I'm sure I'm not the first one to think of this, but this lesson was pleasingly effective. In past years graphing inequalities in two variables was something that seemed easy for the kids in class, but they could never remember what was up when we reviewed it later.

This year, to start off, I had the kids pick the coordinates of three different points (x, y) on the plane, test them in the given inequality, and mark the point with a closed circle if it came out to be true, and an open circle if it came out to be false. After a few minutes I had to say, "If you only got true points, find a point where it comes out to be false." We ended up with a screen that looked like this:

We discussed the significance of the boundary between the blue and white spots, and also tried a few non-integral points. Then I had them try a different one on their own. They developed a very serviceable method for indicating the "true region". One class wanted to write the words "True" and "False" instead of shading, which isn't bad, but I warned them that eventually we'd need to know where two of them overlapped, so they shaded.

This seems like a stunningly obvious approach now that I've tried it, but I thought I'd share since it took me 4 years to think of it. Does anyone do anything similar, or have a better way?