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Section 2.7 Absolute Value Inequalities 131 Student Check 4 Solve each word problem with an absolute value inequality. a. A Gallup Poll, taken in August 2008, revealed that 48% of U.S. workers are completely satisfied with their job. The poll had a margin of error of 5%. What is the range of workers who are completely satisfied with their job? (Source: www.gallup.com) b. A tractor trailer accident at mile marker 51 on I-68 in West Virginia caused a hazardous chemical spill. Emergency management personnel advised that motorists should be at least 8 mi from the site of the spill. What are safe locations on I-68 for motorists? Using Test Points to Solve Absolute Value Inequalities In the first two objectives, compound inequalities were used to solve absolute value inequalities. Absolute value inequalities can be solved using an alternate method involving test points. In this method, the graph of the solution set is first constructed and, from this, the solution set is obtained. We will examine the graph of the solution set of the inequality we solved in Example 2 part (b). The inequality uy + 3u ≥ 4 was shown to have the following solution. t t t c t t –6 –4 –2 2 4 6 8 c t t –12 –10 –8 0 Notice that the number line is separated into regions separated by the numbers -7 and 1. The significance of the numbers -7 and 1 is that they are the solutions of the equation uy + 3u = 4. uy + 3u = 4 y + 3=-4 or y + 3 = 4 y + 3 - 3=-4 -3 y + 3 - 3 = 4 - 3 y=-7 y = 1 The parts of the number line that are shaded include solutions of the inequality. If numbers from these regions are substituted into the inequality, a true statement will result. The values -10 and 3 are included in the shaded regions. When we substitute these numbers in for y, we get u-10 + 3u ≥ 4S u-7u ≥ 4S7 ≥ 4 True u3 + 3u ≥ 4S u6u ≥ 4S6 ≥ 4 True The part of the number line that is not shaded includes numbers that are not solutions of the inequality. If numbers from this region are substituted into the inequality, a false statement will result. The value -4 is in the region that is not shaded. When we substitute this value for y, we get u-4 + 3u ≥ 4S u-1u ≥ 4S1 ≥ 4 False So, this example illustrates that the solutions of the associated absolute value equation separate the number line into intervals of numbers that are or are not solutions of the absolute value inequality. Procedure: Solving an Absolute Value Inequality Using Test Points Step 1: Solve the associated absolute value equation. This equation comes from replacing the inequality symbol with an equals sign. Step 2: Place the solutions of the equation on a number line. Objective 5 ▶ Solve absolute value inequalities using test points.


hendricks_intermediate_algebra_1e_ch1_3
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