3.6 Absolute Value Inequalities Hornick

Warm Up Solve each inequality and graph the solution. 1. x + 7 < x > 1 14x ≥ 28 x < –3 –5 –4 –3–2 – x > –2 –5 –4 –3–2 – x ≥ 2 –5 –4 –3–2 –

Solve compound inequalities in one variable involving absolute-value expressions. Objectives

Example 1 Solve the inequality and graph the solutions. |x|– 3 < –1 –2 <x < 2 +3 |x| < 2 |x|– 3 < –1 Write as a compound inequality. –2 – units

|x – 1| ≤ 2 Example 2 Solve the inequality and graph the solutions. x – 1 ≥ –2 AND x – 1 ≤ ≤ x ≤ 3 –2– –3

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

Example 3 AND Solve the inequality and graph the solutions. 2|x| ≤ 6 –3 ≤ x ≤ 3 |x| ≤ 3 2|x| ≤ 6 2 –2 – units –33

Solve the inequality and graph the solutions. Example 4 OR |x| + 14 ≥ 19 |x| ≥ 5 x ≤ –5 OR x ≥ 5 Write as a compound inequality. –10 –8 –6–4 – units – 14 –14 |x| + 14 ≥ 19

Solve the inequality and graph the solutions. 3 + |x + 2| > 5 Example 5 OR |x + 2| > 2 – 3 – |x + 2| > 5 x –2 x 0 –10 –8 –6–4 –

Example 6: Application A pediatrician recommends that a baby’s bath water be 95°F, but it is acceptable for the temperature to vary from this amount by as much as 3°F. Write and solve an absolute- value inequality to find the range of acceptable temperatures. Graph the solutions. Let t = the actual water temperature. **The difference between t and the ideal temperature is at most 3°F. t – 95 ≤ 3 |t – 95| ≤ 3

Example 6 Continued |t – 95| ≤ 3 t – 95 ≥ –3 AND t – 95 ≤ t ≥ 92 AND t ≤ 98 The range of acceptable temperature is 92 ≤ t ≤

Example 6 Solve the inequality. |x + 4|– 5 > – |x + 4| > –3 Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers. All real numbers are solutions (infinitely many solutions)

Example 7 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. Be careful!

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

Work on Practice 3-6 (Multiples of 3 ONLY) Homework pg 178 #s 1, 2, 3-27(mult of 3), 29-34

Exit Card Solve each inequality and graph the solutions. 1. 3|x| > 15 0 –5– |x + 3| + 1 < 3 –2 0 –1–3–4–5–6 3. A number, n, is no more than 7 units away from 5. Write and solve an inequality to show the range of possible values for n. Solve each inequality. 4. |3x| + 1 < 1 5. |x + 2| – 3 ≥ – 6

Exit Card Solve each inequality and graph the solutions. 1. 3|x| > 15x 5 0 –5– |x + 3| + 1 < 3 –5 < x < –1 –2 0 –1–3–4–5–6 3. A number, n, is no more than 7 units away from 5. Write and solve an inequality to show the range of possible values for n. |n– 5| ≤ 7; –2 ≤ n ≤ 12

Solve each inequality. 4. |3x| + 1 < 1 5. |x + 2| – 3 ≥ – 6all real numbers no solutions Exit Card