Instructor’s Resource Manual Section 0.1 1 CHAPTER 0 Preliminaries 0.1 Concepts Review 1. rational numbers 2. dense 3. If not Q then not P. 4. theorems Problem Set 0.1 1. 4 2(8 11) 6 4 2( 3) 6 4 6 6 16 − − + = − − + = + + = 2. ( ) [ ] [ ] 3 2 4 7 12 3 2 4( 5) 3 2 20 3(22) 66 − − = − − ⎡ ⎤ ⎣ ⎦ = + = = 3. –4[5(–3 12 – 4) 2(13 – 7)] –4[5(5) 2(6)] –4[25 12] –4(37) –148 + + = + = + = = 4. [ ] [ ] ( ) ( ) 5 1(7 12 16) 4 2 5 1(3) 4 2 5 3 4 2 5 1 2 5 2 7 − + − + + = − + + = − + + = + = + = 5. 5 1 65 7 58 – – 7 13 91 91 91 = = 6. 3 3 1 3 3 1 4 7 21 6 3 21 6 42 6 7 43 42 42 42 42 + − = + − − − = − + − = − 7. 1 1 1 1 1 1 1 3 – 4 1 – 3 2 4 3 6 3 2 12 6 1 1 1 1 – 3 2 12 6 1 1 4 – 3 24 24 1 3 1 3 24 24 ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ + = + ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎛ ⎞ = + ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ ⎡ ⎤ = + ⎢ ⎥ ⎣ ⎦ ⎛ ⎞ = = ⎜ ⎟ ⎝ ⎠ 8. 1 2 1 1 1 1 3 5 2 3 5 2 1 5 3 3 5 2 15 15 1 2 1 2 1 2 1 3 5 2 15 3 5 15 1 6 1 1 5 1 3 15 15 3 15 9 ⎡ ⎤ ⎛ ⎞ − − − = − ⎜ ⎟ ⎢ ⎥ ⎡ ⎤ ⎛ ⎞ ⎝ ⎠ ⎣ ⎦ − − ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ ⎡ ⎤ ⎛ ⎞ ⎡ ⎤ = − − = − − ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ ⎣ ⎦ ⎛ ⎞ ⎛ ⎞ = − − = − = − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 9. 2 2 2 1 14 3 3 2 14 2 14 2 14 6 21 21 21 14 5 14 3 2 9 6 21 7 3 49 49 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ = = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 10. 2 2 35 33 5 33 11 7 7 7 7 1 7 1 6 6 2 1 7 7 7 7 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − − − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ = = = − = − ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 11. 7 11 12 11 4 – – 7 7 21 7 7 7 11 12 11 4 15 15 7 21 7 7 7 = = = + + 12. 1 3 7 4 6 7 5 5 2 4 8 8 8 8 8 1 3 7 4 6 7 3 3 2 4 8 8 8 8 8 − + − + = = = + − + − 13. 1 3 2 2 1 1 2 3 2 1 1– 1– 1– – 3 3 3 3 1 = = = = + 14. 3 3 3 2 2 2 5 2 5 7 1 2 2 2 2 6 14 6 20 2 7 7 7 7 + = + = + + − = + = + = 15. ( )( ) ( ) ( ) 2 2 5 3 5 – 3 5 – 3 5 – 3 2 + = = = © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 2 Section 0.1 Instructor’s Resource Manual 16. ( ) ( ) ( )( ) ( ) 2 2 2 5 3 5 2 5 3 3 5 2 15 3 8 2 15 − = − + = − + = − 17. 2 2 (3 4)( 1) 3 3 4 4 3 4 x x x x x x x − + = + − − = − − 18. 2 2 2 (2 3) (2 3)(2 3) 4 6 6 9 4 12 9 x x x x x x x x − = − − = − − + = − + 19. 2 2 (3 – 9)(2 1) 6 3 –18 – 9 6 –15 – 9 x x x x x x x + = + = 20. 2 2 (4 11)(3 7) 12 28 33 77 12 61 77 x x x x x x x − − = − − + = − + 21. 2 2 2 2 4 3 2 3 2 2 4 3 2 (3 1) (3 1)(3 1) 9 3 3 3 3 1 9 6 7 2 1 t t t t t t t t t t t t t t t t t t − + = − + − + = − + − + − + − + = − + − + 22. 3 2 3 2 2 3 2 (2 3) (2 3)(2 3)(2 3) (4 12 9)(2 3) 8 12 24 36 18 27 8 36 54 27 t t t t t t t t t t t t t t t + = + + + = + + + = + + + + + = + + + 23. 2 – 4 ( – 2)( 2) 2 – 2 – 2 x x x x x x + = = + , 2 x ≠ 24. 2 6 ( 3)( 2) 2 3 ( 3) x x x x x x x − − − + = = + − − , 3 x ≠ 25. 2 – 4 – 21 ( 3)( – 7) – 7 3 3 t t t t t t t + = = + + , 3 t ≠ − 26. 2 3 2 2 2 2 2 (1 ) 2 ( 2 1) 2 ( 1) ( 1)( 1) 2 1 x x x x x x x x x x x x x x x x − − = − + − + − − = − − = − − 27. 