Solving Absolute-Value Inequalities 3-Ext Solving Absolute-Value Inequalities Lesson Presentation Holt Algebra 1

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 written as –5 < x < 5, which is the compound inequality x > –5 AND x < 5.

Example 1A: Solving Absolute-Value Inequalities Involving < Solve the inequality and graph the solutions. Then write the solutions as a compound inequality. |x| + 2 ≤ 6 Since 2 is added to |x|, subtract 2 from both sides to undo the addition. |x| + 2 ≤ 6 –2 –2 |x| ≤ 4 Think, “The distance from x to 0 is less than or equal to 4 units.” –5 –4 –3 –2 –1 1 2 3 4 5 4 units x ≥ –4 AND x ≤ 4 Write as a compound inequality. –4 ≤ x ≤ 4

Example 1B: Solving Absolute-Value Inequalities Involving < Solve the inequality and graph the solutions. Then write the solutions as a compound inequality. |x| – 5 < –4 |x| – 5 < –4 Since 5 is subtracted from |x|, add 5 to both sides to undo the subtraction. +5 +5 |x| < 1 Think, “The distance from x to 0 is less than 1unit.” –2 –1 1 2 unit x is between –1 and 1. –1 < x AND x < 1 Write as a compound inequality. –1 < x < 1

Example 1C: Solving Absolute-Value Inequalities Involving < Solve the inequality and graph the solutions. Then write the solutions as a compound inequality. |x + 4| – 1.5 < 3.5 |x + 4| – 1.5 < 3.5 Since 1.5 is subtracted from |x + 4|, add 1.5 to both sides to undo the subtraction. +1.5 +1.5 |x + 4| < 5 5 units Think, “The distance from x to –4 is less than 5 units.” –5 –4 –3 –2 –1 1 2 3 4 5 x + 4 > –5 AND x + 4 < 5 x + 4 is between –5 and 5. –4 –4 –4 –4 x > –9 AND x < 1

Write as a compound inequality. Example 1C Continued x > –9 AND x < 1 Write as a compound inequality. –10 –8 –6 –4 –2 2 4 6 8 10 –9 < x < 1

Think, “The distance from x to 0 is less than 3 units.” Check It Out! Example 1a Solve the inequality and graph the solutions. Then write the solutions as a compound inequality. |x| + 12 < 15 Since 12 is added to |x|, subtract 12 from both sides to undo the addition. |x| + 12 < 15 – 12 –12 |x| < 3 –5 –4 –3 –2 –1 1 2 3 4 5 3 units Think, “The distance from x to 0 is less than 3 units.” x is between –3 and 3. x > –3 AND x < 3 Write as a compound inequality. –3 < x < 3

Think, “The distance from x to 0 is less than 1 unit.” Check It Out! Example 1b Solve the inequality and graph the solutions. Then write the solutions as a compound inequality. |x| – 6 < –5 Since 6 is subtracted from |x|, add 6 to both sides to undo the subtraction. |x| – 6 < –5 + 6 +6 |x| < 1 1 unit Think, “The distance from x to 0 is less than 1 unit.” –2 –1 1 2 x is between –1 and 1. x > –1 AND x < 1 Write as a compound inequality. –1 < x < 1

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 written as the compound inequality x < –5 OR x > 5.

Example 2A: Solving Absolute-Value Inequalities Involving > Solve the inequality and graph the solutions. Then write the solutions as a compound inequality. |x| + 2 > 7 |x| + 2 > 7 Since 2 is added to |x|, subtract 2 from both sides to undo the addition. – 2 –2 |x| > 5 5 units –10 –8 –6 –4 –2 2 4 6 8 10 x < –5 OR x > 5 Write as a compound inequality.

Example 2B: Solving Absolute-Value Inequalities Involving > Solve the inequality and graph the solutions. Then write the solutions as a compound inequality. |x| – 12 ≥ –8 Since 12 is subtracted from |x|, add 12 to both sides to undo the subtraction. |x| – 12 ≥ –8 + 12 +12 |x| ≥ 4 4 units –10 –8 –6 –4 –2 2 4 6 8 10 x ≤ –4 OR x ≥ 4 Write as a compound inequality.

Example 2C: Solving Absolute-Value Inequalities Involving > Solve the inequality and graph the solutions. Then write the solutions as a compound inequality. |x + 3| – 5 > 9 |x + 3| – 5 > 9 Since 5 is subtracted from |x + 3|, add 5 to both sides to undo the subtraction. + 5 +5 |x + 3| > 14 14 units –16 –12 –8 –4 4 8 12 16 x + 3 < –14 OR x + 3 > 14

Solve the two inequalities. – 3 –3 –3 –3 Example 2C Continued x + 3 < –14 OR x + 3 > 14 Solve the two inequalities. – 3 –3 –3 –3 x < –17 OR x > 11 –24 –20 –16 –12 –8 –4 4 8 12 16 –17 11 Graph.

Write as a compound inequality. Check It Out! Example 2a Solve the inequality and graph the solutions. Then write the solutions as a compound inequality. |x| + 10 ≥ 12 |x| + 10 ≥ 12 Since 10 is added to |x|, subtract 10 from both sides to undo the addition. – 10 –10 |x| ≥ 2 2 units –5 –4 –3 –2 –1 1 2 3 4 5 x ≤ –2 OR x ≥ 2 Write as a compound inequality.

Write as a compound inequality. Check It Out! Example 2b Solve the inequality and graph the solutions. Then write the solutions as a compound inequality. |x| – 7 > –1 |x| – 7 > –1 Since 7 is subtracted from |x|, add 7 to both sides to undo the subtraction. +7 +7 |x| > 6 6 units –10 –8 –6 –4 –2 2 4 6 8 10 x < –6 OR x > 6 Write as a compound inequality.

Check It Out! Example 2c Solve the inequality and graph the solutions. Then write the solutions as a compound inequality. Since is added to |x + 2 |, subtract from both sides to undo the addition. 3.5 units –5 –4 –3 –2 –1 1 2 3 4 5 OR

Check It Out! Example 2c Continued OR Solve the two inequalities. OR –10 –8 –6 –4 –2 2 4 6 8 10 1 Graph.