transformation of two dimensional random var

given the following Probability density function:

f x,y(x,y) = { 0.5, ( 0<y<1, 2y-x<2, 2y+x<2 }
0, else

and i need to find f z(z) while z=y-x

i got really confused while trying to calculate the borders of x and y for the integration.

i would be really thankful for someone explaining me how to come to the borders slow and step by step.

i got to the following equations: y < 2-z , y < (2+z)/3 , x < 2-z, x < (2-2z)/3

but i don't know how to resume from here. it said on the solution that borders are 0 to (2-2z)/3 for y and y-z to 2-2y for x. why?

thanks in advance for any help