Showing $ \left| (|a|-|b|)-(|c|-|d|) \right| \le \left| (a-b)-(c-d) \right|$

Question

I am trying to show that for any real $a,b,c,d$ we have

$$ \left| (|a|-|b|)-(|c|-|d|) \right| \le \left| (a-b)-(c-d) \right| $$

At a glance, it seems like this cannot be true, but I haven't been able to come up with a counter example. Furthermore, it is not to hard to establish that

$$ \left| |a-b|-|c-d| \right| \le \left| (a-b)-(c-d) \right|. $$

For example by the (reverse) triangle inequality we have either $$ \left| |a-b|-|c-d| \right| = |a-b|-|c-d| \le |(a-b)-(c-d)| $$ or $$ \left| |a-b|-|c-d| \right| = |c-d|-|a-b| \le |(c-d)-(a-b)| = |(a-b)-(c-d)| .$$

And it seems much more believable that possibly $$ \left| (|a|-|b|)-(|c|-|d|) \right| \le \left| |a-b|-|c-d| \right|$$ could hold, establishing the desired result.

Answer 1

A counterexample is $a=1,b=c=0,d=-1$.