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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, letâ€™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, letâ€™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. Letâ€™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.

 â€¢ If |x| < a, then -a < x < a. â€¢ If |x| ≤ a, then -a ≤ x ≤ a.

â€¢ If |x| < 0, then there is no solution.

If |x| 0, then the solution is x = 0.

â€¢ 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.