So the question asks to express $\, a - |(a-|a|)|$ without using absolute value signs and instead with in multiple cases.
I feel as though I'm on the right track but something just isn't clicking in my head on how to properly do this.
My attempt so far has gone like this.
$a - |(a - |a|)|$
$|a - |a|| < 0 \quad or \quad |a-|a|| >0 $
which then each simplify to
$-a + |a| \quad and \quad a-|a|$
so the whole expression becomes
$a + a - |a| \quad and \quad a -a + |a|$
from here both of these have two cases
$a < 0 \quad then \quad a + a + a =3a$
$a > 0 \quad then \quad a + a - a =a$
$a < 0 \quad then \quad a - a - a =-a$
$a > 0 \quad then \quad a - a + a =a$
The answer given is that: $ a \;\;if\;\; a>0;\quad 3a\;\;if\;\; a<0$
I see how the first interval makes sense, but I don't how I can get $3a$ to appear in the 3rd case since the $a -a$ seems to make it necessarily either $-a$ or $a$.
Thank you in advance for any help.