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.
3. Pick the shading. $y >$ or $y \geq$ shades above. $y <$ or $y \leq$ shades below.
4. If you are unsure, use the test point.

The test point

A point is in the shaded region if it makes the inequality true. So pick an easy point, put its numbers in, and see what happens.

The easiest point to test is $(0, 0)$, because zeros are simple to work with.

Worked example. Is $(0, 0)$ in the solution set of $y < x + 4$?

Put $x = 0$ and $y = 0$ into the inequality:

$0 < 0 + 4$

$0 < 4$

That statement is true, so the point $(0, 0)$ is in the shaded region, and you shade the side of the line that contains the origin.

If the statement had come out false, you would shade the other side instead.