Interval notation turns inequality statements into a clear set of numbers using brackets and parentheses. This Interval Notation Calculator converts common inequalities (like x > 3 or x ≤ 7) into interval notation, correctly handling open vs. closed endpoints.
You’ll enter the inequality type and the endpoint value(s), then the tool outputs the matching interval notation and a plain-English description.
What Is Interval Notation?
Interval notation is a standardized way to describe all real numbers that satisfy an inequality. Instead of writing “x is greater than 3,” you write an interval such as (3, ∞) or [−2, 5).
- Parentheses ( ) mean the endpoint is not included (open interval).
- Brackets [ ] mean the endpoint is included (closed interval).
- ∞ and −∞ represent no bound in that direction.
How Interval Notation Matches Inequalities
Each inequality symbol maps directly to bracket style and direction. The calculator uses these rules to build the correct interval.
| Inequality | Endpoint included? | Interval notation pattern |
|---|---|---|
| x > a | No | (a, ∞) |
| x ≥ a | Yes | [a, ∞) |
| x < a | No | (−∞, a) |
| x ≤ a | Yes | (−∞, a] |
| a < x < b | No for both | (a, b) |
| a ≤ x < b | Left yes, right no | [a, b) |
| a < x ≤ b | Left no, right yes | (a, b] |
| a ≤ x ≤ b | Yes for both | [a, b] |
Variables and Inputs the Calculator Uses
Interval notation depends only on the endpoint value(s) and whether each endpoint is included. The calculator is designed for the most common inequality forms.
- Lower endpoint (a): the smaller bound (used for “greater than” style constraints).
- Upper endpoint (b): the larger bound (used for “less than” style constraints).
- Endpoint inclusion: determined by whether the inequality uses ≤ or ≥.
- Direction: determined by whether the inequality opens to the left (−∞) or right (+∞).
If you enter an interval with two endpoints, the calculator also checks that a ≤ b to avoid impossible solutions like (5, 2).
Reading the Output: Interval Notation and Plain English
The calculator returns both a compact interval notation string and a plain-English sentence. This helps you verify the result quickly and use it in homework or test answers.
- Interval notation shows the exact endpoints and open/closed status.
- Plain-English description confirms the inequality meaning (for example, “greater than 3 but less than or equal to 7”).
Practical Examples (Real Use Cases)
Example 1: Temperature Limits
Suppose a device works for temperatures x > 0 and x ≤ 40. In interval notation, that is (0, 40]. The open parenthesis at 0 matches “greater than,” and the closed bracket at 40 matches “less than or equal to.”
Example 2: Test Score Cutoff
If a passing score is x ≥ 60, the interval notation is [60, ∞). The bracket at 60 means 60 is included, and the infinity symbol indicates there is no upper limit.
Common Mistakes to Avoid
- Using parentheses for ≤ or ≥: ≤ and ≥ require brackets, not parentheses.
- Swapping endpoints in two-number intervals: the lower endpoint must be smaller than the upper endpoint.
- Forgetting the infinity direction: x > a opens to the right, while x < a opens to the left.
- Mixing up (−∞, a) vs. (−∞, a]: the bracket or parenthesis depends on whether the inequality includes a.
Frequently Asked Questions
How do I know whether to use a bracket or a parenthesis in interval notation?
Use parentheses when the endpoint is not included (for > or <). Use brackets when the endpoint is included (for ≥ or ≤). For example, x > 3 becomes (3, ∞) and x ≤ 5 becomes (−∞, 5].
What does infinity mean in interval notation?
Infinity means the interval continues forever in that direction. For x > a, the interval grows without bound to the right, so you write (a, ∞). For x < a, you write (−∞, a). Infinity is never included.
How do I write a two-sided inequality like 2 < x ≤ 8?
Split the inequality into a lower bound and an upper bound. The left side (2 < x) uses a parenthesis at 2. The right side (x ≤ 8) uses a bracket at 8. The result is (2, 8].
Can interval notation describe values of x that are all real numbers?
Yes. If there is no restriction, the solution set is every real number. In interval notation, that is (−∞, ∞). In practice, this comes from inequalities like −∞ < x < ∞ or from equations with no limiting bounds.
What if my lower endpoint is bigger than my upper endpoint?
If you enter a two-endpoint interval where a > b, no real numbers satisfy the inequality. The calculator flags this as invalid input. For example, (5, 2) cannot happen because there is no number greater than 5 and less than 2 at the same time.
Next Steps
Use the calculator above to convert inequalities quickly and accurately. Then practice rewriting a few inequalities by hand, focusing on the two key ideas: open vs. closed endpoints, and the direction of the interval.
