Is the following inequality(that looks like the triangle inequality) valid:
$|a - b| \leq |a| - |b|$
Why?
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It's sometimes called the reverse triangle inequality. The proper form is $$\left| a - b \right| \ge \big||a| - |b|\big|$$ For the proof, consider $$|a| = |a - b + b| \le |a - b| + |b|$$ $$|b| = |b - a + a| \le |a - b| + |a|$$ so that we have $$-|a-b|\le|a|-|b| \le |a - b|$$ |
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No. For example, $|(-2)-3|=5>|-2|-|3|=-1.$ I think you're thinking of $||a|-|b||\le |a- b|.$ |
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The length of any side of a triangle is greater than the absolute difference of the lengths of the other two sides: $$||a|-|b||\leq |a-b|$$ Here is a proof: $$|a+(b-a)|\leq |a|+|b-a|$$ and, (1) $$|a-b|\geq |a|-|b|$$ Interchanging $a$ and $b$, we get also (2) $$|a-b|\geq |b|-|a|$$ Combining (1) and (2) we get our desired result. |
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