how do I solve -3|x+y|-|x+z|
You solve this by noting that |(a+b)| is always positive, whereas (a+b) itself may be either positive or negative depending upon individual values for a and b.
That means that we have 4 cases,
(x+y) +ve, (x+z) +ve
(x+y) +ve, (x+z) -ve
(x+y) -ve, (x+z) +ve
(x+y) -ve, (x+z) -ve
You then consider the value of your expression, without the moduls signs, for each of the 4 cases.
i.e. -3(x+y) - (x+z)