I'm having trouble solving absolute value equations involving multiple absolute value functions added together. For example, take the problem
$|x+3|-|x+1|+ x+2 =0$
If all the outputs of the absolute values are non-negative at the same time:
$x+3-(x+1)+ x+2 =0$
$x+3-x-1+ x+2 =0$
And yet, when you plug in -4 into $|x+3|-|x+1|+ x+2 =0$, it doesn't work.
What confuses me even more is why x=-4 doesn't work when plugged into the original equation, when, if all the outputs of the absolute values are non-negative at the same time, then $x+3−(x+1)+x+2=|x+3|−|x+1|+x+2$.
I'm not just looking for a solution to this problem. I want to know what methods I can use to solve absolute value equations. Because the methods I use to solve absolute value problems involving only 1 absolute value function (or absolute value problems involving the multiplication and division of multiple absolute values) don't work here, as seen above.
Thank you in advance!