Solving Absolute Value Inequalities
Solving an Absolute Value Inequality of the Form
| x| < a
Solving an absolute value inequality is similar to solving an absolute value
equation.
First, let’s solve the equation |x| = 5.
The solution is all numbers 5 units from 0 on the number line.
So, the solutions are x = 5 and x = -5.
Now, let’s solve the inequality |x| = 5.
The solution is all numbers that are less than 5 units from 0.
We can show this on a number line by shading between -5 and 5.
We use an open circle, o, at -5 and 5 to indicate that they are not part of
the solution.
The solution of |x| < 5 can be written as a compound inequality:
-5 < x < 5
The inequality |x| < 5 has infinitely many solutions. Let’s check three of
them.
| Check x = -3
Is |-3| < 5 ?
Is 3 < 5? Yes |
Check x = 0
Is |0| < 5 ?
Is 0 < 5? Yes |
Check x = 4.5
Is |4.5| < 5 ?
Is 4.5 < 5? Yes |
Here is the general principle for absolute value inequalities.
Principle
Absolute Value Inequalities of the Form | x| < a and | x| ≤ a
Let a represent a positive real number.
| • If |x| < a, then -a < x < a. |
 |
| • If |x| ≤ a, then -a
≤ x ≤
a. |
 |
• If |x| < 0, then there is no solution.
If |x| ≤ 0, then the solution is x = 0.
• If |x| < -a, then there is no solution.
If |x| ≤ -a, then there is no solution.
Note:
The absolute value of a number or an expression cannot be
negative.
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