Test Point Method to Solve Inequalities 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.) 2. Plot the boundary points on the number line. This divides the number line into intervals. 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. 4. Test the boundary points in the original inequality. • If the original inequality is strict (< or >), do not include the boundary 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. If a is negative (a < 0), then 1. 0x 0 0x 0 6 a has no solution. 2. 7 a is true for all real numbers. Example 2 Summary 121 Isolate the absolute value. Solve the related equation. 0x 3 0 2 7 0x 3 0 5 0x 3 0 5 x 3 5 or x 3 5 x 8 x 2 or Boundary points Interval I: I II III 2 8 Test x 3: 0 132 3 0 2 7 True Interval II: ? Test x 0: 0 102 3 0 2 7 False Interval III: ? ? Test x 0 0 9: 192 3 2 7 True True False True The solution is Example 3 0x 5 0 7 2 2 8 1, 24 ´ 38, 2. The solution is all real numbers because an absolute value will always be greater than a negative number. 1, 2 Example 4 0x 5 0 6 2 There is no solution because an absolute value cannot be less than a negative number. The solution set is 5 6.
miller_intermediate_algebra_4e_ch1_3
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