The nature of the graph is going to change when either of the expressions in the absolute value signs change sign. Since those expressions change sign when either x or y changes sign, the nature of the graph will be different in each of the four quadrants.
quadrant I: x>=0; y>=0: |x|=x; |y|=y. x-y = 2 --> y = x-2 (red line on graph below)
quadrant II: x<0; y>=0: |x|=-x; |y|=y. -x-y = 2 --> y = -x-2 (blue line on graph below)
quadrant III: x<0; y<0: |x|=-x; |y|=-y. -x+y = 2 --> y = x+2 (black line on graph below)
quadrant IV: x>=0; y<0: |x|=x; |y|=-y. x+y = 2 --> y = -x+2 (green line on graph below)