Question 1124546
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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.<br><pre>
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)
</pre>
{{{drawing(400,400,-10,10,-10,10,
line(0,-10,0,10),line(-10,0,10,0),locate(8,-.5,X),locate(.5,8,Y),
line(1,-.2,1,.2),line(2,-.2,2,.2),line(3,-.2,3,.2),line(-1,-.2,-1,.2),line(-2,-.2,-2,.2),line(-3,-.2,-3,.2),
line(-.2,1,.2,1),line(-.2,2,.2,2),line(-.2,3,.2,3),line(-.2,-1,.2,-1),line(-.2,-2,.2,-2),line(-.2,-3,.2,-3),
locate(-2.5,1,-2),locate(2,-.5,2),locate(.5,2.5,2),locate(-1.5,-1.5,-2),
green(line(2,0,10,-8)),red(line(2,0,10,8)),blue(line(-2,0,-10,8)),line(-2,0,-10,-8)
)}}}