Holt McDougal Algebra 1

3-7 Solving Absolute-Value Inequalities

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

Objectives

When an inequality contains an absolute-value expression, it can be written as a compound inequality. The inequality |x| < 5 describes all real numbers whose distance from 0 is less than 5 units. The solutions are all numbers between –5 and 5, so |x|< 5 can be rewritten as

–5 < x < 5, or as x > –5 AND x < 5.

Holt McDougal Algebra 1

3-7 Solving Absolute-Value Inequalities

LESS THAN = LESS “AND”

Holt McDougal Algebra 1

3-7 Solving Absolute-Value InequalitiesAdditional Example 1A: Solving Absolute-Value

Inequalities Involving <Solve the inequality and graph the solutions.

|x|– 3 < –1

x > –2 AND x < 2

+3 +3|x| < 2

|x|– 3 < –1

–2 –1 0 1 22 units 2 units

Holt McDougal Algebra 1

3-7 Solving Absolute-Value Inequalities

|x – 1| ≤ 2

Additional Example 1B: Solving Absolute-Value Inequalities Involving <

Solve the inequality and graph the solutions.

Holt McDougal Algebra 1

3-7 Solving Absolute-Value Inequalities

Just as you do when solving absolute-value equations, you first isolate the absolute-value expression when solving absolute-value inequalities.

Helpful Hint

Holt McDougal Algebra 1

3-7 Solving Absolute-Value InequalitiesCheck It Out! Example 1a

Solve the inequality and graph the solutions.

2|x| ≤ 6

Holt McDougal Algebra 1

3-7 Solving Absolute-Value InequalitiesCheck It Out! Example 1b

|x + 3|– 4.5 ≤ 7.5Solve each inequality and graph the solutions.

Holt McDougal Algebra 1

3-7 Solving Absolute-Value InequalitiesThe inequality |x| > 5 describes all real numbers whose distance from 0 is greater than 5 units. The solutions are all numbers less than –5 or greater than 5. The inequality |x| > 5 can be rewritten as the compound inequality x < –5 OR x > 5.

Holt McDougal Algebra 1

3-7 Solving Absolute-Value Inequalities

Solve the inequality and graph the solutions.

Additional Example 2A: Solving Absolute-Value Inequalities Involving >

|x| + 14 ≥ 19

|x| ≥ 5x ≤ –5 OR x ≥ 5

–10 –8 –6 –4 –2 0 2 4 6 8 10

5 units 5 units

– 14 –14|x| + 14 ≥ 19

Holt McDougal Algebra 1

3-7 Solving Absolute-Value Inequalities

Solve the inequality and graph the solutions.3 + |x + 2| > 5

Additional Example 2B: Solving Absolute-Value Inequalities Involving >

Holt McDougal Algebra 1

3-7 Solving Absolute-Value InequalitiesCheck It Out! Example 2a

|x| + 10 ≥ 12

Solve each inequality and graph the solutions.

Holt McDougal Algebra 1

3-7 Solving Absolute-Value InequalitiesCheck It Out! Example 2b

Solve the inequality and graph the solutions.

Holt McDougal Algebra 1

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.

Holt McDougal Algebra 1

3-7 Solving Absolute-Value InequalitiesAdditional Example 4A: Special Cases of Absolute-Value

Inequalities

Solve the inequality.|x + 4|– 5 > – 8 |x + 4|– 5 > – 8

+ 5 + 5|x + 4| > –3 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 InequalitiesAdditional Example 4B: Special Cases of Absolute-Value

Inequalities Solve the inequality.|x – 2| + 9 < 7

|x – 2| + 9 < 7 – 9 – 9

|x – 2| < –2 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

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

Remember!

Holt McDougal Algebra 1

3-7 Solving Absolute-Value InequalitiesCheck It Out! Example 4a

Solve the inequality.

|x| – 9 ≥ –11

Holt McDougal Algebra 1

3-7 Solving Absolute-Value InequalitiesCheck It Out! Example 4b

Solve the inequality.

4|x – 3.5| ≤ –8

Holt McDougal Algebra 1

3-7 Solving Absolute-Value InequalitiesLesson Quiz: Part I

Solve each inequality and graph the solutions.1. 3|x| > 15 x < –5 or x > 5

0–5–10 5 10

2. |x + 3| + 1 < 3 –5 < x < –1

–2 0–1–3–4–5–6

Holt McDougal Algebra 1

3-7 Solving Absolute-Value Inequalities

Solve each inequality.3. |3x| + 1 < 1

4. |x + 2| – 3 ≥ – 6 all real numbers

no solutions

Lesson Quiz: Part II