Solving Absolute Value Inequalities
Solving an Absolute Value Inequality of the
Form | x| > a
You have just seen that the solutions of an inequality of the form |x| < a
are numbers that are less than a units from 0 on a number line.
For example, |x| < 3 represents all the numbers between -3 and 3.
The solutions of an inequality of the form |x| > a are the numbers that are
greater than a units from 0 on a number line.
For example, let’s solve this inequality: |x| > 3
The solution includes all numbers whose distance from 0 is greater than 3.
This solution includes two distinct parts:
• Numbers less than -3.
• Numbers greater than +3.
We write the solution as x< -3 or x > 3.
The inequality |x| > 3 has infinitely many solutions so we cannot check
them all. Let’s check one number from each part of the solution:
| Check x = -4 Is |-4| > 3 ?
Is 4 > 3 ? Yes |
Check x = 3.5 Is |3.5| > 3 ?
Is 3.5 > 3 ? Yes |
You may also want to check a value that is not in the solution.
For example, 1 is not part of our solution so it should not satisfy the
original inequality.
Check x = 1.
Is |1| > 3 ?
Is 1 > 3 ? No
Note:
Notice that the solution,
x < -3 or x > 3
uses the word or rather than and. This is
because a number cannot be both less
than -3 and greater than 3.
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