1. Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 3.3 - 1 3.3 Absolute Value Equations and Inequalities

2. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 2 3.3 Absolute Value Equations and Inequalities Summary: Solving Absolute Value Equations and Inequalities 1. To solve | ax + b | = k, solve the following compound equation. Let k be a positive real number, and p and q be real numbers. ax + b = k or ax + b = – k. The solution set is usually of the form {p, q}, which includes two numbers. pq

3. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 3 3.3 Absolute Value Equations and Inequalities Summary: Solving Absolute Value Equations and Inequalities 2. To solve | ax + b | > k, solve the following compound inequality. Let k be a positive real number, and p and q be real numbers. ax + b > k or ax + b < – k. The solution set is of the form (- ∞, p ) U ( q, ∞ ), which consists of two separate intervals. pq

4. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 4 3.3 Absolute Value Equations and Inequalities Summary: Solving Absolute Value Equations and Inequalities 3. To solve | ax + b | < k, solve the three-part inequality Let k be a positive real number, and p and q be real numbers. – k < ax + b < k The solution set is of the form ( p, q ), a single interval. pq

5. 3.3 Absolute Value Equations and Inequalities EXAMPLE 1Solving an Absolute Value Equation Solve |2 x + 3| = 5.

6. 3.3 Absolute Value Equations and Inequalities EXAMPLE 2Solving an Absolute Value Inequality with > Solve |2 x + 3| > 5.

7. 3.3 Absolute Value Equations and Inequalities EXAMPLE 3Solving an Absolute Value Inequality with < Solve |2 x + 3| < 5.

8. 3.3 Absolute Value Equations and Inequalities EXAMPLE 4Solving an Absolute Value Equation That Requires Rewriting Solve the equation | x – 7| + 6 = 9.

9. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 9 3.3 Absolute Value Equations and Inequalities Special Cases for Absolute Value 1. The absolute value of an expression can never be negative: | a | ≥ 0 for all real numbers a. 2. The absolute value of an expression equals 0 only when the expression is equal to 0.

10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 10 3.3 Absolute Value Equations and Inequalities EXAMPLE 6Solving Special Cases of Absolute Value Equations Solve each equation. See Case 1 in the preceding slide. Since the absolute value of an expression can never be negative, there are no solutions for this equation. The solution set is Ø. (a)|2 n + 3| = –7 See Case 2 in the preceding slide. The absolute value of the expres- sion 6 w – 1 will equal 0 only if 6 w – 1 = 0. (b)|6 w – 1| = 0 The solution of this equation is. Thus, the solution set of the original equation is { }, with just one element. Check by substitution. 1 6 1 6

11. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 11 3.3 Absolute Value Equations and Inequalities EXAMPLE 7Solving Special Cases of Absolute Value Inequalities Solve each inequality. The absolute value of a number is always greater than or equal to 0. Thus, | x | ≥ –2 is true for all real numbers. The solution set is (–∞, ∞ ). (a)| x | ≥ –2 Add 1 to each side to get the absolute value expression alone on one side. | x + 5| < –7 (b)| x + 5| – 1 < –8 There is no number whose absolute value is less than –7, so this inequality has no solution. The solution set is Ø.

12. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 3.3 - 12 3.3 Absolute Value Equations and Inequalities EXAMPLE 7Solving Special Cases of Absolute Value Inequalities Solve each inequality. Subtracting 2 from each side gives | x – 9| ≤ 0 (c)| x – 9| + 2 ≤ 2 The value of | x – 9| will never be less than 0. However, | x – 9| will equal 0 when x = 9. Therefore, the solution set is {9}.