PDEs- D'Alembert Solution of the Wave Equation

[Roni1985]

Homework Statement

The question is here:
http://ocw.mit.edu/courses/mathemat...ions-fall-2006/assignments/probwave1solns.pdf

It's a long question and I figured attaching the link here would be better.

I need help with the question on page 4.
when

U_t(x,0)=g(x)
and U(x,0)=0

Homework Equations

The only relevant equation is the integral of Integral[ g(s)ds,s,x-t,x+t]

The Attempt at a Solution

I tried to look at the solutions but I don't understand how to find the boundaries of the integral from the regions.

for example,
to be in region 2, the solution says that x+t>1 and -1<x+t<1
why is that ?
how was this derived ?Thanks,
Roni.

 

[Roni1985]

still can't figure this out :\