Let $x,y\in \mathbb{R}$ and $\left| x \right| \ge \left| y \right|$.
Can we say $\left| {x + y} \right| \ge \left| x \right| - \left| y \right|$?
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2Use $|a|+|b| \ge |a+b|$ and a suitable substitution, followed by some minor reorganisation – Henry Feb 08 '16 at 11:42
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2You might have a look at some posts about the reverse triangle inequality, like here or here. – Martin Sleziak Feb 08 '16 at 12:23
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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.
πr8
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By the triangle inequality, we have $$|a+b| \leq |a| + |b|.$$ Hence, $$|x| = |x+y -y| \leq |x+y| + |-y| = |x+y| + |y|$$ $$\Rightarrow |x| - |y| \leq |x+y|$$
j4GGy
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