The absolute value of a number is its distance from the origin. For every (non-zero) distance there are two numbers with that distance from the origin, a positive and a negative one. That is the only fact peculiar to absolute value equations that you need to understand and appreciate. All absolute value equations in this class can be approached using this simple principle.
For example, if we know that
Let's consider more examples. Suppose
There are two possibilities for the left hand side of this equation:
The above example is typical: we look at all possible sign combinations of the expressions of which we take absolute values, and prune the set by looking for equivalent equations.
Sometimes there are subtleties. Here is another example: Suppose
So we solve an absolute value equation by replacing it with several equations that do not involve absolute values, and solving each one of them. Sometime the replacement equations have solutions that are not satisfied by the original equation. A simple example is given by
Of course in this example it is obvious that there is no solution since the absolute value of anything is never negative. Another less obvious example is given by