back to *Algebra*
# First Degree Absolute Value Inequality Examples

### Absolute Value Inequality Example 1

### Absolute Value Inequality Example 2

### Absolute Value Inequality Example 3

### Absolute Value Inequality Example 4

### Absolute Value Inequality Example 5

- Question:
- Solving the inequality | 3x | < 21 and graph its solution on a number line.

- Solution:
- Properties of an absolute value inequality:
- If a > 0, then
- (i) | x | < a is equivalent to - a < x < a.
- (ii) | x | <= a is equivalent to - a <= x <= a. ( note: x <= a means x less than or equal to a )
- Then the solution of | 3x | < 21 is:
- - 21 < 3x < 21 , divide by 3 on each term
- - 7 < x < 7
- Therefore, the solution set is { x | -7 < x < 7 }

- Question:
- Solving the inequality | 3x - 2 | <= 11 and graph its solution on a number line.

- Solution:
- Properties of an absolute value inequality:
- If a > 0, then | x | <= a is equivalent to - a <= x <= a. ( note: x <= a means x less than or equal to a )
- then the solution of | 3x - 2 | <= 11 is:
- - 11 <= 3x - 2 <= 11 , each term add 2
- - 11 + 2 <= 3x <= 11 + 2
- - 9 <= 3x <= 13 , each term divide by 3
- - 3 <= x <= 13/3
- Therefore, the solution set is { x | -3 <= x <= 13/3 }

- Question:
- Solving the inequality | x + 3 | > 2 and graph its solution on a number line.

- Solution:
- Properties of an absolute value inequality:
- If a > 0, then
- (i) | x | > a is equivalent to x < - a or x > a.
- (ii) | x | >= a is equivalent to x <= - a or x >= a.
- ( note: x <= - a means x less than or equal to - a , x >= a means x large than or equal to a)
- Then the solution of | x + 3 | > 2 is:
- x + 3 < - 2 or x + 3 > 2 (move number to the right side of the inequality)
- x < - 3 - 2 or x > - 3 + 2
- x < - 5 or x > -1
- Therefore, the solution set is { x | x < - 5 or x > -1 }

- Question:
- Solving the inequality | 2x + 5 | >= 9 and graph its solution on a number line.

- Solution
- Properties of an absolute value inequality:
- If a > 0, then | x | >= a is equivalent to x <= - a or x >= a.
- ( note: x <= - a means x less than or equal to - a , x >= a means x large than or equal to a )
- Then the solution of | 2x + 5 | >= 9 is:
- 2x + 5 <= -9 or 2x + 5 >= 9 (move the number to the right side of the inequality)
- 2x <= -5 - 9 or 2x >= -5 + 9
- 2x <= -14 or 2x >= 4 (divide by 2 on each term of the inequality)
- x <= - 7 or x >= 2
- Therefore, the solution set is { x | x <= - 7 or x >= 2 }

- Question:
- Solving the inequality | 3 - 2x | > 7 and graph its solution on a number line.

- Solution
- Properties of an absolute value inequality:
- If a > 0, then | x | > a is equivalent to x < - a or x > a.
- Then the solution of | 3 - 2x | > 7 is:
- 3 - 2x < - 7 or 3 - 2x > 7 (move number to the right side of the inequality)
- - 2x < - 3 - 7 or - 2x > - 3 + 7
- - 2x < - 10 or - 2x > 4
- - x < - 5 or - x > 2
- When both sides of an inequality multiply -1, the inequality change its direction.
- x > 5 or x < - 2
- Therefore, the solution set is { x | x < - 2 or x > 5 }

© Acceler LLC 2022 - www.mathtestpreparation.com