How do you solve absolute value inequalities?

1 Answer
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: 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)