Algebra 1 Chapter 3 Section 7

3-7: Solving Absolute-Value Inequalities 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.

Example 1: Solving Absolute-Value Inequalities Involving < Solve the inequality and graph the solutions. +3 +3 |x| < 2 |x|– 3 < –1 Since 3 is subtracted from |x|, add 3 to both sides to undo the subtraction. {x: –2 < x < 2}. Write as a compound inequality. The solution set is x > –2 AND x < 2 –2 –1 1 2 2 units

Example 2: Solving Absolute-Value Inequalities Involving < Solve the inequality |x – 1| ≤ 2 and graph the solutions. x – 1 ≥ –2 AND x – 1 ≤ 2 Write as a compound inequality. +1 +1 +1 +1 Solve each inequality. x ≥ –1 x ≤ 3 AND {x: –1 ≤ x ≤ 3}. Write as a compound inequality. The solution set is –2 –1 1 2 3 –3

Just as you do when solving absolute-value equations, you first isolate the absolute-value expression when solving absolute-value inequalities. Helpful Hint

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.

Example 3: Solving Absolute-Value Inequalities Involving > Solve the inequality |x| + 14 ≥ 19 and graph the solutions. – 14 –14 |x| + 14 ≥ 19 Since 14 is added to |x|, subtract 14 from both sides to undo the addition. |x| ≥ 5 x ≤ –5 OR x ≥ 5 Write as a compound inequality. The solution set is {x: x ≤ –5 OR x ≥ 5}. 5 units –10 –8 –6 –4 –2 2 4 6 8 10

Example 4: Solving Absolute-Value Inequalities Involving > Solve the inequality 3 + |x + 2| > 5 and graph the solutions. Since 3 is added to |x + 2|, subtract 3 from both sides to undo the addition. |x + 2| > 2 – 3 – 3 3 + |x + 2| > 5 Write as a compound inequality. Solve each inequality. x + 2 < –2 OR x + 2 > 2 –2 –2 –2 –2 x < –4 OR x > 0 Write as a compound inequality. The solution set is {x: x < –4 or x > 0}. –10 –8 –6 –4 –2 2 4 6 8 10

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. Its solution set is ø.

Example 5: Special Cases of Absolute-Value Inequalities Solve the inequality |x + 4|– 5 > – 8 . |x + 4|– 5 > – 8 + 5 + 5 |x + 4| > –3 Add 5 to both sides. Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers. The solution set is all real numbers.

Example 6: Special Cases of Absolute-Value Inequalities Solve the inequality |x – 2| + 9 < 7. |x – 2| + 9 < 7 – 9 – 9 |x – 2| < –2 Subtract 9 from both sides. Absolute-value expressions are always nonnegative. Therefore, the statement is false for all values of x. The inequality has no solutions. The solution set is ø.

An absolute value represents a distance, and distance cannot be less than 0. Remember!