Question

How do you solve absolute value inequalities?

Answer 1

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: `abs(f(x)) < a` `[a in RR]`

Then, either `f(x) < a or -f(x) < a`

Applying the rules of inequalities, either `f(x) < a or f(x) > -a`

Which leads to the compound inequality: `-a < f(x) < 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: `abs(x-3) < 5`

Either `(x-3) < 5 -> x < 8`

Or `-(x-3) < 5 -> (x-3) > -5 -> x > -2`

Thus: `-2 < x < 8`

Which can be expressed in interval notation as: `x in (-2,8)`