Triangle Inequality and absolute value

Question

I'm curious if the triangle inequality (and reverse triangle inequality) still hold if we only take the absolute value of one term. For example:

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

If $b \ge 0$, then $|b|$ is the same due to the definition of absolute value. I am unsure and am having trouble finding (or proving myself) if the inequality still holds if $b \lt 0$.

Answer 1

$a=-1, b=-1$ looks like the counterexample here since

$$||a|-b|=||-1|-(-1)| = |1+1|=2$$ and $$|a-b|=|-1-(-1)|=|0|=0$$

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A more general hint: To find a counterexample in this and similar cases, I would first try the following four options:

$$a=1,b=1\\ a=-1, b=1\\a=1, b=-1\\a=-1, b=-1$$