Sketching Region for |x-y|+|x|-|y| <= 2

[linux666]

i want to sketch the region in the plane consisting of all points (x,y) for

|x-y|+|x|-|y| <= 2

any ideas?
thanks

 

[humanino]

Welcome penguin devil !

Sketch each term separately, it will then be easy for you to sketch the linear combination. We do only provide hints !

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You need 3D sketch of course...

 

[M98Ranger]

To sketch this region, we can start by breaking the inequality into four separate cases based on the absolute value expressions.

Case 1: x ≥ 0, y ≥ 0
In this case, the inequality becomes x-y+x-y ≤ 2, or 2x-2y ≤ 2. Simplifying further, we get x-y ≤ 1. This represents the region above the line y=x-1.

Case 2: x ≥ 0, y < 0
Following the same steps as above, we get x+y+x-y ≤ 2, or 2x ≤ 2. This represents the region to the right of the y-axis and below the line x=1.

Case 3: x < 0, y ≥ 0
Similarly, we get -x-y-x-y ≤ 2, or -2y ≤ 2. Simplifying, we get y ≥ -1. This represents the region to the left of the x-axis and above the line y=-1.

Case 4: x < 0, y < 0
Again, following the same steps, we get -x+y-x+y ≤ 2, or 2y ≤ 2. Simplifying, we get y ≤ 1. This represents the region below the line y=x+1.

Combining all four cases, we get a region in the plane that looks like a diamond shape with its corners at (1,0), (0,1), (-1,0), and (0,-1). See the attached image for a visual representation.

I hope this helps!