I get sup and inf in basic functions, and even using things between absolutes | | . But when the variable you solving for cancels out like this one, I don't know how to work it out:

|x+1| + |2-x| = 3

Normally you would say then:

-3 < x+1 + 2-x < 3 right?

But the problem with this one specifically is that it cancels out. I have the answer in the back of the book (sub = 2 and inf = -1) which i'm guessing they ignored the 3s and took

the |x+1| = 0 so x = -1 and |2-x|=0 and so 2=x

But i don't know what the reasoning would be to do it like that, because whatever value you choose for x, even 10, it just cancels out so that you get 3 in the middle. I don't really get why they did this. I can copy the logic I mentioned here for future problems but I don't really get why or what is the logical way to work it out. Does the triangle inequality come into it?

Any explanations would be most appreciated