In real and complex analysis, we used the alternate triangle inequality $\left|a-b\right| \geq \left||a|-|b|\right|$ a fair bit. I've just been introduced to the notion of real inner product spaces, normed spaces and metric spaces. Using the Cauchy-Schwarz inequality and the fact that every inner product induces a norm, I have derived the "alternate triangle inequality" for normed spaces $$\|v-w\|\geq\left|\|v\|-\|w\|\right|.$$
However, is this form of the triangle inequality used at all for normed or even general Euclidean spaces (like it was for analysis in $\mathbb R$)?
Further, is there a corresponding "alternate triangle inequality" for metrics (distance functions)?