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Algebra
First Degree Absolute Value Inequality Examples
Absolute Value Inequality Example 1
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 }
Absolute Value Inequality Example 2
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 }
Absolute Value Inequality Example 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 }
Absolute Value Inequality Example 4
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 }
Absolute Value Inequality Example 5
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 }