How to Graph Single–Variable Inequalities? (+FREE Worksheet!)

Graphing a single-variable inequality on a number line gives a picture of every value that satisfies the inequality. A dot and an arrow turn abstract notation like \(\color{blue}{x > 4}\) into a visual range that you can read instantly. Mastering this skill is essential for GED Math, because test questions regularly ask you both to graph an inequality and to choose the inequality that matches a given graph.

What Is a Single-Variable Inequality?

A single-variable inequality compares a variable to a number using one of four symbols:

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• \(\color{blue}{>}\) greater than  |  \(\color{blue}{<}\) less than
• \(\color{blue}{\ge}\) greater than or equal to  |  \(\color{blue}{\le}\) less than or equal to

The solution is not one value but a range of values. Graphing shows that entire range on a number line.

How to Graph Single-Variable Inequalities

Step 1: Locate the boundary point

Draw a number line and place a point at the boundary value (the number in the inequality).

Step 2: Open or closed circle?

• > or < (strict) → open circle (boundary not included)
• ≥ or ≤ (or equal to) → closed circle (boundary included)

Step 3: Draw the arrow

• > or ≥ → arrow pointing right (larger values)
• < or ≤ → arrow pointing left (smaller values)

Step 4: Verify with a test point

Pick a number in the shaded region and substitute it into the original inequality. If the statement is true, the graph is correct.

Step-by-Step Summary

1. Identify the boundary number and place it on the number line.
2. Draw an open circle (strict) or closed circle (includes equal to).
3. Draw the arrow: right for >/≥, left for </≤.
4. Test a value: substitute into the inequality and confirm it is true.

Watch: How to Solve and Graph One-Step Inequalities (Math with Mr. J)

Math with Mr. J demonstrates graphing inequalities with clear step-by-step examples:

Worked Examples

Example 1: Graph \(\color{blue}{x > 4}\).

Boundary: 4. Open circle (strict >). Arrow pointing right.
Test: \(\color{blue}{x = 6}\): “6 > 4” ✓  Test boundary exclusion: \(\color{blue}{x = 4}\): “4 > 4” is false ✓ (correctly excluded)

Example 2: Graph \(\color{blue}{x \le -1}\).

Boundary: −1. Closed circle (≤). Arrow pointing left.
Test: x = −3: “−3 ≤ −1” ✓  Test boundary: x = −1: “−1 ≤ −1” ✓ (correctly included)

Example 3: Graph \(\color{blue}{x \ge 2}\).

Boundary: 2. Closed circle. Arrow pointing right.
Solutions: all numbers 2 and greater; e.g., 2, 5, 100.

Example 4: Graph \(\color{blue}{x < -3}\).

Boundary: −3. Open circle. Arrow pointing left.
Test: x = −5: “−5 < −3” ✓  Test: x = −3: “−3 < −3” is false ✓

More Practice: Graphing Inequalities (Khan Academy)

Khan Academy provides additional examples of graphing inequalities with worked solutions:

Exercises

Describe the number line graph for each inequality (circle type, position, arrow direction) and list two solutions.

1. \(\color{blue}{x > 6}\)
2. \(\color{blue}{x \le 3}\)
3. \(\color{blue}{x \ge -4}\)
4. \(\color{blue}{x < 0}\)
5. \(\color{blue}{x \le -2}\)
6. \(\color{blue}{x > -1}\)

Answers

1. Open circle at 6, arrow right. Solutions: 7, 10.
2. Closed circle at 3, arrow left. Solutions: 3, −5.
3. Closed circle \(\color{blue}{\text{ at } -4}\), arrow right. Solutions: −4, 0.
4. Open circle at 0, arrow left. Solutions: −1, −10.
5. Closed circle \(\color{blue}{\text{ at } -2}\), arrow left. Solutions: −2, −7.
6. Open circle \(\color{blue}{\text{ at } -1}\), arrow right. Solutions: 0, 5.

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Frequently Asked Questions

What is the difference between graphing an equation and graphing an inequality?

Graphing an equation gives a single point (or specific points) on a number line. Graphing an inequality gives a range of values, shown with a ray (arrow) extending from the boundary point in one direction.

Why does the direction of the inequality symbol matter?

The symbol tells you which side of the boundary contains solutions. A > or ≥ means solutions are to the right (larger); < or ≤ means solutions are to the left (smaller). Getting the direction wrong gives the opposite set of solutions.

How do I solve an inequality before graphing it?

Treat it like an equation: apply inverse operations to isolate the variable. Important: if you multiply or divide both sides by a negative number, you must flip the inequality symbol. Once the variable is isolated, graph as usual.

Related Topics

• Writing Inequalities From Number Lines
• Solving Word Problems Involving One Step Inequalities
• Solve Integers Inequalities Involving Absolute Values
• How to Solve Word Problems Involving the One Step Equation
• Using Number Lines to Compare and Order Rational Numbers