3 7 Solving AbsoluteValue Inequalities Objectives Solve compound

3 -7 Solving Absolute-Value Inequalities Objectives Solve compound inequalities in one variable involving absolute-value expressions. When an inequality contains an absolute-value expression, it can be written as a compound inequality. The inequality |x| < 5 describes all real numbers whose distance from 0 is less than 5 units. The solutions are all numbers between – 5 and 5, so |x|< 5 can be rewritten as – 5 < x < 5, or as x > – 5 AND x < 5. Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities LESS THAN = LESS “AND” Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Additional Example 1 A: Solving Absolute-Value Inequalities Involving < Solve the inequality and graph the solutions. |x|– 3 < – 1 +3 +3 |x| < 2 x > – 2 AND x < 2 2 units – 2 – 1 2 units 0 Holt Mc. Dougal Algebra 1 1 2

3 -7 Solving Absolute-Value Inequalities Additional Example 1 B: Solving Absolute-Value Inequalities Involving < Solve the inequality and graph the solutions. |x – 1| ≤ 2 Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Helpful Hint Just as you do when solving absolute-value equations, you first isolate the absolute-value expression when solving absolute-value inequalities. Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Check It Out! Example 1 a Solve the inequality and graph the solutions. 2|x| ≤ 6 Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Check It Out! Example 1 b Solve each inequality and graph the solutions. |x + 3|– 4. 5 ≤ 7. 5 Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities The inequality |x| > 5 describes all real numbers whose distance from 0 is greater than 5 units. The solutions are all numbers less than – 5 or greater than 5. The inequality |x| > 5 can be rewritten as the compound inequality x < – 5 OR x > 5. Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Additional Example 2 A: Solving Absolute-Value Inequalities Involving > Solve the inequality and graph the solutions. |x| + 14 ≥ 19 – 14 |x| ≥ 5 x ≤ – 5 OR x ≥ 5 5 units – 10 – 8 – 6 – 4 – 2 0 Holt Mc. Dougal Algebra 1 2 4 6 8 10

3 -7 Solving Absolute-Value Inequalities Additional Example 2 B: Solving Absolute-Value Inequalities Involving > Solve the inequality and graph the solutions. 3 + |x + 2| > 5 Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Check It Out! Example 2 a Solve each inequality and graph the solutions. |x| + 10 ≥ 12 Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Check It Out! Example 2 b Solve the inequality and graph the solutions. Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities When solving an absolute-value inequality, you may get a statement that is true for all values of the variable. In this case, all real numbers are solutions of the original inequality. If you get a false statement when solving an absolute-value inequality, the original inequality has no solutions. Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Additional Example 4 A: Special Cases of Absolute. Value Inequalities Solve the inequality. |x + 4|– 5 > – 8 +5 +5 |x + 4| > – 3 Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers. All real numbers are solutions. Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Additional Example 4 B: Special Cases of Absolute. Value Inequalities Solve the inequality. |x – 2| + 9 < 7 – 9 |x – 2| < – 2 Absolute-value expressions are always nonnegative. Therefore, the statement is false for all values of x. The inequality has no solutions. Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Remember! An absolute value represents a distance, and distance cannot be less than 0. Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Check It Out! Example 4 a Solve the inequality. |x| – 9 ≥ – 11 Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Check It Out! Example 4 b Solve the inequality. 4|x – 3. 5| ≤ – 8 Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Lesson Quiz: Part I Solve each inequality and graph the solutions. 1. 3|x| > 15 – 10 x < – 5 or x > 5 – 5 0 2. |x + 3| + 1 < 3 – 6 – 5 Holt Mc. Dougal Algebra 1 – 4 5 10 – 5 < x < – 1 – 3 – 2 – 1 0

3 -7 Solving Absolute-Value Inequalities Lesson Quiz: Part II Solve each inequality. 3. |3 x| + 1 < 1 no solutions 4. |x + 2| – 3 ≥ – 6 Holt Mc. Dougal Algebra 1 all real numbers