Validity of a certain inequality

Question

I am curious if the inequality below is valid, and if so, is there a name for it? It looks similar to triangle inequality, but it is not same.

$$|x+y+z| \ge |x|-|y|-|z|$$

Answer 1

$$|-y|+|-z|+|x+y+z|\geq |(-y)+(-z)+(x+y+z)| =|x|$$

Don't forget, $|-y|=|y|$...

Answer 2

It's just a direct application of the triangle inequality.

$|x + y + z + (-y) + (-z)| \le |x+y+z + (-z)| + |-y|$ (is the triangle inequality.)

$\le |x+y+z| + |-y| + |-z|$ (is applying the triangle inequality twice.)

$|x + y + z + (-y) + (-z)| = |x|; |-y| = |y|; |-z| = |z|$ so

$|x| \le |x+y+z| + |y| + |z|$ (is just substitution.)

$|x+y+z| \ge |x| - |y| - |z|$ (is just rearranging.)

So this is just another way of stating the Triangle Inequality. It has no other name.

[Needless to say this will be true for $|y| -|x| - |z|$ and $|z| - |x| - |y|$ as well. It worth noting that at most one $|x| - |y| - |z|$, $|y|-|x|-|z|$, $|z|-|x|-|y|$ will be positive and thus a non-trivial result. ... at most one. And only if (wolog $\max(|x|,|y|, |z|) = |x|$), only if $|x| > |y| + |z|$.]