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I was seeing a solved problem and someone said that $|x+y|\geq\big||x|-|y|\big|$ was part of the triangle inequality.

But this isn't the way the triangle inequality is presented. Namely its presented as $$|x+y|\leq|x|+|y|$$

So I was left wondering. Is $|x+y|\geq\big||x|-|y|\big|$ part of the triangle inequality? Is it even true at all?

DLV
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    Yes, because $|x|\leq|x+y|+|-y|$ and $|y|\leq|x+y|+|-x|$ (note that $|x|=|-x|$ and $|y|=|-y|$). – CIJ Sep 17 '15 at 01:36
  • How did you know the inequality direction flips? And why would this imply the inequality I posted above? – DLV Sep 17 '15 at 02:08

2 Answers2

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$|x+y| +|y|=|x+y|+|-y| \geq |(x+y)+(-y)|=|x|$ , so $$|x+y|\geq |x|-|y|.$$ Interchanging $x$ and $y$ in the previous sentence ,we obtain $$|x+y|\geq |y|-|x|.$$ So $$|x+y|\geq \max (|x|-|y|,|y|-|x|)=| |x|-|y| |.$$

DanielWainfleet
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Proof: Let $x,y\in \mathbb{R}$ Then $|x| = |(x + y) - y| \leq |x + y| + |y|$,

and $|y| = |(y + x) - x| \leq |x + y| + |x|$. Then

  1. $|x| - |y| \leq |x + y|$
  2. $|y| - |x| \leq |x + y| \Leftrightarrow -|x + y| \leq |x| - |y|$

Then, the transitive property of inequality implies $-|x + y| \leq |x| - |y| \leq |x + y| \Leftrightarrow ||x| - |y|| \leq |x + y|$.

Mario Alejandro Martinez
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