2 12 4 2 2 2 x x x x + + + + 12 4( 2) 2 ( 2) ( 2) ( 2) 12 4 8 2 6 20 ( 2) ( 2) 2(3 10) ( 2) x x x x x x x x x x x x x x x x x x + = + + + + + + + + + = = + + + = + 28. 2 2 6 2 9 1 y y y + − − 2 2(3 1) (3 1)(3 1) 2(3 1) 2 2(3 1)(3 1) 2(3 1)(3 1) y y y y y y y y y y = + − + − + = + + − + − 6 2 2 2(3 1)(3 1) y y y y + + = + − 8 2 2(3 1)(3 1) y y y + = + − 2(4 1) 4 1 2(3 1)(3 1) (3 1)(3 1) y y y y y y + + = = + − + − 29. a. 0 0 0 ⋅ = b. 0 0 is undefined. c. 0 0 17 = d. 3 0 is undefined. e. 5 0 0 = f. 0 17 1 = 30. If 0 0 a = , then 0 0 a = ⋅ , but this is meaningless because a could be any real number. No single value satisfies 0 0 a = . 31. .083 12 1.000 96 40 36 4 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Instructor’s Resource Manual Section 0.1 3 32. .285714 7 2.000000 1 4 60 56 40 35 50 49 10 7 30 28 2 33. .142857 21 3.000000 2 1 90 84 60 42 180 168 120 105 150 147 3 34. .294117... 17 5.000000... 0.2941176470588235 3 4 160 153 70 68 20 17 30 17 130 119 11 → � 35. 3.6 3 11.0 9 20 18 2 36. .846153 13 11.000000 10 4 60 52 80 78 20 13 70 65 50 39 11 37. x = 0.123123123... 1000 123.123123... 0.123123... 999 123 123 41 999 333 x x x x = = = = = 38. 0.217171717 x = … 1000 217.171717... 10 2.171717... 990 215 215 43 990 198 x x x x = = = = = 39. x = 2.56565656... 100 256.565656... 2.565656... 99 254 254 99 x x x x = = = = 40. 3.929292 x = … 100 392.929292... 3.929292... 99 389 389 99 x x x x = = = = © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 4 Section 0.1 Instructor’s Resource Manual 41. x = 0.199999... 100 19.99999... 10 1.99999... 90 18 18 1 90 5 x x x x = = = = = 42. 0.399999 x = … 100 39.99999... 10 3.99999... 90 36 36 2 90 5 x x x x = = = = = 43. Those rational numbers that can be expressed by a terminating decimal followed by zeros. 44. 1 , p p q q ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ so we only need to look at 1 . q If 2 5 , n m q = ⋅ then 1 1 1 (0.5) (0.2) . 2 5 n m n m q ⎛ ⎞ ⎛ ⎞ = ⋅ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ The product of any number of terminating decimals is also a terminating decimal, so (0.5) and (0.2) , n m and hence their product, 1 , q is a terminating decimal. Thus p q has a terminating decimal expansion. 45. Answers will vary. Possible answer: 0.000001, 12 1 0.0000010819... π ≈ 46. Smallest positive integer: 1; There is no smallest positive rational or irrational number. 47. Answers will vary. Possible answer: 3.14159101001... 48. There is no real number between 0.9999… (repeating 9's) and 1. 0.9999… and 1 represent the same real number. 49. Irrational 50. Answers will vary. Possible answers: and , π π − 2 and 2 − 51. 3 ( 3 1) 20.39230485 + ≈ 52. ( ) 4 2 3 0.0102051443 − ≈ 53. 3 41.123 – 1.09 0.00028307388 ≈ 54. ( ) 1/ 2 3.1415 0.5641979034 − ≈ 55. 2 8.9 1 – 3 0.000691744752 π + π ≈ 56. 