absolute value binomial split into two absolute values

Question

$$ |a-b| = |a|-|b| $$

I think I might missing something with absolute values. Can I split a binomial into two separate absolute values like above?

Answer 1

No, you cannot. For example, Let $a=1$ and $b=-2$. We have $|a-b| = |1-(-2)| = |3|=3$. On the other hand, we have $|a|-|b| = |1|-|-2| = |1|-|2|=1-2=-1$. Clearly, $3 \neq -1$.

It is, however, true that $||a|-|b|| \le |a-b|$. In fact, we can extend this:

$$||a|-|b|| \le |a-b| \le |a|+|b|$$

Answer 2

No. Breaking $|a-b|$ up like that is no good. Try $a = 1$ and $b = 2$. Your equation is true if $a \geq b > 0$.