Solving Equations and Inequalities with Absolute Value Section 3.5 Solving Equations and Inequalities with Absolute Value Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.

Objectives Solve equations with absolute value. Solve inequalities with absolute value.

Equations with Absolute Value For a > 0 and an algebraic expression X: | X | = a is equivalent to X = a or X = a.

Example Solve: The solutions are –5 and 5. To check, note that –5 and 5 are both 5 units from 0 on the number line.

Example Solve: First, add one to both sides to get the expression in the form | X | = a. Let’s check the possible solutions –2 and 8.

Example (continued) The possible solutions are –2 and 8. Check x = –2: TRUE TRUE The solutions are –2 and 8.

More About Absolute Value Equations When a = 0, | X | = a is equivalent to X = 0. Note that for a < 0, | X | = a has no solution, because the absolute value of an expression is never negative. The solution is the empty set, denoted

Inequalities with Absolute Value Inequalities sometimes contain absolute-value notation. The following properties are used to solve them. For a > 0 and an algebraic expression X: | X | < a is equivalent to a < X < a. | X | > a is equivalent to X < a or X > a. Similar statements hold for | X |  a and | X |  a.

Inequalities with Absolute Value For example, | x | < 3 is equivalent to 3 < x < 3 | y | ≥ 1 is equivalent to y ≤ 1 or y ≥ 1 | 2x + 3 | ≤ 4 is equivalent to 4 < 2x + 3 < 4

Example Solve and graph the solution set: The solution set is

Example Solve and graph the solution set: The solution set is