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

Find all numbers whose absolute value is less than 5. Absolute value inequality: Compound inequality:

To Solve Absolute-Value Inequalities 1. Perform inverse operations to isolate the absolute value bars. 2. 3. Solve the compound inequality. 4. Graph the solution set.

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

Write as a compound inequality. +1 +1 +1 +1 Solve each inequality. Additional Example 1B: Solving Absolute-Value Inequalities Involving < 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 ≥ –1 x ≤ 3 AND Write as a compound inequality. –2 –1 1 2 3 –3

Write as a compound inequality. x ≥ –3 AND x ≤ 3 Check It Out! Example 1a Solve the inequality and graph the solutions. 2|x| ≤ 6 2|x| ≤ 6 2 2 Since x is multiplied by 2, divide both sides by 2 to undo the multiplication. |x| ≤ 3 Write as a compound inequality. x ≥ –3 AND x ≤ 3 –2 –1 1 2 3 units –3 3

Write as a compound inequality. x + 3 ≥ –12 AND x + 3 ≤ 12 –3 –3 –3 –3 Check It Out! Example 1b Solve each inequality and graph the solutions. |x + 3|– 4.5 ≤ 7.5 Since 4.5 is subtracted from |x + 3|, add 4.5 to both sides to undo the subtraction. + 4.5 +4.5 |x + 3| ≤ 12 |x + 3|– 4.5 ≤ 7.5 Write as a compound inequality. x + 3 ≥ –12 AND x + 3 ≤ 12 –3 –3 –3 –3 x ≥ –15 AND x ≤ 9 Subtract 3 from both sides of each inequality. –20 –15 –10 –5 5 10 15

Absolute value inequality: Find all numbers whose absolute value is greater than 5. Absolute value inequality: Compound inequality:

Solve the inequality and graph the solutions. Additional Example 2A: Solving Absolute-Value Inequalities Involving > Solve the inequality and graph the solutions. |x| + 14 ≥ 19 – 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. 5 units –10 –8 –6 –4 –2 2 4 6 8 10

Solve the inequality and graph the solutions. Additional Example 2B: Solving Absolute-Value Inequalities Involving > 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. |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. –10 –8 –6 –4 –2 2 4 6 8 10

Solve each inequality and graph the solutions. Check It Out! Example 2a Solve each inequality and graph the solutions. |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 x ≤ –2 OR x ≥ 2 Write as a compound inequality. 2 units –5 –4 –3 –2 –1 1 2 3 4 5

Check It Out! Example 2b Solve the inequality and graph the solutions. Since is added to |x + 2 |, subtract from both sides to undo the addition. Write as a compound inequality. Solve each inequality. OR Write as a compound inequality. x ≤ –6 x ≥ 1

Check It Out! Example 2b Continued Solve the inequality and graph the solutions. –7 –6 –5 –4 –3 1 2 3 –2 –1

Sec. 2-7 Practice B Wksht (1-8) & Sec. 2-7 Practice A Wksht (1-8) Homework: Sec. 2-7 Practice B Wksht (1-8) & Sec. 2-7 Practice A Wksht (1-8)

Additional Example 3: 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 represent the actual water temperature. The difference between t and the ideal temperature is at most 3°F. t – 95 ≤ 3

Additional Example 3 Continued t – 95 ≥ –3 AND t – 95 ≤ 3 Solve the two inequalities. +95 +95 t ≥ 92 AND t ≤ 98 98 100 96 94 92 90 The range of acceptable temperature is 92 ≤ t ≤ 98.

Check It Out! Example 3 A dry-chemical fire extinguisher should be pressurized to 125 psi, but it is acceptable for the pressure to differ from this value by at most 75 psi. Write and solve an absolute-value inequality to find the range of acceptable pressures. Graph the solution. Let p represent the desired pressure. The difference between p and the ideal pressure is at most 75 psi. p – 125 ≤ 75

Check It Out! Example 3 Continued p – 125 ≥ –75 AND p – 125 ≤ 75 Solve the two inequalities. +125 +125 +125 +125 p ≥ 50 AND p ≤ 200 200 225 175 150 125 100 75 50 25 The range of pressure is 50 ≤ p ≤ 200.

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.

Additional Example 4A: 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. All real numbers are solutions.

Additional Example 4B: 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.

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

Check It Out! Example 4a Solve the inequality. |x| – 9 ≥ –11 |x| – 9 ≥ –11 +9 ≥ +9 |x| ≥ –2 Add 9 to both sides. Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers. All real numbers are solutions.

Check It Out! Example 4b Solve the inequality. 4|x – 3.5| ≤ –8 4|x – 3.5| ≤ –8 4 |x – 3.5| ≤ –2 Divide both sides by 4. Absolute-value expressions are always nonnegative. Therefore, the statement is false for all values of x. The inequality has no solutions.

Sec. 2-7 Practice C & Problem Solving Worksheets Homework: Sec. 2-7 Practice C & Problem Solving Worksheets