Prove that $|x - y| \geq |x| - |y|$

Question

For some reason I come to an answer that contradicts that, so I'm wrong.

$-|x| \leq x \leq |x|$ and $-|y| \leq y \leq |y|$

so subtract $x$ from $y$: $-(|x| - |y|) \leq x - y \leq |x| - |y|$

so by the Triangle Inequality: $|x - y| \leq |x| - |y|$

Somewhere I'm going in the wrong direction.

Tell me what's going on there.

P.S. How do I use the proper \leq symbol?

Here, you find a very good answer and explanation: Reverse Triangle Inequality — Jika, Jun 12 '14 at 22:39
@Joe If you want, LaTeX can be written $\LaTeX$ or $\LaTeX$. — Brad, Jun 13 '14 at 00:27

score 1 · Answer 1 · answered Jun 12 '14 at 22:33

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The step where you subtract is wrong. If you have two inequalities $$a \leq a'$$ and $$b \leq b',$$ then if you want to subtract you should make sure to write $$ a - b' \leq a' - b$$ instead of $a-b \leq a'-b'$. (Can you see why? Think about it in full sentences. Subtracting a big number leaves a smaller result than subtracting a small number.)

answered Jun 12 '14 at 22:33

Daan Michiels

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As a general strategy for finding where your mistake is, plug in some easy numbers. – Daan Michiels Jun 12 '14 at 22:34
I just realised that now. So I'm convinced that's the case. I've played around with that for a while now. It hasn't taken me in the direction needed. Could you perhaps give me more information? – Kermit the Hermit Jun 12 '14 at 23:20

score 0 · Answer 2 · answered Jun 12 '14 at 22:30

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Your second step is incorrect. As a counterexample, try $x = -2, y = 3$. Then $$ -(|x|-|y|) = -(2 - 3) = -(-1) = 1 $$ which is definitely greater than $$ x-y = -2 -3 = -5 $$

answered Jun 12 '14 at 22:30

cnick

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score 0 · Accepted Answer · answered Jun 12 '14 at 23:29

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$$|x|=|x-y+y|\le|x-y|+|y|\implies |x|-|y|\le |x-y|$$

answered Jun 12 '14 at 23:29

DonAntonio

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Great, makes sense. Thanks DonAntonio! – Kermit the Hermit Jun 13 '14 at 01:22

Prove that $|x - y| \geq |x| - |y|$

3 Answers3