In several proofs in a functional analysis class I'm going through, I've seen the professor use a common technique when dealing with proofs involving inequalities. The technique/trick goes something like this where you add and subtract the same term to an initial value and then proceed to derive an inequality. (pulled from the wikipedia proof of the reverse triangle in-equality):
$\lvert|x|\rvert$ = $\lvert|(x-y) + y |\rvert$ $\le$ $\lvert|x-y|\rvert$ + $\lvert|y|\rvert$ => $\lvert|x|\rvert$ - $\lvert|y|\rvert$ $\le$ $\lvert|x-y|\rvert$
$\lvert|y|\rvert$ = $\lvert|(y-x) + x|\rvert$ $\le$ $\lvert|y-x|\rvert$ + $\lvert|x|\rvert$ => $\lvert|x|\rvert$ - $\lvert|y|\rvert$ $\ge$ -$\lvert|x-y|\rvert$
The rest of the proof follows from combining the above two statements and can be found here: https://en.wikipedia.org/wiki/Triangle_inequality#Reverse_triangle_inequality
My question is if someone could provide some sort of generalization of when to use this technique of adding and subtracting the same term in inequality proofs. Said similarly, if someone has any conceptual understanding of the technique and why its useful in many scenarios that would much appreciated. I realize a hard and fast rule of when this technique is useful is very unlikely, I'm mostly just looking for some additional clarity of how it works.