# Triangle Inequality and absolute value

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 $$b \lt 0$$.

• Did you test your hypothesis for some simple cases? – Martin R Apr 23 at 7:36
• So why didn't you just test it for some $b\lt 0$...??? – CiaPan Apr 23 at 7:43
• 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. – tau Apr 23 at 7:50

## 1 Answer

$$a=-1, b=-1$$ 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:

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