# 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.

## 1 Answer

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.

• I have heard of a method involving testing points. Are you familiar with this? – Trogdor Oct 19 '14 at 3:14
• No. It sounds like dividing into cases, with the testing points being points away from where the absolute value changes, but I am guessing. – Ross Millikan Oct 19 '14 at 3:16