Insert absolute value to another absolute value

Question

Not knowing anything about the values of $a,b$, is the next move legal? $$||a|-|b|| + |b| = | |a|-|b|+|b| |$$

Answer 1

Why would it be?

Let $|a| - |b| = c$.

The triangle inequality says:

$||a| - |b| + |b|| = |c + |b|| \le |c| + ||b||= ||a| - |b|| + |b|$.

equality does not always hold.

As $|x + y| \le |x| + |y|$ has $|x + y| = |x| + |y|$ only if i) one of $x$ or $y$ is $0$ or ii) $x$ and $y$ have he same parity (both greater or both less than $0$).

[Actually, $|x+y| = |x| + |y|$ if $x \ge 0; y\ge 0$ or $x \le 0; y \le 0$ or $|x+y| = ||x| -|y||$ if $y< 0 < x$ or $x < 0 < y$]

So

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

If $|a| - |b| \ge 0$ (or in other words if $|a| \ge |b|$) and $|b| \ge 0$ then $||a| - |b| + |b|| =|a|$ and $||a| - |b|| + |b| = |a| - |b| + |b| = |a|$.

but if $|a| - |b| < 0$ (or in other words if $|a| < |b|$) then $||a| - |b| + |b|| = |a|$ and $||a| - |b|| + |b| = |b| - |a| + |b| = 2|b| - |a| > |b| > |a|$.

Answer 2

$||a|-|b||+|b|=||a|-|b|+|b||$ is true if and only if $|a|\geq |b|$

Before continuing too much further, let us go ahead and make the observation that the right hand side simply simplifies to $|a|$.

$\implies)$ Suppose that $|a|\geq |b|$. We have then $||a|-|b||=|a|-|b|$ and so the left hand side simplifies to $||a|-|b||+|b|=|a|-|b|+|b|=|a|$ which is indeed equal to the right hand side.

$\impliedby)$ Suppose that $|a|<|b|$. We have $||a|-|b||+|b|\geq |b|>|a|$ and so the left hand side is strictly greater than the right hand side and cannot be equal.