So I was given the task of proving the following variant of the triangle inequality using only the properties of the absolute value:
$\vert\lvert x\rvert -\lvert y \rvert \rvert \leq \lvert x+y\rvert$
Now my initial instinct was to divide it to four cases based on the possible values of x and y, and show that in each case I get a truth expression, but it seems a bit "brute force" to me and I'm not sure this is what my instructor meant. Since I'm not allowed to use the actual triangle inequality and must adhere to absolute value properties, is there any other solution?
(Of course he'll accept this answer if it's correct, but I want to learn what it is I'm missing).