What is the best method for solving absolute value inequalities?

The first most important part about solving absolute value inequalities is to understand that there are two types of inequalities. These are |ax+b|c. These inequalities have totally different solutions. This is because of the difference between the two basic inequalities |x|c. Absolute value is distance from 0. If |x|c, then the distance between x and 0 is greater than c. So x is either far away from 0 to the left or far away from 0 to the right. Now there are 2 intervals: (-infinity,-c) ; (c,infinity). We show how to "see" and solve these 2 types step-by-step.

Finally, we remember that |x|=|-x|. This means that we do not solve absolute value inequalities with a negative x inside! We simply rewrite it: |2-3x|=|3x-2|. This means we never have to worry about dividing by a negative number.