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

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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:

(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

Identify the boundary number and place it on the number line.
Draw an open circle (strict) or closed circle (includes equal to).
Draw the arrow: right for >/≥, left for </≤.
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.

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

Answers

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

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Frequently Asked Questions

How do you graph single-variable inequalities?

Place the boundary number on a number line, choose an open or closed circle, then shade the side that makes the inequality true. For x < 3, draw an open circle at 3 and shade to the left. For x ≥ −1, draw a closed circle at −1 and shade to the right. Check by testing a shaded number in the original inequality.

What is graphing inequalities in math?

Graphing an inequality means drawing every value that makes it true, since an inequality usually has a whole range of solutions rather than one. With one variable, you show that range on a number line using a circle at the boundary and a shaded ray. For example, x > 4 includes 4.1, 5 and 100, so the graph shades everything to the right of 4.

How do you graph one-variable inequalities with an open or closed circle?

Use an open circle when the symbol is strict, < or >, because the boundary value is not part of the solution. Use a closed (filled) circle for ≤ or ≥, because the boundary is included. So x > 2 gets an open circle at 2, while x ≥ 2 gets a solid dot at 2. Both shade to the right, toward larger numbers.

How do you graph inequalities with variables on both sides?

Solve the inequality first so the variable is alone, then graph the result. For 3x + 2 > x + 8, subtract x to get 2x + 2 > 8, subtract 2 to get 2x > 6, and divide by 2, giving x > 3. Draw an open circle at 3 and shade right. Remember to flip the sign if you ever divide by a negative.

Can you graph inequalities on a graphing calculator?

Yes. Many graphing calculators and online graphing tools let you type an inequality such as y > 2x − 1 and shade the solution region. For a single-variable inequality, it is usually faster to solve it and draw a number line by hand, which is what most tests, including GED Math, expect you to read and interpret.

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

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