If $\left| x \right| \ge \left| y \right|$ then Can we say $\left| {x + y} \right| \ge \left| x \right| - \left| y \right|$?

Question

Let $x,y\in \mathbb{R}$ and $\left| x \right| \ge \left| y \right|$.

Can we say $\left| {x + y} \right| \ge \left| x \right| - \left| y \right|$?

Answer 1

Yep - triangle inequality gives that $\vert(x+y)+(-y)\vert\le\vert x+y\vert+\vert(-y)\vert$

$$\implies\vert x \vert \le \vert x+y \vert+\vert y \vert$$

$$\implies \vert x+y \vert \ge \vert x \vert - \vert y \vert$$

Perhaps worth noting that the restriction $\vert x \vert \ge \vert y \vert$ isn't needed.

Answer 2

By the triangle inequality, we have $$|a+b| \leq |a| + |b|.$$ Hence, $$|x| = |x+y -y| \leq |x+y| + |-y| = |x+y| + |y|$$ $$\Rightarrow |x| - |y| \leq |x+y|$$