Represent inequalities on a number line

An inequality in one variable is represented as the set of values that make the comparison true: an open endpoint marks an excluded boundary for $<$ or $>$ , a closed endpoint marks an included boundary for $\leq$ or $\geq$ , and a shaded ray shows values extending in the appropriate direction. The representation includes rational-number solutions and compound inequalities formed by joining or intersecting regions; systems in two variables and more advanced nonlinear or absolute-value cases are outside this scope.

Detailed Explanation: Represent inequalities on a number line

To represent an inequality on a number line:

Find the boundary value by solving the inequality.
Use an open circle for $<$ or $>$ because the boundary is not included.
Use a closed circle for $\leq$ or $\geq$ because the boundary is included.
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.

Learn by doing: Represent inequalities on a number line

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Function Domain/Range Definition - Inequality to Number Line (Without Union)

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