Graphing Linear Inequalities

Solid or dashed boundary

The boundary line is the graph of the matching equation — for $y < 2x + 1$, that's the line $y = 2x + 1$, drawn with its slope and $y$-intercept as usual.

Strict symbols $<$ and $>$ mean points on the line itself are not solutions, so draw it dashed. The symbols $\leq$ and $\geq$ include equal, so points on the line are solutions — draw it solid.

This is the same idea as open versus closed circles on a number line, stretched into two dimensions: dashed is the open circle, solid is the closed one.

Which side to shade

With the inequality solved for $y$, the symbol tells you directly: $y >$ or $y \geq$ shades above the line (larger $y$-values), and $y <$ or $y \leq$ shades below (smaller $y$-values).

Here's $y < 2x + 1$ finished: the boundary $y = 2x + 1$ is dashed because $<$ is strict, and the shading sits below the line because $y$ is less than the expression.

Check with a test point

Not sure about the shading, or the inequality isn't solved for $y$? Test a point. Pick any point not on the boundary — $(0, 0)$ is the easiest whenever the line misses the origin — and substitute it into the inequality.

If the statement comes out true, shade the side containing your test point. If it comes out false, shade the other side. For $y < x + 4$, testing $(0, 0)$ gives $0 < 4$, which is true — so the origin's side gets the shading.

Worked examples

Try one yourself

Common questions