Question

How do you graph `abs(x+y)>1`?

Answer 1

There's plenty of ways one could do this, but let's just start by thinking about what this could look like.

We will think about the boundary of this region: `|x+y| = 1`.

Here we think about the two cases of absolute value: positive and negative.

If `x+y>0`, the boundary is simple:
`|x+y| = x+y = 1 implies y = -x + 1`
which is just a line going down from (0,1) to (1,0).

If `x+y<0`, the boundary is another line:
`|x+y| = -(x+y) = 1 implies y = -x - 1`
which is a line going from (-1,0) to (0, -1).

Therefore, there are three regions:
- Below the left line
- Between the lines
- Above the right line

It is clear that the more outside of those you get, the more `>1` you'll get, hence it is the regions outside the strip that are included.

In case my description wasn't sufficient, here's the plot:
graph{|x+y|=1 [-10, 10, -5, 5]}