2 4 (6 2) 3.661591807 π π − ≈ 57. Let a and b be real numbers with b a < . Let n be a natural number that satisfies a b n − < / 1 . Let } : { b n k k S > = . Since a nonempty set of integers that is bounded below contains a least element, there is a S k ∈ 0 such that b n k > / 0 but b n k ≤ − /)1 ( 0 . Then a n b n n k n k > − > − = − 1 1 1 0 0 Thus, b a n k ≤ < −1 0 . If b n k < −1 0 , then choose n k r 1 0 − = . Otherwise, choose n k r 2 0 − = . Note that 1 a b r n < − < . Given a b < , choose r so that 1 a r b < < . Then choose 2 3 , r r so that 2 1 3 a r r r b < < < < , and so on. 58. Answers will vary. Possible answer: 3 120 in ≈ 59. ft 4000 mi 5280 21,120,000 ft mi r = × = equator 2 2 (21,120,000) 132,700,874 ft r π π = = ≈ 60. Answers will vary. Possible answer: beats min hr day 70 60 24 365 20 yr min hr day year 735,840,000 beats × × × × = 61. 2 2 3 16 12 (270 12) 2 93,807,453.98 in. V r h ⎛ ⎞ = π = π ⋅ ⋅ ⎜ ⎟ ⎝ ⎠ ≈ volume of one board foot (in inches): 3 1 12 12 144 in. × × = number of board feet: 93,807,453.98 651,441 board ft 144 ≈ © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Instructor’s Resource Manual Section 0.1 5 62. 2 2 3 (8.004) (270) (8) (270) 54.3 ft. V π π = − ≈ 63. a. If I stay home from work today then it rains. If I do not stay home from work, then it does not rain. b. If the candidate will be hired then she meets all the qualifications. If the candidate will not be hired then she does not meet all the qualifications. 64. a. If I pass the course, then I got an A on the final exam. If I did not pass the course, thn I did not get an A on the final exam. b. If I take off next week, then I finished my research paper. If I do not take off next week, then I did not finish my research paper. 65. a. If a triangle is a right triangle, then 2 2 2. a b c + = If a triangle is not a right triangle, then 2 2 2. a b c + ≠ b. If the measure of angle ABC is greater than 0o and less than 90o, it is acute. If the measure of angle ABC is less than 0o or greater than 90o, then it is not acute. 66. a. If angle ABC is an acute angle, then its measure is 45o. If angle ABC is not an acute angle, then its measure is not 45o. b. If 2 2 a b < then . a b < If 2 2 a b ≥ then . a b ≥ 67. a. The statement, converse, and contrapositive are all true. b. The statement, converse, and contrapositive are all true. 68. a. The statement and contrapositive are true. The converse is false. b. The statement, converse, and contrapositive are all false. 69. a. Some isosceles triangles are not equilateral. The negation is true. b. All real numbers are integers. The original statement is true. c. Some natural number is larger than its square. The original statement is true. 70. a. Some natural number is not rational. The original statement is true. b. Every circle has area less than or equal to 9π. The original statement is true. c. Some real number is less than or equal to its square. The negation is true. 71. a. True; If x is positive, then 2 x is positive. b. False; Take 2 x = − . Then 2 0 x > but 0 x < . c. False; Take 1 2 x = . Then x x < = 4 1 2 d. True; Let x be any number. Take 2 1 y x = + . Then 2 y x > . e. True; Let y be any positive number. Take 2 y x = . Then 0 x y < < . 72. a. True; ( ) ( ) 1 : 0 1 x x x x + − < + + − < b. False; There are infinitely many prime numbers. c. True; Let x be any number. Take 1 1 y x = + . Then 1 y x > . d. True; 1/ n can be made arbitrarily close to 0. e. True; 1/ 2n can be made arbitrarily close to 0. 