3-7 Solving Absolute-Value Inequalities Warm Up Lesson Presentation Holt Algebra 1 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1

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

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

Like Back to the Future II STEP 1: Get the | | by themselves. STEP 2: Drop the bars and set up 2 separate equations. 1st Equation is set up normally 2nd 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.

|x| ≥ |x| < When the bars are by themselves. Will always look like an “OR” inequalities. Works like a restraining order. |x| < Will always look like an “AND” inequalities. Works like a leash.

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

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

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.

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

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.

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.