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 $b \lt 0$.