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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 $b \lt 0$.

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    $\begingroup$ Did you test your hypothesis for some simple cases? $\endgroup$ – Martin R Apr 23 at 7:36
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    $\begingroup$ So why didn't you just test it for some $b\lt 0$...??? $\endgroup$ – CiaPan Apr 23 at 7:43
  • $\begingroup$ oops, sorry, looks like i made a computation mistake early on when testing it (-10 and -3) and then tried to find a way to prove it before double checking my computation. $\endgroup$ – tau Apr 23 at 7:50
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$a=-1, b=-1$ looks like the counterexample here since

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


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

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

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