Skill: Graph inequalities on a number line

An inequality describes many possible values. To graph it on a number line:

1. Find the boundary number.
2. Use an open circle if the inequality is $<$ or $>$. The boundary is not included.
3. Use a closed circle if the inequality is $\le$ or $\ge$. The boundary is included.
4. Shade in the direction of the values that make the inequality true.

Example: Graph $(x \le 3)$

Step 1: Find the boundary number.

The boundary number is $3$.

Step 2: Choose the endpoint.

The symbol is $\le$, which means “less than or equal to.” Since $x$ is allowed to equal $3$, use a closed circle at $3$.

Step 3: Choose the shading direction.

Numbers less than $3$ are to the left of $3$, so shade to the left.

<==========●---------------->
           3

The graph represents all numbers that are less than or equal to $3$, such as $3$, $2$, $0$, and $(-4)$.

To check, try a shaded value and an unshaded value:

• $(2 \le 3)$ is true, so $2$ belongs in the shaded part.
• $(5 \le 3)$ is false, so $5$ does not belong in the shaded part.