Solving AbsoluteValue Inequalities 3 7 Solving Inequalities 3

Solving. Absolute-Value Inequalities 3 -7 Solving Inequalities 3 -7 Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1 Algebra Holt Mc. Dougal Algebra 11

3 -7 Solving Absolute-Value Inequalities Objectives Solve compound inequalities in one variable involving absolute-value expressions. Holt Mc. Dougal Algebra 1

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. 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

3 -7 Solving Absolute-Value Inequalities STEP 1: Get the | | by themselves. STEP 2: Drop the bars and set up 2 separate equations. 1 st Equation is set up normally 2 nd Equation same value inside bars, outside of the bars need to opposite. Like Back to the Future II Everything is different except what was in the bars. Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities When the bars are by themselves. |x| ≥ Will always look like an |x| < Will always look like an Holt Mc. Dougal Algebra 1 “OR” inequalities. Works like a restraining order. “AND” inequalities. Works like a leash.

3 -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

3 -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

3 -7 Solving Absolute-Value Inequalities Check It Out! Example 1 a Solve the inequality and graph the solutions. 2|x| ≤ 6 2 2 |x| ≤ 3 x ≥ – 3 AND x ≤ 3 3 units – 3 – 2 – 1 Holt Mc. Dougal 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 Additional Example 2 A: Solving Absolute-Value Inequalities Involving > Solve the inequality and graph the solutions. |x| + 14 ≥ 19 – 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 – 10 – 8 – 6 – 4 – 2 0 Holt Mc. Dougal Algebra 1 2 4 6 8 10

3 -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

3 -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. Remember absolute values must be positive numbers. Holt Mc. Dougal Algebra 1

3 -7 Solving Absolute-Value Inequalities Remember! An absolute value represents a distance, and distance cannot be less than 0. Holt Mc. Dougal Algebra 1

3 -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

3 -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

3 -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