Graphical Inequalities and Regions in the Plane
A linear inequality in two variables picks out a half-plane, and several of them together leave a region whose corners answer the question being asked. Drawing the boundary as solid or dashed is what records whether the boundary itself counts.
What a learner can do afterwards
- Shades the region satisfying y < 2x + 1 and says why the boundary line is dashed
- Finds the region satisfying three inequalities at once and lists the integer points inside it
- Writes down the inequalities that describe a region already shaded on a grid
1 · Read
An inequality in two variables gives you a whole half plane, not a line. The boundary line y equals 2x plus 1 splits the grid, and every point on one side passes while the other side fails. Pick a test point such as (0, 0), which is off the line, and shade the side that works. Line, test, shade: that is the whole game.
The line style carries meaning. A strict inequality like y less than 2x plus 1 excludes its boundary, so you draw it dashed. An inclusive one like y less than or equal to 2x plus 1 includes it, so you draw it solid. One more trap: multiplying or dividing by a negative flips the symbol, so watch each step as you rearrange.
Stack three constraints and shade each on the same grid: x at least 0, y at least 0, and x plus y at most 3. Keep only the overlap where every test passes, a small triangle. Hunt integer points by listing pairs: x equals 0 gives 4 points, x equals 1 gives 3, x equals 2 gives 2, x equals 3 gives 1. That is 10 points in total.
Reading backwards uses the same skills in reverse. Given a shaded polygon, find each edge line, decide dashed or solid from whether the edge counts, then test a point for the direction. A region on or below the line y equals 4 minus x, edge included, is y less than or equal to 4 minus x.
Draw the boundary with the right dash, test a point, and shade or read the side that passes.
2 · Watch
Take it off screen
Where it sits
This opens up
Nothing builds on it yet.
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.