Section 1.7 Absolute Value Inequalities 107 2. Solving Absolute Value Inequalities by the Test Point Method For the problems in Examples 1 and 2, the absolute value inequality was converted to an equivalent compound inequality. However, sometimes students have difficulty setting up the appropriate compound inequality. To avoid this problem, you may want to use the test point method to solve absolute value inequalities. Solving Inequalities by Using the Test Point Method Step 1 Find the boundary points of the inequality. (Boundary points are the real solutions to the related equation and points where the inequality is undefined.) Step 2 Plot the boundary points on the number line.This divides the Write as two equations. number line into intervals. Step 3 Select a test point from each interval and substitute it into the original inequality. • If a test point makes the original inequality true, then that interval is part of the solution set. Step 4 Test the boundary points in the original inequality. 16 72 • If the original inequality is strict or , do not include the boundary points in the solution set. • If the original inequality is defined using or , then include the boundary points that are well defined within the inequality. Note: Any boundary point that makes an expression within the inequality undefined must always be excluded from the solution set. To demonstrate the use of the test point method, we will repeat the absolute value inequalities from Examples 1 and 2. Notice that regardless of the method used, the absolute value is always isolated first before any further action is taken. Solving an Absolute Value Inequality by the Test Point Method Solve the inequality by using the test point method. Solution: Isolate the absolute value. Step 1: Solve the related equation. These are the only boundary points. Step 2: Plot the boundary points. 03w 1 0 11 3w 1 11 or 3w 1 11 3w 10 or 3w 12 w 10 3 or w 4 0 3w 1 0 6 11 0 3w 1 0 4 6 7 03w 1 0 4 6 7 Example 5
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