In this post I will talk about the absolutely trivial inequality

\[x \leq \lvert x \rvert .\]This is easily seen. If $x$ is positive, it is equal to its absolute value. If it is negative, it is of course less than its absolute value.

I have to admit that I probably didn’t understand the reverse triangle inequality

\[\lvert \lvert x \rvert - \lvert y \rvert \rvert \leq \lvert x-y\rvert\]or at least I definitely did not think it through. But actually, as a weaker inequality,

\[\lvert x \rvert - \lvert y \rvert \leq \lvert x-y\rvert .\]Had I realised this sooner I probably would have spent less time banging my head against a simple problem.