Sign In
An absolute value inequality is an inequality that involves the absolute value of an expression containing a variable. As with other inequalities, absolute value inequalities can be strict or non-strict.
| Strict Absolute Value Inequalities | Non-Strict Absolute Value Inequalities | ||
|---|---|---|---|
| |x+2| > 5 | |x+7| < 5 | |2x| ≥ 10 | |x-2| ≤ 4 |
If a≥ 0, an absolute value inequality of the form |x|< a can be seen as the set of all numbers that are greater than - a and less than a. Similarly, an absolute value inequality of the form |x|≤ a can be seen as the set of all numbers that are greater than or equal to - a and less than or equal to a. c|c |x|< a & |x|≤ a ⇕ & ⇕ - a < x < a & - a ≤ x ≤ a Likewise, if a≥ 0, an absolute value inequality of the form |x|> a can be seen as the set of all numbers that are less than - a or greater than a. Similarly, an absolute value inequality of the form |x|≥ a can be seen as the set of all numbers that are less than or equal to - a or greater than or equal to a. c|c |x|> a & |x|≥ a ⇕ & ⇕ x < - a or x > a & x≤ - a or x≥ a As with other inequalities, absolute value inequalities can be represented by an interval on a number line. Open points at the ends of the interval represent strict inequalities where the corresponding values are not included in the interval. Conversely, closed points represent non-strict inequalities and the corresponding values are included in the interval.
If the absolute value inequality involves an expression of the form |x-b| with b>0 rather than just |x|, the interval will be translated b units to the right. Likewise, if the expression is of the form |x+b|, the interval will be translated b units to the left.
