Which Side of the Boundary Line to Shade
A linear inequality in two variables splits the plane into two halves along a boundary line; you shade the half whose points make the inequality true, found by testing one point not on the line.
A line like y = -2x + 5 has exactly one job: split the coordinate plane into two regions. Every point in one region makes y < -2x + 5 true, and every point in the other makes it false. The line itself is the border between yes and no. That's why graphing an inequality is really two steps: draw the boundary, then figure out which side to color in.
To find the right side, pick any point not on the line, usually (0,0) because it's easy to compute with, and plug its coordinates into the inequality. If the statement comes out true, shade the side containing that point. If it comes out false, shade the other side. This works every time because the truth value can't change anywhere within one region, only at the boundary itself.
The line style carries its own meaning: strict inequalities (< or >) get a dashed line, because points exactly on it don't satisfy the inequality, while ≤ or ≥ get a solid line, since those points do count. This same logic shows up next in systems of inequalities, where you shade two regions separately and look for where they overlap, and in absolute value inequalities, which split into two linear cases, each with its own boundary and shading rule.
Key terms
- Boundary line
- The line from replacing the inequality symbol with an equals sign, marking where the two regions meet.
- Half-plane
- One of the two regions a line divides the coordinate plane into.
- Test point
- A point not on the boundary line, plugged into the inequality to check which half-plane satisfies it.
- Strict inequality
- An inequality using < or >, where points on the boundary line are not included, shown with a dashed line.
- Inclusive inequality
- An inequality using ≤ or ≥, where points on the boundary line are included, shown with a solid line.
Watch it explained
- How it worksHow does testing the origin in an inequality tell you instantly which half-plane gets shaded?
- How it worksHow do you decide between drawing a solid boundary line and drawing a dashed one?
- Worked exampleGraph y less than -2x + 5 by drawing a dashed boundary and testing the point (0, 0).
- Common mistakeWhy doesn't a greater-than symbol always mean shading upward when the inequality is not solved for y?
- Going furtherWhat happens to the shaded region when you multiply both sides by -1 and flip the symbol?
Check yourself
End of topic.