What is the difference between solving absolute value equations and absolute value inequalities?

The difference between the two is the symbols used.

In absolute value equations you use an equal sign to show that the two sides are equal. (Ex. 10x+7 = 37) In absolute value inequalities you use a greater than or less than sign to show that one side is greater than or less than the other side. (Ex. 13x+9 > 7x+15)