73. a. If n is odd, then there is an integer k such that 2 1. n k = + Then 2 2 2 2 (2 1) 4 4 1 2(2 2 ) 1 n k k k k k = + = + + = + + b. Prove the contrapositive. Suppose n is even. Then there is an integer k such that 2 . n k = Then 2 2 2 2 (2 ) 4 2(2 ) n k k k = = = . Thus 2 n is even. 74. Parts (a) and (b) prove that n is odd if and only if 2 n is odd. 75. a. 243 3 3 3 3 3 = ⋅ ⋅ ⋅ ⋅ b. 2 124 4 31 2 2 31 or 2 31 = ⋅ = ⋅ ⋅ ⋅ © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 6 Section 0.2 Instructor’s Resource Manual c. 2 2 5100 2 2550 2 2 1275 2 2 3 425 2 2 3 5 85 2 2 3 5 5 17 or 2 3 5 17 = ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 76. For example, let 2 3; A b c d = ⋅ ⋅ then 2 2 4 6 A b c d = ⋅ ⋅ , so the square of the number is the product of primes which occur an even number of times. 77. 2 2 2 2 2 ; 2 ; 2 ; p p q p q q = = = Since the prime factors of p2 must occur an even number of times, 2q2 would not be valid and 2 p q = must be irrational. 78. 2 2 2 2 3 ; 3 ; 3 ; p p q p q q = = = Since the prime factors of 2 p must occur an even number of times, 2 3q would not be valid and 3 p q = must be irrational. 79. Let a, b, p, and q be natural numbers, so a b and p q are rational. a p aq bp b q bq + + = This sum is the quotient of natural numbers, so it is also rational. 80. Assume a is irrational, 0 p q ≠ is rational, and p r a q s ⋅ = is rational. Then q r a p s ⋅ = ⋅ is rational, which is a contradiction. 81. a. – 9 –3; = rational b. 3 0.375 ; 8 = rational c. (3 2)(5 2) 15 4 30; = = rational d. 2 (1 3) 1 2 3 3 4 2 3; + = + + = + irrational 82. a. –2 b. –2 c. x = 2.4444...; 10 24.4444... 2.4444... 9 22 22 9 x x x x = = = = d. 1 e. n = 1: x = 0, n = 2: 3 , 2 x = n = 3: 2 – , 3 x = n = 4: 5 4 x = The upper bound is 3. 2 f. 2 83. a. Answers will vary. Possible answer: An example is 2 { : 5, a rational number}. S x x x = < Here the least upper bound is 5, which is real but irrational. b. True 0.2 Concepts Review 1. [ 1,5);( , 2] − −∞ − 2. b > 0; b < 0 3. (b) and (c) 4. 1 5 x − ≤ ≤ Problem Set 0.2 1. a. b. c. d. © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Instructor’s Resource Manual Section 0.2 7 e. f. 2. a. (2,7) b. [ 3,4) − c. ( , 2] −∞ − d. [ 1,3] − 3. 7 2 5 2 ;( 2, ) x x x − < − − < − ∞ 4. ( ) 3 5 4 6 1 ; 1, x x x − < − < ∞ 5. 7 – 2 9 3 –5 2 5 5 – ; – , 2 2 x x x x ≤ + ≤ ⎡ ⎞ ≥ ∞⎟ ⎢⎣ ⎠ 6. 5 3 6 4 1 ;( ,1) x x x − > − > −∞ 7. 4 3 2 5 6 3 3 2 1;( 2, 1) x x x − < + < − < < − < < − − 8. 3 4 9 11 6 4 20 3 3 5; ,5 2 2 x x x − < − < < < ⎛ ⎞ < < ⎜ ⎟ ⎝ ⎠ 9. –3 1– 6 4 –4 –6 3 2 1 1 2 – ; – , 3 2 2 3 x x x < ≤ < ≤ ⎡ ⎞ > ≥ ⎟ ⎢⎣ ⎠ 10. 4 5 3 7 1 3 2 1 2 2 1 ; , 3 3 3 3 x x x < − < − < − < ⎛ ⎞ > > − − ⎜ ⎟ ⎝ ⎠ 11. x2 + 2x – 12 < 0; 2 –2 (2) – 4(1)(–12) –2 52 2(1) 2 –1 13 x ± ± = = = ± ( ) ( ) – –1 13 – –1– 13 0; x x ⎡ ⎤ ⎡ ⎤ + < ⎣ ⎦ ⎣ ⎦ ( ) –1– 13, –1 13 + 12. 2 5 6 0 ( 1)( 6) 0; x x x x − − > + − > ( , 1) (6, ) −∞ − ∪ ∞ 13. 2x2 + 5x – 3 > 0; (2x – 1)(x + 3) > 0; 1 ( , 3) , 2 ⎛ ⎞ −∞ − ∪ ∞ ⎜ ⎟ ⎝ ⎠ 14. 2 4 5 6 0 3 (4 3)( 2) 0; ,2 4 x x x x − − < ⎛ ⎞ + − < − ⎜ ⎟ ⎝ ⎠ 15. 4 0; – 3 x x + ≤ [–4, 3) © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 8 Section 0.2 Instructor’s Resource Manual 16. 