Is the following inequality(that looks like the triangle inequality) valid:
$|a - b| \leq |a| - |b|$
Why?
Is the following inequality(that looks like the triangle inequality) valid:
$|a - b| \leq |a| - |b|$
Why?
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 $-|a-b|\le|a|-|b| \le |a - b|$
No. For example, $|(-2)-3|=5>|-2|-|3|=-1.$
I think you're thinking of $||a|-|b||\le |a- b|.$
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.