2 7 Solving AbsoluteValue Inequalities Warm Up Solve

2 -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. 14 x ≥ 28 3. 5 + 2 x > 1 Holt Mc. Dougal Algebra 1 x≥ 2 x > – 2 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5

2 -7 Solving Absolute-Value Inequalities Additional Example 1 A: 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. |x|– 3 < – 1 +3 +3 |x| < 2 x > – 2 AND x < 2 2 units – 2 – 1 Write as a compound inequality. 2 units 0 Holt Mc. Dougal Algebra 1 1 2

2 -7 Solving Absolute-Value Inequalities Additional Example 1 B: 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 Solve each inequality. x ≥ – 1 AND – 3 – 2 – 1 0 Holt Mc. Dougal Algebra 1 x ≤ 3 Write as a compound inequality. 1 2 3

2 -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 Mc. Dougal Algebra 1

2 -7 Solving Absolute-Value Inequalities Check It Out! Example 1 b Solve each inequality and graph the solutions. |x + 3|– 4. 5 ≤ 7. 5 Since 4. 5 is subtracted from | + 3|, add 4. 5 to both sides to undo the subtraction. |x + 3|– 4. 5 ≤ 7. 5 + 4. 5 +4. 5 |x + 3| ≤ 12 x + 3 ≥ – 12 AND x + 3 ≤ 12 – 3 – 3 x ≥ – 15 AND x≤ 9 – 20 – 15 – 10 – 5 Holt Mc. Dougal Algebra 1 0 5 10 15 Write as a compound inequality. Subtract 3 from both sides of each inequality.

2 -7 Solving Absolute-Value Inequalities Holt Mc. Dougal Algebra 1

2 -7 Solving Absolute-Value Inequalities Additional Example 2 B: 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. 3 + |x + 2| > 5 – 3 |x + 2| > 2 Write as a compound inequality. x + 2 < – 2 OR x + 2 > 2 Solve each inequality. – 2 – 2 x < – 4 OR x > 0 Write as a compound inequality. – 10 – 8 – 6 – 4 – 2 0 Holt Mc. Dougal Algebra 1 2 4 6 8 10

2 -7 Solving Absolute-Value Inequalities Check It Out! Example 2 b 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 Holt Mc. Dougal Algebra 1 x≥ 1

2 -7 Solving Absolute-Value Inequalities 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 absolutevalue 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 Holt Mc. Dougal Algebra 1 ≤ 75

2 -7 Solving Absolute-Value Inequalities How can I write an absolute-value equation from a graph? 1. Is it a greater than or less than inequality? 2. What is the distance from the end to the midpoint? 3. Where is the midpoint 4. **Remember the cookie cutter!*** Holt Mc. Dougal Algebra 1 25 50 75 100 125 150 175 200 225

2 -7 Solving Absolute-Value Inequalities Write the absolute-value inequality for this graph 90 92 Holt Mc. Dougal Algebra 1 94 96 98 100

2 -7 Solving Absolute-Value Inequalities 3 -5 0 Holt Mc. Dougal Algebra 1 17 5 10 15 20

2 -7 Solving Absolute-Value Inequalities Write the absolute-value inequality for this graph – 7 – 6 – 5 – 4 – 3 – 2 – 1 0 Holt Mc. Dougal Algebra 1 1 2 3

2 -7 Solving Absolute-Value Inequalities Write the absolute-value inequality for this graph -10 0 Holt Mc. Dougal Algebra 1 10 20 30 40

2 -7 Solving Absolute-Value Inequalities 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. Holt Mc. Dougal Algebra 1

2 -7 Solving Absolute-Value Inequalities Additional Example 4 A: Special Cases of Absolute. Value Inequalities Solve the inequality. |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 Mc. Dougal Algebra 1

2 -7 Solving Absolute-Value Inequalities Additional Example 4 B: Special Cases of Absolute. Value Inequalities Solve the inequality. |x – 2| + 9 < 7 – 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 Mc. Dougal Algebra 1

2 -7 Solving Absolute-Value Inequalities Remember! An absolute value represents a distance, and distance cannot be less than 0. Holt Mc. Dougal Algebra 1

2 -7 Solving Absolute-Value Inequalities Check It Out! Example 4 a Solve the inequality. |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. Holt Mc. Dougal Algebra 1

2 -7 Solving Absolute-Value Inequalities Check It Out! Example 4 b Solve the inequality. 4|x – 3. 5| ≤ – 8 4 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. Holt Mc. Dougal Algebra 1

2 -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 Mc. Dougal Algebra 1

2 -7 Solving Absolute-Value Inequalities Lesson Quiz: Part II Solve each inequality. 4. |3 x| + 1 < 1 no solutions 5. |x + 2| – 3 ≥ – 6 Holt Mc. Dougal Algebra 1 all real numbers