Skill: Represent inequalities on a number line

To represent an inequality on a number line:

1. Find the boundary value by solving the inequality.
2. Use an open circle for $<$ or $>$ because the boundary is not included.
3. Use a closed circle for $\leq$ or $\geq$ because the boundary is included.
4. Shade in the direction of all the values that make the inequality true.

Example

Represent $2x+1\leq 4$ on a number line.

Step 1: Solve the inequality.

$$2x+1\leq 4$$

Subtract $1$ from both sides:

$$2x\leq 3$$

Divide both sides by $2$:

$$x\leq \frac{3}{2}$$

Step 2: Mark the boundary.

The boundary value is $\frac{3}{2}$. Since the inequality is $\leq$, include $\frac{3}{2}$ with a closed circle.

Step 3: Shade the correct direction.

The solutions are all numbers less than or equal to $\frac{3}{2}$, so shade to the left.

$$\leftarrow\!\!\! \text{ shaded region }\!\!\!\bullet$$ $$\hspace{5.5cm}\frac{3}{2}$$

The graph represents all values $x\leq \frac{3}{2}$. For example, $1$, $0$, and $-\frac{5}{2}$ are included, but $2$ is not.