Help Graphing Multivariable Inequality with Absolute Value

[pearapricot]

Question: Sketch the region in the plane consisting of all points (x,y) such that [FONT=Helvetica Neue, Arial, sans-serif]|x - y| + |x| - |y| >= 2[/FONT]
[FONT=Helvetica Neue, Arial, sans-serif]I recognised that there are 4 possible conditions:
+x,+y
-x,-y
+x,-y
-x,+y
And solved to get the following equations:
[/FONT]y >= x-2
y <= x+2y <= -x+2
y >= -x-2
After plotting the equations I was left with a confusing 'X' shape. Is there a better way to approach this? I'm confused!
[FONT=Helvetica Neue, Arial, sans-serif]
[/FONT]

 

[Ishuda]

Question: Sketch the region in the plane consisting of all points (x,y) such that |x - y| + |x| - |y| >= 2
I recognised that there are 4 possible conditions:
+x,+y
-x,-y
+x,-y
-x,+y
And solved to get the following equations:
y >= x-2
y <= x+2y <= -x+2
y >= -x-2
After plotting the equations I was left with a confusing 'X' shape. Is there a better way to approach this? I'm confused!

You really have 6 conditions: Three each for x-y>0 (you can't have x<0 and y>0) and x-y<0 (you can't have x>0 and y<0). Those 6 will lead to other conditions, for example for x-y>0, x>0 and y<0 [The lower right quadrant], you will also have to have x$\displaystyle \ge$

 

[pearapricot]

You really have 6 conditions: Three each for x-y>0 (you can't have x<0 and y>0) and x-y<0 (you can't have x>0 and y<0). Those 6 will lead to other conditions, for example for x-y>0, x>0 and y<0 [The lower right quadrant], you will also have to have x$\displaystyle \ge$

How do you graph this? I don't really understand

 

[Ishuda]

How do you graph this? I don't really understand

You don't really graph a function for the answer, you 'color areas' where the inequality is satisfied. As an example let x-y be greater than zero, that is, the area under the line y=x which includes part of the first quadrant (x$\displaystyle \ge$0 & y$\displaystyle \ge$0), part of the second quadrant (x$\displaystyle \le$0 & y$\displaystyle \le$0) and the complete fourth quadrant (x$\displaystyle \ge$0 & y$\displaystyle \le$ 0). So we have three cases. Looking at the case for the fourth quadrant, we have
f(x) = |x-y| + |x| - |y| = x - y + x - (-y) = 2 x
The area where f(x)$\displaystyle \ge$2 is the area bounded by
x=2, x=$\displaystyle \infty$, and y=0
and that is the area you would 'color'

Do the same for the remainder of the conditions (including x-y$\displaystyle \le$0). Some parts of the areas may coincide or be connected.

 

[pka]

How do you graph this? I don't really understand

Have a look at this.