6 min read
How to Graph Inequalities on a Number Line
A visual guide to open and closed circles, shading direction, and compound inequality graphs — with worked examples for every pattern.
Definition and Core Idea
A number line graph of an inequality shows every value that satisfies the inequality as a shaded region. The boundary value is marked with a circle, and the shading extends in the direction of all values that make the inequality true.
The circle type carries the most important information. An open circle means the boundary value is not part of the solution — it corresponds to a strict inequality symbol (> or <). A closed circle means the boundary value is included — it corresponds to an inclusive symbol (≥ or ≤). Getting the circle type wrong changes the actual solution set, not just the appearance of the graph.
The shading direction follows directly from the symbol. Greater than means shade to the right. Less than means shade to the left. Compound inequalities either shade a bounded region between two boundaries or shade two separate rays in opposite directions.
Rules, Forms, and Patterns
Open circle ○
Used for strict symbols > and <. The boundary value is not part of the solution. Corresponds to a parenthesis in interval notation.
Closed circle ●
Used for inclusive symbols ≥ and ≤. The boundary value is part of the solution. Corresponds to a bracket in interval notation.
Compound inequality
AND inequalities shade a bounded region between two circles. OR inequalities shade two separate rays, each with its own circle type.
Worked Example
Prompt
Identify the boundary values and symbols. Left boundary: -2, symbol: ≤ (inclusive) → closed circle ●. Right boundary: 5, symbol: < (strict) → open circle ○.
Mark the circles on the number line. Place a closed circle at -2 and an open circle at 5.
Shade the region between the circles. This is an AND inequality — shade all values between -2 and 5. The shading does not extend beyond either boundary.
Result
Use the Calculator for This Topic
A concept becomes durable only when you can move from the rule back into a fresh problem. The calculator is useful here because it lets you test the exact pattern from this article, compare your work with the step list, and verify the final graph or notation.
Use the number line inequality calculator to check endpoint style and shading instantly. If you want extra practice with split regions, compare the output with the compound inequality calculator, then translate the same answer with the interval notation calculator.
Suggested input
Enter -2 ≤ x < 5 to see the mixed open and closed circle pattern from the worked example.
Try x > 3 or x ≤ -1 to see two separate shaded rays on the same number line.
Enter any inequality from your homework to check the circle type and shading direction before drawing it by hand.
The circle type and the interval notation bracket are the same decision
An open circle on the number line and a parenthesis in interval notation both mean the same thing: the boundary value is excluded. A closed circle and a bracket both mean the boundary value is included. The two representations are different ways of showing the same fact.
This connection makes it easy to check your work. If you draw an open circle at 3 but write [3, +∞) in interval notation, the representations disagree and one of them is wrong. The circle and the bracket on the same side should always match.
OR inequalities produce two separate rays, not one connected region
A compound OR inequality such as x < -2 or x ≥ 4 has two separate solution regions that do not connect. On the number line, this looks like two rays pointing in opposite directions with a gap between them. Students sometimes try to shade the region between the two boundaries, which is the opposite of the correct answer — the gap is exactly the region that does not satisfy the inequality.
The easiest check is to test a value in the gap. For x < -2 or x ≥ 4, try x = 0: 0 < -2 is false and 0 ≥ 4 is false, so 0 is not in the solution. The gap should not be shaded.
Put The Rule Into Practice
Concept pages are useful only if they transfer back into actual problem solving. After reading this guide, the best next step is to try several inequalities with different circle types, shading directions, and compound structures so you can see the pattern rather than memorize one worked example.
The calculator pages linked here are meant to shorten that feedback loop. You can test a new inequality, inspect the number line output, and compare the graph with the interval notation to confirm that your visual model is consistent.
Once the picture is clear, convert the same answer with inequality to interval notation and compare it with the interval notation guide. The graph and the notation should tell the same story.
Common Mistakes To Avoid
Drawing an open circle for ≥ or ≤ — the inclusive symbols always use a closed circle.
Shading the wrong direction — greater than shades right, less than shades left.
Shading the middle region for an OR inequality instead of the two outer rays.
FAQ
How do I know which direction to shade?
Shade in the direction of the values that satisfy the inequality. For x > 3, values greater than 3 are to the right, so shade right. For x ≤ -1, values less than or equal to -1 are to the left, so shade left.
What does the number line graph look like for x ≠ 2?
Place an open circle at 2 and shade everything to the left and right — the entire number line except the point at 2. In interval notation this is (-∞, 2) ∪ (2, +∞).