3 2 2 0; , (1, ) 1 3 x x − ⎛ ⎤ ≥ −∞ ∪ ∞ ⎜ ⎥ − ⎝ ⎦ 17. 2 5 2 5 0 2 5 0; x x x x < − < − < 2 (– , 0) , 5 ⎛ ⎞ ∞ ∪ ∞ ⎜ ⎟ ⎝ ⎠ 18. 7 7 4 7 7 0 4 7 28 0; 4 x x x x ≤ − ≤ − ≤ ( ) 1 ,0 , 4 ⎡ ⎞ −∞ ∪ ∞⎟ ⎢⎣ ⎠ 19. 1 4 3 2 1 4 0 3 2 1 4(3 2) 0 3 2 9 12 2 3 0; , , 3 2 3 4 x x x x x x ≤ − − ≤ − − − ≤ − − ⎛ ⎞ ⎡ ⎞ ≤ −∞ ∪ ∞ ⎜ ⎟ ⎟ ⎢ − ⎝ ⎠ ⎣ ⎠ 20. 3 2 5 3 2 0 5 3 2( 5) 0 5 2 7 7 0; 5, 5 2 x x x x x x > + − > + − + > + − − ⎛ ⎞ > − − ⎜ ⎟ + ⎝ ⎠ 21. ( 2)( 1)( 3) 0;( 2,1) (3,8) x x x + − − > − ∪ 22. 3 1 (2 3)(3 1)( 2) 0; , ,2 2 3 x x x ⎛ ⎞ ⎛ ⎞ + − − < −∞ − ∪ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 23. 2 (2 -3)( -1) ( -3) 0; x x x ≥ [ ) 3 – , 3, 2 ⎛ ⎤ ∞ ∪ ∞ ⎜ ⎥ ⎝ ⎦ 24. ( ) ( ) 2 (2 3)( 1) ( 3) 0; 3 ,1 1, 3, 2 x x x − − − > ⎛ ⎞ −∞ ∪ ∪ ∞ ⎜ ⎟ ⎝ ⎠ 25. 3 2 – – 2 – 5 6 0 ( 5 – 6) 0 ( 1)( – 6) 0; x x x x x x x x x < < + < ( , 1) (0,6) −∞ − ∪ 26. 3 2 2 2 1 0 ( 1)( 1) 0 ( 1)( 1) 0; x x x x x x x − − + > − − > + − > ( 1,1) (1, ) − ∪ ∞ 27. a. False. b. True. c. False. © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Instructor’s Resource Manual Section 0.2 9 28. a. True. b. True. c. False. 29. a. ⇒ Let a b < , so 2 ab b < . Also, 2 a ab < . Thus, 2 2 a ab b < < and 2 2 a b < . ⇐ Let 2 2 a b < , so a b ≠ Then ( ) ( ) 2 2 2 2 2 0 2 2 2 a b a ab b b ab b b b a < − = − + < − + = − Since 0 b > , we can divide by 2b to get 0 b a − > . b. We can divide or multiply an inequality by any positive number. 1 1 1 a a b b b a < ⇔ < ⇔ < . 30. (b) and (c) are true. (a) is false: Take 1, 1 a b = − = . (d) is false: if a b ≤ , then a b − ≥ − . 31. a. 3x + 7 > 1 and 2x + 1 < 3 3x > –6 and 2x < 2 x > –2 and x < 1; (–2, 1) b. 3x + 7 > 1 and 2x + 1 > –4 3x > –6 and 2x > –5 x > –2 and 5 – ; 2 x > ( ) 2, − ∞ c. 3x + 7 > 1 and 2x + 1 < –4 x > –2 and 5 – ; 2 x < ∅ 32. a. 2 7 1 or 2 1 3 2 8 or 2 2 4 or 1 x x x x x x − > + < > < > < ( ,1) (4, ) −∞ ∪ ∞ b. 2 7 1 or 2 1 3 2 8 or 2 2 4 or 1 ( ,4] x x x x x x − ≤ + < ≤ < ≤ < −∞ c. 2 7 1 or 2 1 3 2 8 or 2 2 4 or 1 ( , ) x x x x x x − ≤ + > ≤ > ≤ > −∞ ∞ 33. a. 2 2 – 3 2 2 – – 3 2 ( 1)( 2 – 7) 1 3 5 – 7 1 2 – 5 – 6 0 ( 3)( 1)( – 2) 0 x x x x x x x x x x x x x x + + ≥ + ≥ + ≥ + + ≥ [ 3, 1] [2, ) − − ∪ ∞ b. 4 2 4 2 2 2 2 2 8 2 8 0 ( 4)( 2) 0 ( 2)( 2)( 2) 0 x x x x x x x x x − − − − ≥ ≥ + ≥ + + − ≥ ( , 2] [2, ) −∞ − ∪ ∞ c. 2 2 2 2 2 2 2 ( 1) 7( 1) 10 0 [( 1) 5][( 1) 2] 0 ( 4)( 1) 0 ( 2)( 1)( 1)( 2) 0 x x x x x x x x x x − − − + + + < + − + − < < + + − − < ( 2, 1) (1,2) − − ∪ 34. a. 01 .2 1 99 .1 < < x x x 01 .2 1 99 .1 < < 1 99 .1 < x and x 01 .2 1< 99 .1 1 < x and 01 .2 1 > x 99 .1 1 01 .2 1 < < x ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 99 .1 1 , 01 .2 1 b. 01 .3 2 1 99 .2 < + < x ) 2 ( 01 .3 1 ) 2 ( 99 .2 + < < + x x 1 98 .5 99 .2 < + x and 02 .6 01 .3 1 + < x 99 .2 98 .4 − < x and 01 .3 02 .5 − > x 99 .2 98 .4 01 .3 02 .5 − < < − x ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − 99 .2 98 .4 , 01 .3 02 .5 35. 2 5; 2 5 or 2 5 3 or 7 x x x x x − ≥ − ≤ − − ≥ ≤ − ≥ ( , 3] [7, ) −∞ − ∪ ∞ © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 10 Section 0.2 Instructor’s Resource Manual 36. 2 1; x + < –1 2 1 –3 –1 x x < + < < < (–3, –1) 37. 