Any Simpler Approach to Solve This Question?

Screenshot 2021-08-23 at 11.19.05 PM

Hi,
Could anyone please advise on this question?
I wonder if there is any other (simpler) way to solve this other than my approach.
I basically divided into four situations:

(1) 1-x>=0, y-1>=0
(2) 1-x>=0, y-1<0
(3) 1-x<0, y-1>=0
(4) 1-x<0, y-1<0

Then, I solved each equation with an algebra.

(1) y=x ; when 1>=x, y>=1
(2) y=-x+2 ; when 1>= x, y<1
(3) y=x ; when 1<x, y>=1
(4) y=-x+2 ; when 1<x, y<1

And then, I drew a graph for each situation.

I got the question right, but it took me like 7 minutes, so my approach definitely was a time sinkhole.

Does anyone know a better approach to solve this than mine? Would really appreciate your tips!

A post was merged into an existing topic: Area between lines