Interval Notation Calculator
Interval notation is a concise way to represent all the numbers that satisfy a condition. If you are solving inequalities, working with domain and range, or describing solution sets, you need to know how to write and read interval notation correctly.
What Is Interval Notation?
Interval notation describes a set of numbers using a pair of numbers and two symbols. The left symbol shows whether the smaller value is included; the right symbol shows whether the larger value is included. Everything between those endpoints is included unless the intervals are separated by a union.
For example, [2, 7) represents all numbers from 2 to 7, including 2 but excluding 7. It is much shorter than writing “x ≥ 2 and x < 7” or “2 ≤ x < 7.”
Symbols in Interval Notation
Brackets and Parentheses
[Square bracket] means the number is included (closed endpoint).
( Parenthesis ) means the number is not included (open endpoint).
| Inequality | Interval Notation | Meaning |
|---|---|---|
| x > 5 | (5, ∞) | Numbers greater than 5, not including 5 |
| x ≥ 5 | [5, ∞) | Numbers greater than or equal to 5 |
| x < 5 | (-∞, 5) | Numbers less than 5, not including 5 |
| x ≤ 5 | (-∞, 5] | Numbers less than or equal to 5 |
| 2 < x < 8 | (2, 8) | Numbers between 2 and 8, excluding both |
| 2 ≤ x ≤ 8 | [2, 8] | Numbers between 2 and 8, including both |
Infinity
Infinity (∞) always uses a parenthesis in interval notation because infinity is not a real number and cannot be included. Write (-∞, 10] to mean all numbers up to and including 10.
How to Write Interval Notation?
Step 1: Identify the endpoints. Find the smallest and largest values (or use -∞ and ∞ if the interval extends without bound).
Step 2: Determine inclusion. If the endpoint is included in the solution, use a bracket [ or ]. If it is excluded, use a parenthesis ( or ).
Step 3: Write the interval. Place the smaller value on the left, the larger on the right, separated by a comma.
Example 1: The inequality x ≥ -3 includes all numbers from -3 onward.
- Smaller endpoint: -3 (included, so use [ )
- Upper endpoint: ∞ (always open, so use ) )
- Result: [-3, ∞)
Example 2: The inequality 4 < x ≤ 12 includes numbers strictly greater than 4 and up to and including 12.
- Left endpoint: 4 (excluded, so use ( )
- Right endpoint: 12 (included, so use ] )
- Result: (4, 12]
Example 3: All real numbers.
- Result: (-∞, ∞)
Using the Union Symbol for Multiple Intervals
When a solution includes two or more separate ranges, use the union symbol (∪) to combine them.
For example, |x| > 3 means x < -3 or x > 3, which in interval notation is (-∞, -3) ∪ (3, ∞).
The solution includes two intervals with a gap between them. Nothing in the interval [-3, 3] is part of the solution.
Interval Notation vs. Other Notations
Inequality notation: x > 2
Interval notation: (2, ∞)
Set-builder notation: {x | x > 2}
All three mean the same thing–numbers greater than 2. Interval notation is the shortest and most commonly used in calculus and higher mathematics.
Common Mistakes to Avoid
- Using a bracket with infinity: (-∞, 5] ✓ vs. (-∞, 5) ✗ (infinity always uses parenthesis)
- Reversing the order: Always write the smaller value on the left: (2, 8) ✓ not (8, 2) ✗
- Confusing the symbols: Remember, [ and ] are closed (included); ( and ) are open (excluded)
- Forgetting the union symbol: If there are two separate regions, write (1, 3) ∪ (5, 7), not (1, 3)(5, 7)
Domain and Range Using Interval Notation
In functions, the domain is the set of all possible input values (x-values) and the range is the set of all possible output values (y-values). Both are often expressed in interval notation.
If a function is defined for x ≥ 0 and y can be any real number, write:
- Domain: [0, ∞)
- Range: (-∞, ∞)
This content is educational. For academic assignments, verify your interval notation with your instructor’s requirements, as some contexts may have specific conventions.