4 5 10; 10 4 5 10 15 4 5 15 5 15 5 ; , 4 4 4 4 x x x x + ≤ − ≤ + ≤ − ≤ ≤ ⎡ ⎤ − ≤ ≤ − ⎢ ⎥ ⎣ ⎦ 38. 2 –1 2; x > 2x – 1 < –2 or 2x – 1 > 2 2x < –1 or 2x > 3; 1 3 1 3 – or , – , – , 2 2 2 2 x x ⎛ ⎞ ⎛ ⎞ < > ∞ ∪ ∞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 39. 2 5 7 7 2 2 5 7 or 5 7 7 7 2 2 2 or 12 7 7 7 or 42; x x x x x x x − ≥ − ≤ − − ≥ ≤ − ≥ ≤ − ≥ ( , 7] [42, ) −∞ − ∪ ∞ 40. 1 1 4 x + < 1 1 1 4 2 0; 4 x x − < + < − < < –8 < x < 0; (–8, 0) 41. 5 6 1; 5 6 1 or 5 6 1 5 5 or 5 7 7 7 1 or ;( ,1) , 5 5 x x x x x x x − > − < − − > < > ⎛ ⎞ < > −∞ ∪ ∞ ⎜ ⎟ ⎝ ⎠ 42. 2 – 7 3; x > 2x – 7 < –3 or 2x – 7 > 3 2x < 4 or 2x > 10 x < 2 or x > 5; ( ,2) (5, ) −∞ ∪ ∞ 43. 1 3 6; 1 1 3 6 or 3 6 1 1 3 0 or 9 0 x x x x x − > − < − − > + < − > 1 3 1 9 0 or 0; 1 1 ,0 0, 3 9 x x x x + − < > ⎛ ⎞ ⎛ ⎞ − ∪ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 44. 5 2 1; x + > 5 5 2 –1 or 2 1 5 5 3 0 or 1 0 3 5 5 0 or 0; x x x x x x x x + < + > + < + > + + < > 5 (– , – 5) – , 0 (0, ) 3 ⎛ ⎞ ∞ ∪ ∪ ∞ ⎜ ⎟ ⎝ ⎠ 45. 2 3 4 0; x x − − ≥ 2 3 (–3) – 4(1)(–4) 3 5 –1,4 2(1) 2 x ± ± = = = ( 1)( 4) 0;( , 1] [4, ) x x + − = −∞ − ∪ ∞ 46. 2 2 4 ( 4) 4(1)(4) 4 4 0; 2 2(1) ( 2)( 2) 0; 2 x x x x x x ± − − − + ≤ = = − − ≤ = 47. 3x2 + 17x – 6 > 0; 2 –17 (17) – 4(3)(–6) –17 19 1 –6, 2(3) 6 3 x ± ± = = = (3x – 1)(x + 6) > 0; 1 (– , – 6) , 3 ⎛ ⎞ ∞ ∪ ∞ ⎜ ⎟ ⎝ ⎠ 48. 2 14 11 15 0; x x + − ≤ 2 11 (11) 4(14)( 15) 11 31 2(14) 28 3 5 , 2 7 x x − ± − − − ± = = = − 3 5 3 5 0; , 2 7 2 7 x x ⎛ ⎞⎛ ⎞ ⎡ ⎤ + − ≤ − ⎜ ⎟⎜ ⎟ ⎢ ⎥ ⎝ ⎠⎝ ⎠ ⎣ ⎦ 49. 3 0.5 5 3 5(0.5) 5 15 2.5 x x x − < ⇒ − < ⇒ − < 50. 2 0.3 4 2 4(0.3) 4 18 1.2 x x x + < ⇒ + < ⇒ + < © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Instructor’s Resource Manual Section 0.2 11 51. 2 6 2 6 12 6 x x x ε ε ε − < ⇒ − < ⇒ − < 52. 4 2 4 2 8 2 x x x ε ε ε + < ⇒ + < ⇒ + < 53. 3 15 3( 5) 3 5 5 ; 3 3 x x x x ε ε ε ε ε δ − < ⇒ − < ⇒ − < ⇒ − < = 54. 4 8 4( 2) 4 2 2 ; 4 4 x x x x ε ε ε ε ε δ − < ⇒ − < ⇒ − < ⇒ − < = 55. 6 36 6( 6) 6 6 6 ; 6 6 x x x x ε ε ε ε ε δ + < ⇒ + < ⇒ + < ⇒ + < = 56. 5 25 5( 5) 5 5 5 ; 5 5 x x x x ε ε ε ε ε δ + < ⇒ + < ⇒ + < ⇒ + < = 57. C d π = –10 0.02 –10 0.02 10 – 0.02 10 0.02 – 0.0064 C d d d ≤ π ≤ ⎛ ⎞ π ≤ ⎜ ⎟ π ⎝ ⎠ ≤ ≈ π π We must measure the diameter to an accuracy of 0.0064 in. 58. ( ) 5 50 1.5, 32 50 1.5; 9 C F − ≤ − − ≤ ( ) 5 32 90 1.5 9 122 2.7 F F − − ≤ − ≤ We are allowed an error of 2.7� F. 59. 2 2 2 2 – – 2 – –1 2 – 3 –1 2 – 6 ( –1) (2 – 6) 2 1 4 24 36 3 22 35 0 (3 – 7)( – 5) 0; x x x x x x x x x x x x x x < < < + < + + > > 7 – , (5, ) 3 ⎛ ⎞ ∞ ∪ ∞ ⎜ ⎟ ⎝ ⎠ 60. ( )2 2 2 1 1 (2 1) 1 x x x x − ≥ + − ≥ + 2 2 2 4 4 1 2 1 3 6 0 3 ( 2) 0 x x x x x x x x − + ≥ + + − ≥ − ≥ ( ,0] [2, ) −∞ ∪ ∞ 61. 2 2 2 2 2 2 2 3 10 4 6 10 (4 6) ( 10) 16 48 36 20 100 15 68 64 0 (5 4)(3 16) 0; x x x x x x x x x x x x x x − − − < + − < + − < + + < + + − < + − < 4 16 – , 5 3 ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ 62. ( )( ) 2 2 2 2 2 3 1 2 6 3 1 2 12 (3 1) (2 12) 9 6 1 4 48 144 5 54 143 0 5 11 13 0 x x x x x x x x x x x x x x − < + − < + − < + − + < + + − − < + − < 11,13 5 ⎛ ⎞ − ⎜ ⎟ ⎝ ⎠ © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 12 Section 0.2 Instructor’s Resource Manual 63. ( ) 2 2 2 2 2 2 and Order property: when is positive. Transitivity x y x x x y x y y y x y xz yz z x y x y x x < ⇒ ≤ < < ⇔ < ⇒ < ⇒ < = Conversely, ( ) ( )( ) 2 2 2 2 2 2 2 2 2 – 0 Subtract from each side. – 0 Factor the difference of two squares. – 0 This is the only factor that can be negative. Add t x y x y x x x y y x y x y x y x y y < ⇒ < = ⇒ < ⇒ + < ⇒ < ⇒ < o each side. 64. ( ) ( ) 2 2 0 , a b a a and b b < < ⇒ = = so ( ) ( ) 2 2 , a b < and, by Problem 63, a b a b < ⇒ < . 