Vivekanand Vellanki
1

Lets take an example.

Suppose you want to find all points where x\ge0. Here are the steps involved:

1. Draw the line without the inequality, that is draw the line x=0
2. This line divides the plane into two portions, the top-half and the bottom-half

Your question is: "How do I know whether you should shade the top-half or the bottom-half?"

I find the following way the easiest:

1. Pick two points, one belonging to the top-half and another belonging to the bottom-half
2. In this case, let me pick the points (1, 0) from the top-half and (-1, 0) from the bottom half
3. Re-visit the equation and see which of the two points makes the inequality true
4. Taking (-1, 0) and looking at the equation and substituting 1 for x, we get -1\ge0. Clearly, this is false. Hence, we found a point in the bottom-half that makes the equation false. This implies that all points in the bottom-half make the equation false. Hence, the bottom-half should not be shaded
5. Taking (1, 0) and looking at the equation and substituting 1 for x, we get 1\ge0. Clearly, this is true. Hence, (1, 0) makes the equation true. The top-half has to be shaded

To verify, pick another point from the top-half (1, 1) and confirm that this point also makes the equation true.