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Note that left side is part of reverse triangle inequality $$||x|-|y||≤|x - y|$$ And the right side is part of triangle inequality $$|x+y|≤|x|+|y|$$

I tried to solve it in two different ways, but I could not get the final answer in any of them. First: As $|x-y|=|x+(-y)|$, I've applied the triangle inequality: $$|x+(-y)|≤|x|+|(-y)|$$ If I could define $| y | = | -y |$ (can I do this?), we would have $$|x+(-y)|≤|x|+|y|$$ In reverse triangle inequality, we have that $$||x|-|y||≤|x-y| $$ $$ ||x|-|y||≤|x|+|y|$$ $$||x|-|y||≤|x+(-y)|=|x-y|$$ Is this right? I don't know if I can do this.

The other way that I tried to solve was: $$||x|-|y||≤|x+y|$$ $$||x|-|y||≤|x|+|y|$$ $$||x|-|y||≤|x|+|y|$$ Let $|x|-|y|=t$, $$|t|≤|x|+|y|$$ Now we have $$|x|≥|t|-|y|$$ And applying the reverse triangle inequality again, we have $$|x|≥|t-y|$$

But I think this isn't the right way to solve the problem...

Somos
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João Pedro
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  • From $|x-y|+|y|ge |x|$ it follows that $|x-y|\ge |x|-|y|$. Reversing the role of $x$ and $y$ we get $|x-y|=|y-x|\ge |x|-|y|$. Combining, $|x-y|\ge ||x|-|y||$. Now substitute $-y$ in place of $y$ and you get your result. –  May 23 '19 at 12:56
  • Just replace $y$ with $-y$ in reverse triangle inequailty. – Somos May 23 '19 at 15:44

1 Answers1

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Apply the triangle inequalty to $y = x + y - x$ and $x = y + x - y$ and use the fact that $|y - x| = |x - y|.$

ncmathsadist
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