Section 1.7 Absolute Value Inequalities 105 Solving an Absolute Value Inequality Solve the inequality. Solution: Write the inequality with the absolute value on the left. Isolate the absolute value. The inequality is in the form , where 0x 0 t 5. a Rewrite in the equivalent form 3 1 ` 1 1 ` 1 2 2 t 5 ` t 5 ` 3 1 2 t 5 2 or 1 2 t 3 or 1 2 t 7 x a or x a. Solve the compound inequality. 1 2 t 5 2 x 12 ` 1 2 t 5 ` 2 3 1 ` 1 2 t 5 ` Example 2 TIP: It is generally easier to solve an absolute value inequality if the absolute value appears on the left-hand side of the inequality. Clear fractions. 2 a1 2 tb 2132 or 2 a1 2 tb 2172 t 6 or t 14 5t 0 t 6 or t 146 6 14 The solution is or, equivalently in interval notation, 314, 1 2. , 6 4 ´ Skill Practice Solve the inequality. Write the solution in interval notation. 2. 5 6 1 ` 1 By definition, the absolute value of a real number will always be nonnegative. Therefore, the absolute value of any expression will always be greater than a negative number. Similarly, an absolute value can never be less than a negative number. Let a represent a positive real number.Then • The solution to the inequality 0 x 0 7 a is all real numbers, 1, 2 . • There is no solution to the inequality Solving Absolute Value Inequalities Example 3 Solve the inequalities. a. b. Solution: a. 03d 5 0 03d 5 0 7 7 4 7 6 4 03d 5 0 7 6 4 0 3d 5 0 6 3 No solution, 5 6 0 x 0 6 a. 3 c 1 ` Isolate the absolute value. An absolute value expression cannot be less than a negative number.Therefore, there is no solution. Answer 2. 1, 92 h 115, 2
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