65. a. – (– ) – a b a b a b a b = + ≤ + = + b. – – – a b a b a b ≥ ≥ Use Property 4 of absolute values. c. ( ) a b c a b c a b c a b c + + = + + ≤ + + ≤ + + 66. 2 2 2 2 2 1 1 1 1 2 2 3 3 1 1 2 3 1 1 2 3 1 1 2 3 x x x x x x x x x x ⎛ ⎞ − = + − ⎜ ⎟ ⎜ ⎟ + + + + ⎝ ⎠ ≤ + − + + = + + + = + + + by the Triangular Inequality, and since 2 3 0, x + > 2 1 1 2 0 0, 0. 2 3 x x x + > ⇒ > > + + 2 3 3 x + ≥ and 2 2, x + ≥ so 2 1 1 3 3 x ≤ + and 1 1 , 2 2 x ≤ + thus, 2 1 1 1 1 2 3 2 3 x x + ≤ + + + 67. 2 2 – 2 (–2) 9 9 x x x x + = + + 2 2 2 – 2 –2 9 9 9 x x x x x ≤ + + + + 2 2 2 2 2 – 2 2 9 9 9 9 x x x x x x x + ≤ + = + + + + Since 2 2 1 1 9 9, 9 9 x x + ≥ ≤ + 2 2 2 9 9 x x x + + ≤ + 2 2 – 2 9 9 x x x + ≤ + 68. 2 2 2 2 7 2 7 4 4 7 15 x x x x x ≤ ⇒ + + ≤ + + ≤ + + = and 2 1 1 x + ≥ so 2 1 1. 1 x ≤ + Thus, 2 2 2 2 2 7 1 2 7 1 1 15 1 15 x x x x x x + + = + + + + ≤ ⋅ = 69. 4 3 2 1 1 1 1 2 4 8 16 x x x x + + + + 4 3 2 4 3 2 1 1 1 1 2 4 8 16 1 1 1 1 1 since 1. 2 4 8 16 1 1 1 1 So 1.9375 2. 2 4 8 16 x x x x x x x x x ≤ + + + + ≤ + + + + ≤ + + + + ≤ < © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Instructor’s Resource Manual Section 0.3 13 70. a. 2 2 0 (1 ) 0 0 or 1 x x x x x x x x < − < − < < > b. 2 2 0 ( 1) 0 0 1 x x x x x x x < − < − < < < 71. 0 a ≠ ⇒ 2 2 2 1 1 0 – – 2 a a a a ⎛ ⎞ ≤ = + ⎜ ⎟ ⎝ ⎠ so, 2 2 2 2 1 1 2 or 2 a a a a ≤ + + ≥ . 72. and 2 2 2 a b a a a b a b b b a a b b a b a b < + < + + < + < + < + < < 73. 2 2 2 2 0 and a b a ab ab b a ab b a ab b < < < < < < < < 74. ( ) ( ) ( ) 2 2 2 2 2 2 2 1 1 2 2 4 1 1 1 1 0 2 4 2 4 4 1 0 ( ) which is always true. 4 ab a b ab a ab b a ab b a ab b a b ≤ + ⇔ ≤ + + ⇔ ≤ − + = − + ⇔ ≤ − 75. For a rectangle the area is ab, while for a square the area is 2 2 . 2 a b a + ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ From Problem 74, 2 1 ( ) 2 2 a b ab a b ab + ⎛ ⎞ ≤ + ⇔ ≤ ⎜ ⎟ ⎝ ⎠ so the square has the largest area. 76. 2 3 99 1 0; ( , 1] x x x x + + + + + ≤ −∞ − … 77. 1 1 1 1 10 20 30 R ≤ + + 1 6 3 2 60 1 11 60 60 11 R R R + + ≤ ≤ ≥ 1 1 1 1 20 30 40 1 6 4 3 120 120 13 R R R ≥ + + + + ≥ ≤ Thus, 60 120 11 13 R ≤ ≤ 78. 2 2 2 2 2 4 ; 4 (10) 400 4 400 0.01 4 100 0.01 0.01 100 4 A r A r r r π π π π π π π = = = − < − < − < 2 0.01 0.01 100 4 4 0.01 0.01 100 100 4 4 0.00004 in r r π π π π δ − < − < − < < + ≈ 0.3 Concepts Review 1. 2 2 ( 2) ( 3) x y + + − 2. (x + 4)2 + (y – 2)2 = 25 3. 2 5 3 7 , (1.5,5) 2 2 − + + ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ 4. d b c a − − © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 14 Section 0.3 Instructor’s Resource Manual Problem Set 0.3 1. 2 2 (3 –1) (1–1) 4 2 d = + = = 2. 2 2 ( 3 2) (5 2) 74 8.60 d = − − + + = ≈ 3. 2 2 (4 – 5) (5 8) 170 13.04 d = + + = ≈ 4. 