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Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 1 of 14 Absolute Value Equations and Inequalities Objective 1: Solving Absolute Value Equations Example: Solve 7 2 x . Write your answer as a solution set. Example: Solve 8 5 2 x . Write your answer as a solution set. Definition of Absolute Value 0 0 x if x x x if x 1. Isolate the absolute value 2. Read it as “distance from zero is ___” 3. Set up the two equations 4. Solve 5. Check

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• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 1 of 14

Absolute Value Equations and Inequalities

Objective 1: Solving Absolute Value Equations

Example: Solve 7 2x . Write your answer as a solution set.

Example: Solve 8 5 2x . Write your answer as a solution set.

Definition of Absolute Value

0

0

x if xx

x if x

1. Isolate the absolute value

2. Read it as “distance from zero is ___”

3. Set up the two equations

4. Solve

5. Check

• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 2 of 14

Example: Solve 3 4

53

Example: Solve 6 4 12x . Write your answer as a solution set.

Example: Solve 4 5 2 9x . Write your answer as a solution set.

• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 3 of 14

Pause the video and try these problems.

Solve each of the following equations. Write your answer as a solution set.

1. 4 7x

2. 2 5x

3. 4 1 7

3

x

Objective 2: More Absolute Value Equations

• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 4 of 14

Example: Solve 8 3 5 16x x . Write your answer as a solution set.

Example: Solve 8 2 4 4x . Write your answer as a solution set.

Example: Solve 9 3x . Write your answer as a solution set.

• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 5 of 14

Example: Solve 2 5 6x x . Write your answer as a solution set.

Pause the video and try these problems.

Solve each of the following equations. Write your answer as a solution set.

1. 3 5 8 2x

2. 3 7 4 5x x

• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 6 of 14

3. 9 2 3 9x

4. 2 25 3x x x x

Objective 3: Absolute Value Inequalities with < or

• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 7 of 14

Example: Solve the inequality. Graph the solution and write it in both set builder notation and interval notation.

5 4 8x

Example: Solve the inequality. Graph the solution and write it in both set builder notation and interval notation.

4 2x

1. Isolate the absolute value and put it

on the left

2. Read it as “distance from zero is ___”

3. Draw a graph as an aid

4. Put the expression that is inside the

absolute value where the graph is

5. Set up the compound inequality

6. Solve

If a > 0,

x a is equivalent to a x a

• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 8 of 14

Example: Solve the inequality. Graph the solution and write it in both set builder notation and interval notation.

3 4 1 7x

Pause the video and try these problems.

Solve each inequality. Graph the solution and write it in both set builder notation

and interval notation.

1. 3 5 8x

2. 18 3 4x

• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 9 of 14

Objective 4: Absolute Value Inequalities with > or >. Example: Solve the inequality. Graph the solution and write it in both set builder notation and interval notation.

6x

Example: Solve the inequality. Graph the solution and write it in both set builder notation and interval notation.

3 6 8x

Example: Solve the inequality. Graph the solution and write it in both set builder notation and interval notation.

2 3 1 4 6x

• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 10 of 14

Example: Solve the inequality. Graph the solution and write it in both set builder notation and interval notation.

4 2 12x

Pause the video and try these problems.

Solve each inequality. Graph the solution and write it in both set builder notation

and interval notation.

1. 3 5 2 7x

• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 11 of 14

2. 5 82 1x

Objective 5: More Absolute Value Inequalities We have covered

x c

x c

x c

x c

where c is a positive number. Let us now look at what happens if c is a negative number, or zero.

Example: Solve.

3x

3x

Example: Solve.

5x

5x

• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 12 of 14

Example: Solve.

0x

0x

Example: Solve.

0x

0x

Example: Solve the inequality. Graph the solution and write it in both set builder notation and interval notation.

2 3 2 5 0x

Example: Solve the inequality. Graph the solution and write it in both set builder notation and interval notation.

3 5 5x

• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 13 of 14

Example: Solve the inequality. Graph the solution and write it in both set builder notation and interval notation.

4 5 2 0x

Example: Solve the inequality. Graph the solution and write it in both set builder notation and interval notation.

4 3 7 7x

Pause the video and try these problems.

Solve each inequality. Graph the solution set and write it in both interval notation

and set builder notation if possible.

1. 3 1 7 0x

• Cypress College Math Department – CCMR Notes Absolute Value Equations and Inequalities, Page 14 of 14

2. 7 3 3x

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