Logic behind absolute value problem
I'm trying to nail down the logic behind absolute value inequalities. Given an inequality like
the normal process is to use cases and the definition of absolute value to determine the solution set.
However, upon thinking about the process a little more I've confused myself. When faced with this problem the goal is to find the values of that make the relation true. But by using cases I am assuming that is nonnegative in one case and negative in another and then I proceed to find the solutions that satisfy each case. Is this logic circular since the sign of depends on ? It seems like I'm assuming something about before I find it. Can someone explain the logic here? Thanks for the help.