2 2 ( 1 6) (5 3) 49 4 53 7.28 d = − − + − = + = ≈ 5. 2 2 1 (5 2) (3 – 4) 49 1 50 d = + + = + = 2 2 (5 10) (3 8) 25 25 50 2 2 2 ( 2 10) (4 8) 3 144 16 160 so the triangle is isosceles. 1 2 d d d d = − + − = + = = − − + − = + = = 6. 2 2 (2 4) ( 4 0) 4 16 20 a = − + − − = + = 2 2 (4 8) (0 2) 16 4 20 b = − + + = + = 2 2 (2 8) ( 4 2) 36 4 40 c = − + − + = + = 2 2 2, a b c + = so the triangle is a right triangle. 7. (–1, –1), (–1, 3); (7, –1), (7, 3); (1, 1), (5, 1) 8. ( ) 2 2 2 2 2 2 ( 3) (0 1) ( 6) (0 4) ; 6 10 12 52 6 42 7 7,0 x x x x x x x x − + − = − + − − + = − + = = ⇒ 9. –2 4 –2 3 1 , 1, ; 2 2 2 + + ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 2 1 25 2 (1 2) – 3 9 3.91 2 4 d ⎛ ⎞ = + + = + ≈ ⎜ ⎟ ⎝ ⎠ 10. 1 2 3 6 3 9 midpoint of , , 2 2 2 2 AB + + ⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 4 3 7 4 7 11 midpoint of , , 2 2 2 2 CD + + ⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 2 2 3 7 9 11 2 2 2 2 4 1 5 2.24 d ⎛ ⎞ ⎛ ⎞ = − + − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = + = ≈ 11 (x – 1)2 + (y – 1)2 = 1 12. 2 2 2 2 2 ( 2) ( 3) 4 ( 2) ( 3) 16 x y x y + + − = + + − = 13. 2 2 2 2 2 2 2 2 2 ( 2) ( 1) (5 2) (3 1) 9 16 25 ( 2) ( 1) 25 x y r r r x y − + + = − + + = = + = − + + = © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. Instructor’s Resource Manual Section 0.3 15 14. 2 2 2 ( 4) ( 3) x y r − + − = 2 2 2 2 2 2 (6 4) (2 3) 4 1 5 ( 4) ( 3) 5 r r x y − + − = = + = − + − = 15. 1 3 3 7 center , (2, 5) 2 2 + + ⎛ ⎞ = = ⎜ ⎟ ⎝ ⎠ 1 1 2 2 radius (1– 3) (3 – 7) 4 16 2 2 1 20 5 2 = + = + = = 2 2 ( – 2) ( – 5) 5 x y + = 16. Since the circle is tangent to the x-axis, 4. r = 2 2 ( 3) ( 4) 16 x y − + − = 17. 2 2 2 10 – 6 –10 0 x x y y + + + = 2 2 2 2 2 2 2 – 6 0 ( 2 1) ( – 6 9) 1 9 ( 1) ( – 3) 10 x x y y x x y y x y + + = + + + + = + + + = center (–1, 3); radius 10 = = 18. 2 2 2 2 2 2 6 16 ( 6 9) 16 9 ( 3) 25 x y y x y y x y + − = + − + = + + − = center (0, 3); radius 5 = = 19. 2 2 –12 35 0 x y x + + = 2 2 2 2 2 2 –12 –35 ( –12 36) –35 36 ( – 6) 1 x x y x x y x y + = + + = + + = center (6, 0); radius 1 = = 20. 2 2 2 2 2 2 10 10 0 ( 10 25) ( 10 25) 25 25 ( 5) ( 5) 50 x y x y x x y y x y + − + = − + + + + = + − + + = ( ) center 5, 5 ; radius 50 5 2 = − = = 21. 2 2 4 16 15 4 6 0 x x y y + + + + = 2 2 3 9 9 4( 4 4) 4 15 16 2 16 4 x x y y ⎛ ⎞ + + + + + = − + + ⎜ ⎟ ⎝ ⎠ 2 2 3 13 4( 2) 4 4 4 x y ⎛ ⎞ + + + = ⎜ ⎟ ⎝ ⎠ 2 2 3 13 ( 2) 4 16 x y ⎛ ⎞ + + + = ⎜ ⎟ ⎝ ⎠ center = 3 2, ; 4 ⎛ ⎞ − − ⎜ ⎟ ⎝ ⎠ radius = 13 4 22. 2 2 2 2 105 4 16 4 3 0 16 3 9 4( 4 4) 4 4 64 105 9 16 16 16 x x y y x x y y + + + + = ⎛ ⎞ + + + + + ⎜ ⎟ ⎝ ⎠ = − + + 2 2 2 2 3 4( 2) 4 10 8 3 5 ( 2) 8 2 x y x y ⎛ ⎞ + + + = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ + + + = ⎜ ⎟ ⎝ ⎠ 3 5 10 center 2, ; radius 8 2 2 ⎛ ⎞ = − − = = ⎜ ⎟ ⎝ ⎠ 23. 2 –1 1 2 –1 = 24. 7 5 2 4 3 − = − 25. –6 – 3 9 –5 – 2 7 = 26. 6 4 1 0 2 − + = − 27. 5 – 0 5 – 0 – 3 3 = 28. 6 0 1 0 6 − = + 29. 2 1( 2) 2 2 4 0 y x y x x y − = − − − = − + + − = 30. 4 1( 3) 4 3 7 0 y x y x x y − = − − − = − + + − = 31. 2 3 2 – 3 0 y x x y = + + = 32. 0 5 0 5 0 y x x y = + + − = © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher.