Calculator Input
Use one piece for a single interval. Use two pieces for a union.
Example Data Table
| Example | Inequality | Interval Notation | Set Builder | Notes |
|---|---|---|---|---|
| 1 | 2 ≤ x < 7 | [2, 7) | {x | 2 ≤ x < 7} | Closed left endpoint and open right endpoint. |
| 2 | x > -3 | (-3, ∞) | {x | x > -3} | Positive infinity always uses a parenthesis. |
| 3 | x ≤ 4 | (-∞, 4] | {x | x ≤ 4} | Negative infinity also always uses a parenthesis. |
| 4 | x = 5 | [5] | {x | x = 5} | A single point needs both endpoints included. |
| 5 | x < -1 or x ≥ 3 | (-∞, -1) ∪ [3, ∞) | {x | x < -1 or x ≥ 3} | Separate regions are joined with a union. |
| 6 | -2 < x ≤ 2 | (-2, 2] | {x | -2 < x ≤ 2} | Mixed endpoints create a half-open interval. |
Formula Used
Closed interval: [a, b] ↔ a ≤ x ≤ b
Open interval: (a, b) ↔ a < x < b
Left unbounded: (-∞, b] ↔ x ≤ b, and (-∞, b) ↔ x < b
Right unbounded: [a, ∞) ↔ x ≥ a, and (a, ∞) ↔ x > a
Union rule: A ∪ B means values that satisfy A or B.
Simplification rule: Overlapping or touching intervals merge when no gap remains between them.
How to Use This Calculator
- Enter the variable symbol you want displayed.
- Choose one piece or two pieces for a union.
- Type the lower and upper bounds for each piece.
- Select whether each endpoint is included or excluded.
- Check the infinity boxes for unbounded sides.
- Press calculate to see the interval, inequality, and set forms.
- Review the Plotly graph to confirm open and closed endpoints.
- Export the result with the CSV or PDF button.
FAQs
1. What does interval notation represent?
Interval notation describes a range of values on the number line. Brackets include endpoints. Parentheses exclude endpoints. Unions combine separate valid ranges into one expression.
2. Why do infinity symbols always use parentheses?
Infinity is not a reachable endpoint. It represents unbounded growth or decrease, not a fixed number. Because it cannot be included, interval notation always uses parentheses with infinity.
3. When should I use a union symbol?
Use a union when the solution has separate pieces. For example, x < 2 or x ≥ 6 becomes two intervals joined by ∪ because there is a gap between them.
4. Can this calculator simplify overlapping intervals?
Yes. If two pieces overlap or touch without leaving a gap, the calculator merges them into one simpler equivalent interval. That helps produce cleaner final notation.
5. What is a single-point interval?
A single-point interval contains exactly one value. It occurs when both endpoints are equal and included. In this tool, that appears as [a] and the inequality becomes x = a.
6. What is the difference between interval and set-builder notation?
Interval notation focuses on endpoints and range symbols. Set-builder notation states the logical condition directly, such as {x | 2 ≤ x < 5}. Both describe the same solution set.
7. Why is my midpoint undefined?
Midpoints only exist for finite intervals with two numeric endpoints. If an interval extends to infinity, there is no center value, so the midpoint is reported as undefined.
8. Does endpoint inclusion affect interval length?
No. Length depends on the difference between numeric endpoints. Open and closed versions of the same finite interval have the same length, even though their endpoint membership differs.