 SOLVE EQUATIONS WITH ABSOLUTE VALUE.  SOLVE INEQUALITIES WITH ABSOLUTE VALUE. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 3.5 Solving Equations and Inequalities with Absolute Value

Equations with Absolute Value Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley For a > 0 and an algebraic expression x: | x | = a is equivalent to x =  a or x = a.

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Solve:

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Solve:

More About Absolute Value Equations Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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. Similar statements hold for | x |  a and | x |  a.

Inequalities with Absolute Value Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 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 Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Solve: Solve and graph the solution set:

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Solve: Solve and graph the solution set:

Absolute Value Inequalities Absolute Value Equations Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Practice