Graphing Inequalities on a Number Line Calculator
Visualize any linear inequality on a number line with open/closed circles, shading direction, and interval notation.
Enter a simple inequality (e.g., x > 3) or a compound inequality (e.g., -2 <= x < 5) to see it graphed on a number line.
Graphing Inequalities on a Number Line Calculator
Visualize any linear inequality on a number line with open/closed circles, shading direction, and interval notation.
About the Number Line Inequality Calculator
An inequality on a number line is one of the most fundamental concepts in algebra and pre-calculus. While an equation such as x = 5 has exactly one solution, an inequality like x > 5 has infinitely many solutions — all real numbers greater than 5. Graphing the solution set on a number line transforms an abstract collection of numbers into an intuitive visual shape: a dot or circle marking where the boundary falls, and a shaded arrow or segment indicating which numbers satisfy the inequality.
The two most important visual elements are the boundary point and the shading. The boundary point is the number that appears in the inequality, and it is marked on the number line with a circle. Whether that circle is open or closed depends on the type of inequality symbol. A strict inequality (< or >) means the boundary number itself is NOT a solution, so an open circle ○ is drawn to show the number is excluded. A non-strict (inclusive) inequality (≤ or ≥) means the boundary IS a solution, so a closed (filled) circle ● is drawn to show the number is included. The shading — or arrow — then extends in the direction containing all the other solutions: to the right for > or ≥, to the left for < or ≤.
A compound inequality such as −3 ≤ x < 5 combines two inequalities. The solution set is the set of all numbers that satisfy both simultaneously. Graphically, it produces two boundary circles — one closed at −3 and one open at 5 — with shading in the region between them. This region forms a bounded interval, unlike a simple inequality whose solution extends to infinity in one direction.
Interval notation is a compact way to write the same solution set using parentheses and brackets. A parenthesis ( or ) is used at a boundary that is NOT included (corresponding to an open circle), and a square bracket [ or ] is used at a boundary that IS included (corresponding to a closed circle). The symbol ∞ always carries a parenthesis because infinity is never actually reached. For example, x > 3 is written (3, ∞); x ≤ −1 is written (−∞, −1]; and −2 ≤ x < 7 is written [−2, 7).
Inequalities appear throughout mathematics and the real world. Speed limits define the range v ≤ 65 mph. Budget constraints set an amount 0 ≤ s ≤ 50. Quality-control tolerances require a dimension L to fall in an interval like 4.98 ≤ L ≤ 5.02. Eligibility requirements for voting, driving, or retirement create age inequalities like a ≥ 18. Understanding how to graph and read inequalities on a number line is an essential skill for algebra, calculus, data analysis, and everyday decision-making.
This calculator parses both simple and compound linear inequalities, renders the graph description clearly with open and closed circle notation, and provides the solution in interval notation — all in one step.
Number Line Inequality Examples
Five common inequalities showing both simple and compound cases with their interval notation.
| Inequality | Interval Notation | Graph Description |
|---|---|---|
| x > 3 | (3, ∞) | Open circle at 3 (not included), arrow pointing right toward positive infinity. |
| y <= -2 | (−∞, −2] | Closed circle at −2 (included), arrow pointing left toward negative infinity. |
| -1 < z <= 4 | (−1, 4] | Open circle at −1, closed circle at 4, shading between the two boundary points. |
| x >= 0 | [0, ∞) | Closed circle at 0 (the origin is included), arrow pointing right. |
How to Use the Number Line Inequality Calculator
- Type your inequality in the input field. Supported formats include 'x > 5', 'y <= -1.5', '-3 < z <= 3', and similar linear inequalities. You may use any variable name.
- Use <= for ≤ and >= for ≥, or type the Unicode symbols directly if your keyboard supports them.
- Click 'Graph Inequality'. The tool parses the expression and displays the graph description with open/closed circles, the shading direction, and the interval notation.
- Read the interval notation in the result — parentheses exclude the boundary, brackets include it, and ∞ always has a parenthesis.
- Click Reset to clear the field and enter a new inequality.
Number Line Inequality Calculator FAQ
What is the difference between an open and a closed circle?
An open circle ○ on the boundary point means that number is NOT included in the solution set — used for strict inequalities < and >. A closed circle ● means the boundary number IS included — used for non-strict inequalities ≤ and ≥.
How do I write less than or equal to in the calculator?
Type <= for ≤ (less than or equal to) and >= for ≥ (greater than or equal to). The calculator also accepts the Unicode characters ≤ and ≥ directly.
What is a compound inequality?
A compound inequality combines two inequalities with the same variable, such as -3 < x ≤ 5. It means x must satisfy both conditions simultaneously. On the number line it produces two boundary points and a shaded region between them.
How does interval notation relate to the graph?
Interval notation uses parentheses ( ) for excluded boundaries (open circles) and square brackets [ ] for included boundaries (closed circles). The notation (3, ∞) corresponds to an open circle at 3 with shading to the right, exactly what you see on the number line.
Can I enter inequalities with coefficients, like 2x > 6?
This calculator is designed for inequalities already solved for the variable, such as x > 3. To graph 2x > 6 you would first divide both sides by 2 to get x > 3, then enter that result.
What does the infinity symbol ∞ in interval notation mean?
The infinity symbol ∞ indicates that the solution set extends without bound in that direction. It always gets a parenthesis — never a bracket — because infinity is not a real number that can be 'reached' or 'included' as an endpoint.