CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and Models 3.3 Analyzing Graphs of Quadratic Functions 3.4 Solving Rational Equations and Radical Equations 3.5 Solving Equations and Inequalities with Absolute Value Copyright © 2009 Pearson Education, Inc.

3.5 Solving Equations and Inequalities with Absolute Value Solve equations with absolute value. Solve inequalities with absolute value. Copyright © 2009 Pearson Education, Inc.

Equations with Absolute Value For a > 0 and an algebraic expression X: | X | = a is equivalent to X = a or X = a. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solve: Solution: The solutions are –5 and 5. To check, note that –5 and 5 are both 5 units from 0 on the number line. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solve: Solution: First, add one to both sides to get the expression in the form | X | = a. Let’s check the possible solutions –2 and 8. Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example (continued) The possible solutions are –2 and 8. Check x = –2: Check x = 8: TRUE TRUE The solutions are –2 and 8. Copyright © 2009 Pearson Education, Inc.

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 Copyright © 2009 Pearson Education, Inc.

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. Copyright © 2009 Pearson Education, Inc.

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 Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solve: Solve and graph the solution set: Solution: The solution set is Copyright © 2009 Pearson Education, Inc.

Copyright © 2009 Pearson Education, Inc. Example Solve: Solve and graph the solution set: Solution: The solution set is Copyright © 2009 Pearson Education, Inc.