How to Write Inequalities from Number Lines?

A number line graph of an inequality shows you immediately which values satisfy the inequality — every point shaded (or the arrow pointing toward) is a solution. Reading those visual clues and writing the correct inequality notation is a skill that appears on every GED Math exam. This lesson teaches you to decode any number line graph and write the corresponding inequality in under thirty seconds.

What Does a Number Line Inequality Look Like?

An inequality graphed on a number line has three components:

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• A point (the boundary value) with either an open circle (ˆ) or a closed circle (•)
• An arrow or shading extending left or right from that point
• The direction: \(\color{blue}{\text{ right } = \text{ greater }}\) than; \(\color{blue}{\text{ left } = \text{ less }}\) than

Rules for Reading Number Line Inequalities

Rule 1: Open circle vs. closed circle

• Open circle (hollow): the boundary value is not included → use \(\color{blue}{>}\) or \(\color{blue}{<}\)
• Closed circle (filled): the boundary value is included → use \(\color{blue}{\ge}\) or \(\color{blue}{\le}\)

Rule 2: Direction of the arrow

• Arrow pointing right (toward larger numbers) → greater than: \(\color{blue}{x > a}\) or \(\color{blue}{x \ge a}\)
• Arrow pointing left (toward smaller numbers) → less than: \(\color{blue}{x < a}\) or \(\color{blue}{x \le a}\)

Quick reference table

Step-by-Step Summary

1. Identify the boundary number (the point on the number line).
2. Check the circle: \(\color{blue}{\text{ open } = \text{ strict }}\) inequality (> or <); closed = “or equal to” (≥ or ≤).
3. Check the arrow direction: right = > or ≥; left = < or ≤.
4. Write the inequality: \(\color{blue}{x [\text{ symbol }] [\text{ boundary number }]}\).

Watch: Writing Inequalities From Number Lines (Math with Mr. J)

Math with Mr. J explains exactly how to read a number line and write the inequality:

Worked Examples

Example 1: Open circle at 3, arrow pointing right. Write the inequality.

Open \(\color{blue}{\text{ circle } = \text{ strict }}\); right \(\color{blue}{\text{ arrow } = \text{ greater }}\) than.
x > 3

Example 2: Closed circle \(\color{blue}{\text{ at } -2}\), arrow pointing left. Write the inequality.

Closed circle = “or equal to”; left \(\color{blue}{\text{ arrow } = \text{ less }}\) than.
x ≤ −2

Example 3: Open circle at 0, arrow pointing left. Write the inequality.

Open \(\color{blue}{\text{ circle } = \text{ strict }}\); left \(\color{blue}{\text{ arrow } = \text{ less }}\) than.
x < 0

Example 4: Closed circle at 5, arrow pointing right. Write the inequality, then list three solutions.

\(\color{blue}{x \ge 5}\). Three solutions: 5, 7, 100 (any number 5 or greater).

More Practice: Graphing Inequalities on Number Lines (Math with Mr. J)

This companion video covers the reverse skill — drawing a number line graph from a written inequality:

Exercises

Write the inequality shown by each description of a number line graph.

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

Answers

1. \(\color{blue}{x > 4}\)
2. \(\color{blue}{x \le -1}\)
3. \(\color{blue}{x < -5}\)
4. \(\color{blue}{x \ge 7}\)
5. \(\color{blue}{x < 2}\)
6. \(\color{blue}{x \ge 0}\)

Frequently Asked Questions

How do you write inequalities from number lines?

Look at three things: the boundary number, the circle, and the arrow. The boundary number goes in the inequality. An open circle means the number is not included (< or >), and a closed circle means it is (≤ or ≥). An arrow pointing right means greater than, and left means less than. So a closed circle at −2 with the arrow pointing left gives x ≤ −2.

How do you write an absolute value inequality from a number line?

Find the midpoint of the two boundary points and the distance from the midpoint to each end. If the graph is shaded between closed circles at −1 and 7, the center is 3 and the distance is 4, so |x − 3| ≤ 4. If the shading goes outward from open circles at −1 and 7, the values are more than 4 away from 3, so |x − 3| > 4.

What do inequality signs look like on a number line?

Each sign has its own picture. x > 5 is an open circle at 5 with shading to the right. x ≥ 5 uses a filled circle at 5, shaded right. x < 5 is an open circle with shading to the left, and x ≤ 5 is a filled circle, shaded left. A handy memory aid: the filled circle matches the extra line under ≤ and ≥, meaning the boundary itself counts.

How are number lines used for inequalities?

A number line shows every solution of an inequality at once, which a single number cannot do. For x > 1, the solutions include 1.5, 2, 10, and 1,000, so the graph shades everything to the right of 1. To test your graph, pick a point in the shaded part and plug it in: if x = 3, then 3 > 1 is true, so the shading is correct.

What are the ways to write inequalities?

The same solution set can be written several ways. With symbols: x ≥ 4. In words: x is at least 4, or x is no less than 4. In interval notation: [4, ∞), where the square bracket means 4 is included. On a number line: a closed circle at 4 shaded to the right. Word problems often use phrases like at most (≤) and more than (>).

Related Topics

• Graphing Single Variable Inequalities
• Solving Word Problems Involving One Step Inequalities
• How to Solve Word Problems Involving the One Step Equation
• Using Number Lines to Represent Integers
• Absolute Value Definition