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:
- 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
| Circle | Arrow direction | Inequality |
|---|---|---|
| Open | Right | \(\color{blue}{x > a}\) |
| Closed | Right | \(\color{blue}{x \ge a}\) |
| Open | Left | \(\color{blue}{x < a}\) |
| Closed | Left | \(\color{blue}{x \le a}\) |
Step-by-Step Summary
- Identify the boundary number (the point on the number line).
- Check the circle: \(\color{blue}{\text{ open } = \text{ strict }}\) inequality (> or <); closed = “or equal to” (≥ or ≤).
- Check the arrow direction: right = > or ≥; left = < or ≤.
- 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.
- Open circle at 4, arrow pointing right.
- Closed circle \(\color{blue}{\text{ at } -1}\), arrow pointing left.
- Open circle \(\color{blue}{\text{ at } -5}\), arrow pointing left.
- Closed circle at 7, arrow pointing right.
- Open circle at 2, arrow pointing left.
- Closed circle at 0, arrow pointing right.
Answers
- \(\color{blue}{x > 4}\)
- \(\color{blue}{x \le -1}\)
- \(\color{blue}{x < -5}\)
- \(\color{blue}{x \ge 7}\)
- \(\color{blue}{x < 2}\)
- \(\color{blue}{x \ge 0}\)
Frequently Asked Questions
What is the difference between an open and a closed circle on a number line?
An open circle means the boundary value is not a solution; the inequality is strict (> or <). A closed (filled) circle means the boundary value is a solution; the inequality includes “or equal to” (≥ or ≤).
How do I remember which symbol to use?
The arrowhead of < and > points in the direction of the smaller number. The direction of the number line arrow matches: \(\color{blue}{\text{ left } = \text{ smaller }}\) values, so left arrow = < (less than); right arrow = > (greater than).
Can an inequality have solutions on both sides of the number line?
A standard single inequality graphs on one side. However, a compound inequality such as \(\color{blue}{-2 < x \le 5}\) is shaded between two boundary points. You will study those in compound inequality lessons.
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