Section 4: Absolute Value Inequalities

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 4.4 Absolute Value Inequalities 391 Version: Fall 2007 4.4 Absolute Value Inequalities In the last section, we solved absolute value equations. In this section, we turn our attention to inequalities involving absolute value. Solving | x | < a The solutions of | x | < a again depend upon the value and sign of the number a . To solve | x | < a graphically, we must determine where the graph of the left-hand side lies below the graph of the right-hand side of the inequality | x | < a . There are three cases to consider. Case I: a < 0 In this case, the graph of y = a lies strictly below the x -axis. As you can see in Figure 1 (a), the graph of y = | x | never lies below the graph of y = a . Hence, the inequality | x | < a has no solutions. Case II: a = 0 In this case, the graph of y = 0 coincides with the x -axis. As you can see in Figure 1 (b), the graph of y = | x | never lies strictly below the x -axis. Hence, the inequality | x | < 0 has no solutions. Case III: a > 0 In this case, the graph of y = a lies strictly above the x -axis. In Figure 1 (c), the graph of y = | x | and y = a intersect at x = a and x = a . In Figure 1 (c), we also see that the graph of y = | x | lies strictly below the graph of y = a when x is in-between a and a ; that is, when a < x < a . In Figure 1 (c), we’ve dropped dashed vertical lines from the points of intersec- tion of the two graphs to the x -axis. On the x -axis, we’ve shaded the solution of | x | < a , that is, a < x < a . x y y = | x | y = a x y y = | x | y = a 0 x y y = | x | y = a - a a (a) (b) (c) Figure 1. The solution of | x | < a has three cases. Copyrighted material. See: 1
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392 Chapter 4 Absolute Value Functions Version: Fall 2007 This discussion leads to the following key property. Property 1. The solution of | x | < a depends upon the value and sign of a . Case I: a < 0 The inequality | x | < a has no solution. Case II: a = 0 The inequality | x | < 0 has no solution. Case III: a > 0 The inequality | x | < a has solution set { x : a < x < a } . Let’s look at some examples. l⚏ Example 2. Solve the inequality | x | < 5 for x . The graph of the left-hand side of | x | < 5 is the “V” of Figure 1 (a). The graph of the right-hand side of | x | < 5 is a horizontal line located 5 units below the x -axis. This is the situation shown in Figure 1 (a). The graph of y = | x | is therefore never below the graph of y = 5 . Thus, the inequality | x | < 5 has no solution. An alternate approach is to consider the fact that the absolute value of x is always nonnegative and can never be less than 5 . Thus, the inequality | x | < 5 has no solution. l⚏ Example 3. Solve the inequality | x | < 0 for x . This is the case shown in Figure 1 (b). The graph of y = | x | is never strictly below the x -axis. Thus, the inequality | x | < 0 has no solution. l⚏ Example 4. Solve the inequality | x | < 8 for x . The graph of the left-hand side of | x | < 8 is the “V” of Figure 1 (c). The graph of the right-hand side of | x | < 8 is a horizontal line located 8 units above the x -axis. This is the situation depicted in Figure 1 (c). The graphs intersect at ( 8 , 8) and (8 , 8) and the graph of y = | x | lies strictly below the graph of y = 8 for values of x in-between 8 and 8. Thus, the solution of | x | < 8 is 8
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