is the following true: $|a-b| = ||a|-|b||$?

Question

is $|a-b| = \bigg||a|-|b|\bigg|$ ?

I have tried a few examples and they seems to come out true, but I can't find any rule stating it.

Is it true for all $a$ and $b$? Or am I missing something?

Please notice I'm not talking about $|a-b| = |a| - |b|$, I know this one does not hold.

Answer 1

No, it's not. Take $a=1$, $b=-1$. Then $\lvert a-b\rvert = 2$ but $\lvert\lvert a\rvert-\lvert b\rvert\rvert = 0$.

(You only have an inequality, the reverse triangle inequality: $\lvert a-b\rvert \geq \lvert\lvert a\rvert-\lvert b\rvert\rvert $.)

Answer 2

It is true if and only if $ab\ge 0$.

Answer 3

if we take $a=2$ and $b=-1$ I get: $$|2-(-1)|=3$$ and $$ ||2|-|-1||=1 $$

Answer 4

Squaring, you get

$$(a-b)^2=(|a|-|b|)^2,$$ which simplifies to $$ab=|a||b|,$$ not a valid identity.