# Absolute value inequality example

I have to express this without absolute value: $$a-|a-|a||$$ The answer is $$\begin{cases}a&a\ge0\\ 3a&a<0\end{cases}$$ All examples went OK, but I dont know this one. I might understand answer by intuition but I can't do it rigorously. Thanks.
• If $a>0$ then $|a|=a$ so your expression is $a-|a-a|=a$. The case $a<0$ is similar. – lulu Oct 25 '16 at 11:15
• $$a\gt 0\Rightarrow |a|=a\Rightarrow a-|a-a|=a-0=a$$ $$a\lt 0\Rightarrow |a|=-a\Rightarrow a-|a-(-a)|=a-|2a|=a-(-2a)=3a$$ – Piquito Oct 25 '16 at 11:29