187 1.8 Absolute Value Equations and Inequalities 80. Wind Power Extraction Tests When a model kite was flown in crosswinds in tests to determine its limits of power extraction, it attained speeds of 98 to 148 ft per sec in winds of 16 to 26 ft per sec. Using x as the variable in each case, write absolute value inequalities that correspond to these ranges. Relating Concepts For individual or collaborative investigation (Exercises 83–86) To see how to solve an equation that involves the absolute value of a quadratic polynomial, such as x2 - x = 6, work Exercises 83–86 in order. 83. For x2 - x to have an absolute value equal to 6, what are the two possible values that x may assume? (Hint: One is positive and the other is negative.) 84. Write an equation stating that x2 - x is equal to the positive value found in Exercise 83, and solve it using the zero-factor property. 85. Write an equation stating that x2 - x is equal to the negative value found in Exercise 83, and solve it using the quadratic formula. (Hint: The solutions are not real numbers.) 86. Give the complete solution set of x2 - x = 6, using the results from Exercises 84 and 85. (Modeling) Carbon Dioxide Emissions When humans breathe, carbon dioxide is emitted. In one study, the emission rates of carbon dioxide by college students were measured during both lectures and exams. The average individual rate RL (in grams per hour) during a lecture class satisfied the inequality RL - 26.75 … 1.42, whereas during an exam, the rate RE satisfied the inequality RE - 38.75 … 2.17. (Data from Wang, T. C., ASHRAE Trans., 81 (Part 1), 32.) Use this information to solve each problem. 81. Find the range of values for RL and RE. 82. The class had 225 students. If TL and TE represent the total amounts of carbon dioxide in grams emitted during a 1-hour lecture and a 1-hour exam, respectively, write inequalities that model the ranges for TL and TE. Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) 87. 3x2 + x = 14 88. 2x2 - 3x = 5 89. 3x2 - 14x = 5 90. 2x2 - 7x = 3 91. x2 + 5x + 5 = 1 92. x2 + 7x + 11 = 1 93. x2 - 9 = x + 3 94. 2x2 + 3 = x + 4 95. 4x2 - 23x - 6 = 0 96. 6x3 + 23x2 + 7x = 0 97. x2 + 1 - 2x = 0 98. ` x2 + 2 x ` - 11 3 = 0 99. x4 + 2x2 + 1 60 100. x2 + 10 60 101. ` x - 4 3x + 1 ` Ú 0 102. ` 9 - x 7 + 8x ` Ú 0
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