180 CHAPTER 1 Equations and Inequalities Relating Concepts For individual or collaborative investigation (Exercises 95–98) Inequalities that involve more than two factors, such as 13x - 421x + 221x + 62 … 0, can be solved using an extension of the method shown in Examples 5 and 6. Work Exercises 95–98 in order, to see how the method is extended. 95. Use the zero-factor property to solve 13x - 421x + 221x + 62 = 0. 96. Plot the three solutions in Exercise 95 on a number line, using closed circles because of the nonstrict inequality, …. 97. The number line from Exercise 96 should show four intervals formed by the three points. For each interval, choose a test value from the interval and decide whether it satisfies the original inequality. 98. On a single number line, do the following. (a) Graph the intervals that satisfy the inequality, including endpoints. This is the graph of the solution set of the inequality. (b) Write the solution set in interval notation. Use the technique described in Exercises 95–98 to solve each inequality. Give the solution set in interval notation. 99. 12x - 321x + 221x - 32 Ú 0 100. 1x + 5213x - 421x + 22 Ú 0 103. 1x + 1221x - 32 60 104. 1x - 5221x + 12 60 105. x3 + 4x2 - 9x Ú 36 106. x3 + 3x2 - 16x … 48 107. x21x + 422 Ú 0 108. -x212x - 322 … 0 1.8 Absolute Value Equations and Inequalities ■ Basic Concepts ■ Absolute Value Equations ■ Absolute Value Inequalities ■ Special Cases ■ Absolute Value Models for Distance and Tolerance Basic Concepts Recall that the absolute value of a number a, written ∣ a ∣, gives the undirected distance from a to 0 on a number line. By this definition, the equation x = 3 can be solved by finding all real numbers at a distance of 3 units from 0. As shown in Figure 19, two numbers satisfy this equation, -3 and 3, so the solution set is 5-3, 36. 0 –3 3 Distance is less than 3. Distance is greater than 3. Distance is greater than 3. Distance is less than 3. Distance is 3. Distance is 3. Figure 19 Similarly, x 63 is satisfied by all real numbers whose undirected distances from 0 are less than 3. As shown in Figure 19, this is the interval -3 6x 63, or 1-3, 32. 101. 4x - x3 Ú 0 102. 16x - x3 Ú 0
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