Solve each inequality. Graph the solutions.

57. | 3 x − 4 | + 5 ≤ 27 vertical line 3 x minus 4 vertical line plus 5 less than or equal to 27
58. | 2 x + 3 | − 6 ≥ 7 vertical line 2 x plus 3 vertical line negative 6 greater than or equal to 7
59. − 2 | x + 4 | < 22 negative 2 vertical line x plus 4 vertical line less than 22
60. 2 | 4 t − 1 | + 6 > 20 2 vertical line 4 t minus 1 vertical line plus 6 greater than 20
61. | 3 z + 15 | ≥ 0 vertical line 3 z plus 15 vertical line greater than or equal to 0
62. | − 2 x + 1 | > 2 vertical line negative 2 x plus 1 vertical line greater than 2
63. 1 9 | 5 x − 3 | − 3 ≥ 2 1 ninth , vertical line 5 x minus 3 vertical line negative 3 greater than or equal to 2
64. 1 11 | 2 x − 4 | + 10 ≤ 11 1 eleventh absolute value of , 2 x minus 4 , end absolute value , . plus 10 less than or equal to 11
65. | x − 3 2 | + 2 < 6 absolute value of . fraction x minus 3 , over 2 end fraction , end absolute value , . plus 2 less than 6
66. | x + 5 3 | − 3 > 6 absolute value of . fraction x plus 5 , over 3 end fraction , end absolute value , . minus 3 greater than 6
67. Writing Describe the differences in the graphs of | x | < a vertical line x vertical line less than eh  and | x | > a , vertical line x vertical line greater than eh comma  where a is a positive real number.
68. Open-Ended Write an absolute value inequality for which every real number is a solution. Write an absolute value inequality that has no solution.

Write an absolute value inequality to represent each situation.

69. Cooking Suppose you used an oven thermometer while baking and discovered that the oven temperature varied between +5 and − 5 negative 5  degrees from the setting. If your oven is set to 350°, let t be the actual temperature.
70. Time Workers at a hardware store take their morning break no earlier than 10 A.M. and no later than noon. Let c represent the time the workers take their break.
71. Climate A friend is planning a trip to Alaska. He purchased a coat that is recommended for outdoor temperatures from − 15 ° F negative 15 degrees cap f  to 45°F. Let t represent the temperature for which the coat is intended.

Write an absolute value inequality and a compound inequality for each length x with the given tolerance.

72. a length of 36.80 mm with a tolerance of 0.05 mm
73. a length of 9.55 mm with a tolerance of 0.02 mm
74. a length of 100 yd with a tolerance of 4 in.

Is the absolute value inequality or equation always, sometimes, or never true? Explain.

75. | x | = − 6 vertical line x vertical line equals negative 6
76. − 8 > | x | negative 8 greater than vertical line x vertical line
77. | x | = x vertical line x vertical line equals x
78. | x | + | x | = 2 x vertical line x vertical line plus vertical line x vertical line equals 2 x
79. | x + 2 | = x + 2 vertical line x plus 2 vertical line equals x plus 2
80. ( | x | ) 2 < x 2 open vertical line x vertical line , close squared , less than , x squared
81. Error Analysis A classmate wrote the solution to the inequality | − 4 x + 1 | > 3 vertical line negative 4 x plus 1 vertical line greater than 3  as shown. Describe and correct the error.

    
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