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How can I prove the following 3 claims? Or, if possible, could you provide suggestions for how I could prove them?

For real numbers $x$ and $y$:

  • Claim 1. $|x| − |y| ≤ |x − y|$

  • Claim 2. $|x-y| ≤ |x|+|y|$

  • Claim 3. $|x| − |y| ≤ |x + y|$

FD_bfa
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MINTTTTT
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  • Welcome to stakexchange. Try the cases depending on the signs of $x$ and $y$ and the signs of the various sums and differences. (There are faster ways, but you will learn a lot this way.) – Ethan Bolker Apr 19 '22 at 15:52
  • Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. – Community Apr 19 '22 at 15:54
  • Have you made a search here on MSE. I'm pretty sure that your questions are duplicates. – callculus42 Apr 19 '22 at 16:04

1 Answers1

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It is enough to prove the second one (which is called the triangle inequality) because the remaining inequalities follows immediately.

Indeed,

  1. Let's show that $|x|-|y|\leq |x-y|$. Using the triangle inequality we have, $$|x|=|(x-y)+y|\leq |x-y|+|y|.$$

  2. Let's show that $|x|-|y|\leq |x+y|$. Using the triangle inequality we have, $$|x|=|(x+y)+(-y)|\leq |x+y|+|-y|=|x+y|+|y|.$$

In both of them I've used that $|a-b|\leq |a|+|b|$. So it suffices to prove the second one.

RFZ
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  • For the second problem it is also an immediate consequence of your first calculation by adding $-\vert y \vert$ to both sides and extracting the appropriate inequality. – CyclotomicField Apr 19 '22 at 16:02