Interval Notation Calculator
This interval notation calculator converts both ways: enter an inequality such as −2 < x ≤ 5 or x < −1 or x ≥ 3, or an interval such as (−∞, 4] ∪ (6, ∞), and get the interval notation, the inequality, set-builder notation, the type of interval and its length.
How it is calculated
Interval notation writes a set of numbers with its two bounds. A square bracket [ ] means the bound is included (≤, ≥), a round bracket ( ) means it is not (<, >). So −2 < x ≤ 5 is written (−2, 5].
Examples
| