# Solving absolute value equation, different methods.

I'm interested to know how people solve absolute value equations differently and how many methods there are out there. For example, say I wish to solve

$|x-2|-|x-3|=|x+4|$.

How would you solve it personally, and how many other ways can you think of? For example, I would personally solve it graphically, but another way would be to simplify the LHS into cases, and then solve the cases individually.

You have named the main approaches. In this particular example, I would focus on the left side. If $x$ is greater than $3$, the left is $1$. If $x$ is less than $2$, the left is $-1$. In between, it is between $-1$ and $1$. The right is only in that range when $x \in [-5,-3]$, where the left is $-1$. Since the right is positive, there is no solution.