College Algebra 9th txtbk

7. 8. 3 4
57. 2(1 3t) 2 2 58. 2(3 2x) 2 6 5
59. 2(2t 3) 2 7 3 60. 2(3m 5) 2 4
9. (6) (2) 10. (2) (6)
SECTION 1–3 Absolute Value in Equations and Inequalities 73
In Problems 7–14, simplify, and write without absolute value
signs. Do not replace radicals with decimal approximations.
55. 2x 2 6 2
56. 2m 2 7 3
11. 5 15 12. 17 2
In Problems 61–64, solve and write answers in inequality notation.
13. 15 5 14. 2 17
Round decimals to three significant digits.
61. 2.25 1.02x 1.64
In Problems 15–20, use the number line shown to find the
62. 0.962 0.292x 2.52
indicated distances.
63. 21.7 11.3x 15.2
A
B O
C D
x
64. 195 55.5x 315
10 5 0
5
10
15. d(B, O) 16. d(A, B) 17. d(O, B)
18. d(B, A) 19. d(B, C) 20. d(D, C)
Write each of the statements in Problems 21–30 as an absolute
value equation or inequality.
21. x is 4 units from 3.
22. y is 3 units from 1.
23. m is 5 units from 2.
24. n is 7 units from 5.
25. x is less than 5 units from 3.
26. z is less than 8 units from 2.
27. p is more than 6 units from 2.
28. c is no greater than 7 units from 3.
29. q is no less than 2 units from 1.
30. d is no more than 4 units from 5.
In Problems 31–42, solve, interpret geometrically, and graph.
When applicable, write answers using both inequality notation
and interval notation.
31. y 5 3 32. t 3 4 33. y 5 6 3
34. t 3 6 4 35. y 5 7 3 36. t 3 7 4
37. u 8 3 38. x 1 5 39. u 8 3
40. x 1 5 41. u 8 3 42. x 1 5
In Problems 43–60, solve the equation or inequality. Write solutions
to inequalities using both inequality and interval notation.
43. 2x 11 13 44. 5x 20 5
45. 100 40t 7 60 46. 150 20y 6 10
47. 4x 7 13 48. 8x 3 91
49. 2w 3 4 6 2 50. 1 3z 5 6 1
51. 0.2u 1.7 0.5 52. 0.5v 2.5 7 1.6
53. 9 5 C 32 6 31 54. 5 9 (F 32) 6 40
Problems 6568 involve expressions that are important in the
study of limits in calculus. First, provide a verbal translation of
the inequality. Then solve and graph, writing your solution in
interval notation.
65. 0 6 x 3 6 0.1 66. 0 6 x 5 6 0.5
67. 0 6 x a 6 1
10
68. 0 6 x 8 6 d
In Problems 69–76, for what values of x does each hold?
69. x 2 2x 7 70. x 4 3x 8
71. 3x 5 2x 6 72. 7 2x 5 x
73. x x 3 3 74. x x 5 5
75. 3 x 2(4 x)
76. 5 2x 4(x 5)
x
77. What are the possible values of
x ?
x 1
78. What are the possible values of
x 1 ?
79. Explain why ax b 6 3 has no solution for any values of
a and b.
80. Explain why ax b 7 3 has solution all real numbers for
any values of a and b.
81. Prove that b a a b for all real numbers a and b.
[Hint: Apply Definition 1 and use cases.]
82. Prove that x 2 x 2 for all real numbers x.
83. Prove that the average of two numbers is between the two numbers;
that is, if m 6 n, then
84. Prove that for m 6 n,
d am, m n
2
85. Prove that m m.
m 6 m n
2
6 n
b d a m n , nb
2
86. Prove that m n if and only if m n or m n.