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3-7 Solving Absolute-Value Inequalities Warm Up Solve each inequality and graph the solution. 1. x + 7 < 4 x < –3 –5 –4 –3 –2 –1 0 1 2 3 4 5 2. 14x ≥ 28 3. 5 + 2x > 1 Holt McDougal Algebra 1 x≥2 x > –2 –5 –4 –3 –2 –1 0 1 2 3 4 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 3-7 Solving Absolute-Value Inequalities Objectives Solve compound inequalities in one variable involving absolute-value expressions. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Example 1 Solve the inequality and graph the solutions. |x|– 3 < –1 |x|– 3 < –1 +3 +3 |x| < 2 Since 3 is subtracted from |x|, add 3 to both sides to undo the subtraction. x > –2 AND x < 2 Write as a compound inequality. 2 units –2 –1 2 units 0 Holt McDougal Algebra 1 1 2 3-7 Solving Absolute-Value Inequalities Example 2 Solve the inequality and graph the solutions. |x – 1| ≤ 2 x – 1 ≥ –2 AND x – 1 ≤ 2 Write as a compound inequality. +1 +1 +1 +1 Solve each inequality. x ≥ –1AND –3 –2 –1 0 Holt McDougal Algebra 1 x ≤ 3 Write as a compound inequality. 1 2 3 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 McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Example 3 AND Solve the inequality and graph the solutions. 2|x| ≤ 6 2|x| ≤ 6 2 2 |x| ≤ 3 x ≥ –3 AND x ≤ 3 3 units –3 –2 –1 Holt McDougal Algebra 1 Since x is multiplied by 2, divide both sides by 2 to undo the multiplication. Write as a compound inequality. 3 units 0 1 2 3 3-7 Solving Absolute-Value Inequalities Example 4 OR Solve the inequality and graph the solutions. |x| + 14 ≥ 19 |x| + 14 ≥ 19 – 14 –14 |x| ≥ 5 x ≤ –5 OR x ≥ 5 Since 14 is added to |x|, subtract 14 from both sides to undo the addition. Write as a compound inequality. 5 units 5 units –10 –8 –6 –4 –2 0 Holt McDougal Algebra 1 2 4 6 8 10 3-7 Solving Absolute-Value Inequalities Example 5 OR Solve the inequality and graph the solutions. 3 + |x + 2| > 5 Since 3 is added to |x + 2|, subtract 3 from both sides to undo the addition. 3 + |x + 2| > 5 –3 –3 |x + 2| > 2 Write as a compound inequality. x + 2 < –2 OR x + 2 > 2 Solve each inequality. –2 –2 –2 –2 x < –4 OR x > 0 Write as a compound inequality. –10 –8 –6 –4 –2 0 Holt McDougal Algebra 1 2 4 6 8 10 3-7 Solving Absolute-Value Inequalities 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 absolutevalue inequality to find the range of acceptable temperatures. Graph the solutions. Let t represent the actual water temperature. The difference between t and the ideal temperature is at most 3°F. t – 95 Holt McDougal Algebra 1 ≤ 3 3-7 Solving Absolute-Value Inequalities Example 6 Continued t – 95 ≤ 3 |t – 95| ≤ 3 t – 95 ≥ –3 AND t – 95 ≤ 3 +95 +95 +95 +95 t ≥ 92 AND t ≤ 98 90 92 94 96 98 Solve the two inequalities. 100 The range of acceptable temperature is 92 ≤ t ≤ 98. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Example 6 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. All real numbers are solutions. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Example 7 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. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Remember! An absolute value represents a distance, and distance cannot be less than 0. Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities • Work on Practice 3-6 (Multiples of 3 ONLY) • Homework page 128 #s 1-27 odd do not do #23 Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Lesson Quiz: Part I Solve each inequality and graph the solutions. 1. 3|x| > 15 –10 –5 x < –5 or x > 5 0 2. |x + 3| + 1 < 3 –6 –5 –4 5 10 –5 < x < –1 –3 –2 –1 0 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 Holt McDougal Algebra 1 3-7 Solving Absolute-Value Inequalities Lesson Quiz: Part II Solve each inequality. 4. |3x| + 1 < 1 no solutions 5. |x + 2| – 3 ≥ – 6 Holt McDougal Algebra 1 all real numbers