 4 4 2 2 2x5 2 x2 x x x x 45. 4x3 7x2 1 4x3 7x2 1 13 13 13 13 x x x x 46. 2x5 3x3 5x 1 2x5 3x3 5x 1 32 32 32 32 32 x x x x x 4x313 7x213 47. 50. 51. 7x 53 2x72 3x32 1 x13 3 5x2 x4 3 5x2 x4 x x x x 49. 4x83 2x532 3x332 5x132 x32 1 x13 3 x 48. x3 5x4 x 5x2 x3 5x4 2 2 2 3x 3x 3x 3 3 5x212 x412 3 5x32 x72 x12 2x2 332xx 13 3x 12x2 32 x 2 33x 124xx 1 3x2 3 x 132 x 16 4x2 4x 3x2 9 x2 33x 14 7x2 4x 9 x2 33x 14 x53x2 142x x2 135x4 x 4x2 146x2 5x2 1 x52 x10 x 46x2 5x2 5 x10x2 14 11x2 5 x6x2 14 6x 1327x2 2 9x3 2x36x 126 6x 126x 127x2 2 189x3 2x 3 2 6x 1 6x 16 162x3 12x 27x2 2 162x3 36x 6x 14 27x2 24x 2 6x 14 5 1 x12 x32 1208 52. 53. Appendix A Review of Fundamental Concepts of Algebra 4x2 9122 2x 312 4x2 9128x 24x2 9124x2 9 2x2x 3 4x2 9122 4x2 9 24x2 9 4x2 6x 4x2 932 29 6x 4x2 932 62x 3 4x2 932 x 234x 323 x 313x 214 x 214x 323x 2 x 3 x 2342 x 264 x2x3 x 214x 323x 264 1 x 323x 274 54. 2x 112 x 22x 112 2x 112 55. 56. x 2 2x 112 2x 1 x 2 2x 112 2x 112 2x 1 x 2 2x 112 x3 2x 112 23x 113 2x 1133x 1233 3x 12323x 1 2x 1 3x 123 3x 123 6x 2 2x 1 3x 1233x 123 4x 3 3x 143 x 1122x 3x2122 6x 2x 3x212 x 12x 3x2121 3x 2x 3x212 x 12 x 12 2x 3x212x 11 3x 2x 3x2 x 12 x 3x2 1 3x 2x 3x2 2x 3x212x 12 1 4x 2x 3x212 x 12 Appendix A.7 57. 58. Errors and the Algebra of Calculus 1209 1 1 1 1 1 x2 4122x x2 412 x2 412 22x x2 412 2 x2 1 x x2 41 x x2 4 1 1 2x2x 5 2x2 6 2x 2 6 2x 5 x2 62x 5 59. x2 512 323x 2 4x2 10x 2x2 12 6x2 10x 12 2 2 x 62x 5 x 62x 5 23x2 5x 6 x2 62x 5 3 3x 232 12 12x 2 9 5122x x2 5123x 212 xx2 5123x 232 2 9 2 x2 5123x 212 xx2 5123x 232 2 2 1 x2 5123x 2129x2 51 2x3x 21 2 1 x2 5123x 2129x2 45 6x2 4x 2 213x 2 60. 3x 2123x 6121 x 63 32 3x 21215x2 4x 45 2x2 512 3 33x 212x 612 32x 6 3x 2 32 3 3 x 6123x 23223x 2 x 652 2 61. t (a) x2 4 2 3x 6126x 4 x 652 23x 232 4 x2 4 6 1 x t 0.5 1.70 (b) She should swim to a point about 2 mile down the coast to minimize the time required to reach the finish line. 1.0 1.72 1 2 1 3 1 xx 412 x 4x2 8x 2012 xx 2 412 x 4x 2 8x 2012 2 6 6 6 1.5 1.78 1 3xx2 412 x 4x2 8x 2012 6 2.0 1.89 2.5 2.02 1 3x x4 6 x2 412 x2 8x 2012 3.0 2.18 3.5 2.36 3xx2 8x 20 x 4x2 4 6x2 4 x2 8x 20 4.0 2.57 (c) 1210 Appendix A 62. (a) y1 x2 13x 2 Review of Fundamental Concepts of Algebra 1232x x2 1132x 2xx2 123 2xx2 123 x2 2x 123 x 2 1 12 0 1 2 5 2 y1 8.7 2.9 1.1 0 2.9 8.7 12.5 x2 3x2 1 3 3 y2 8.7 2.9 1.1 0 2.9 8.7 12.5 3 x x2 (b) 2 1 4x2 3 3 2x4x2 3 3x2 123 y2 63. True. x1 y2 1 1 2 x y 64. False. Cannot move term-by-term from denominator to numerator. x xy2 y2 x2 1 1 x2y 1 1 2 1 y 1 yx y x2 2 2 x y xy 66. False. x2 9 does not factor into x 3x 3. 65. True. 1 x 4 1 x 4 x 4 x 4 x 4 x2 9 x 3x 3 x 3 x 3 x 16 67. x n x 3n x 3n x 3 x 3 x 3x 3x 3 x9 68. xn2n x2nn x2n x2n 2x2n 2 2 2 There is no error. Add exponents when multiplying powers with like bases. xn x 3n x 4n 69. x2n y2n x n y n2 70. When squaring binomials, there is also a middle term. x n y n2 x2n 2x ny n y 2n x2n x3n x2n3n x5n 3n 3n 3n 2 2 x x x x x x2 There is no error. 71. The two answers are equivalent and can be obtained by factoring. 1 10 2x 1 152 16 2x 132 60 2x 13262x 1 10 1 60 2x 13212x 4 4 60 2x 1323x 1 1 15 2x 1323x 1 (a) 2 3 x2x 2 2 332 15 2x 352 15 2x 332 5x 2x 3 2 15 2x 3323x 3 2 15 2x 3323x 1 25 2x 332x 1 (b) 2 3 x4 2 2 x32 15 4 x52 15 4 x325x 4 x 2 15 4 x324x 4 152 4 x324x 1 158 4 x32x 1 2