# How do you solve absolute value inequalities?

Jul 22, 2018

There are many facets to this question. I will address a simply one and let the questioner expand to a specific example he/she may have.

Assume: $\left\mid f \left(x\right) \right\mid < a$ $\left[a \in \mathbb{R}\right]$

Then, either $f \left(x\right) < a \mathmr{and} - f \left(x\right) < a$

Applying the rules of inequalities, either $f \left(x\right) < a \mathmr{and} f \left(x\right) > - a$

Which leads to the compound inequality: $- a < f \left(x\right) < a$

Therefore in solving absolute value inequalities of this and similar forms simply consider both positive and negative possibilities of the function and solve for each.

Example: $\left\mid x - 3 \right\mid < 5$

Either $\left(x - 3\right) < 5 \to x < 8$

Or $- \left(x - 3\right) < 5 \to \left(x - 3\right) > - 5 \to x > - 2$

Thus: $- 2 < x < 8$

Which can be expressed in interval notation as: $x \in \left(- 2 , 8\right)$