I was never really taught how to deal with the absolute value function, but I need it from time to time. So here are a few hints.

## Solving Equations

Lets say you want to solve the equation

for \(x\). Then you need to realize that this equation is equivalent to two equations:

you can solve both of them independantly. You can get 0, 1 or 2 solutions when the absolute function is involved:

or shorter

## Solving Inequalities

Lets say you want to solve the inequality

for \(x\). Again, this inequality is equivalent to the two inequalities

You can solve both of them independantly for \(x\):

leading to

Note that both inequalities have to be fulfilled at the same time! Just try it for \(a = 0\) and \(b = -5\)!

## Derivatives

The function \(f(x) = |x|\) is equivalent to \(f(x) = \sqrt{x^2}\). Hence you can derive the absolute value by deriving the root of the square function of its argument. And the chain rule, of course:

Note that the derivative is not devined at 0.