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

x . Write your answer as a solution set.

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

Restart when you are ready to check your answers.

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

Restart when you are ready to check your answers.

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

7. Graph your answer

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

Restart when you are ready to check your answers.

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

Restart when you are ready to check your answers.

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

3. 3 2 0x