Graphing Inequalities — Core Skills

Graphing Inequalities: The Basics

Part 1 of 2 — The Basics

An equation like $y = 2x + 1$ draws a line. An inequality like $y > 2x + 1$ draws a whole region — a shaded area on one side of that line.

Every one of these graphs has exactly two decisions to make.

Decision 1: Solid line or dashed line?

The line itself is called the boundary line. You draw it by pretending the inequality sign is an equals sign.

• $\geq$ or $\leq$ (the ones with the little line underneath) mean the points on the line count. Draw a solid line.
• $>$ or $<$ (no line underneath) mean the points on the line do not count. Draw a dashed line.

An easy way to remember it: the line under the sign becomes a solid line on the graph.

Decision 2: Shade above or shade below?

Get $y$ by itself on the left, then look at the sign:

• $y >$ or $y \geq$ means shade above the line.
• $y <$ or $y \leq$ means shade below the line.

"Greater" means higher up on the graph, so you shade upward.

Worked example

Graph $y < 2x + 1$.

Step 1. Draw the boundary line $y = 2x + 1$. The sign is $<$ with no line underneath, so the boundary line is dashed.

Step 2. The sign is $y <$, so shade below the line.

That is the finished graph: a dashed line with everything below it shaded.

Graphing Inequalities: Practice

Part 2 of 2 — Practice

The steps, every time

1. Make sure $y$ is alone on the left side.
2. Pick the line type. A sign with a line underneath ($\geq$ or $\leq$) gets a solid boundary line. A sign without one ($>$ or $<$) gets a dashed boundary line.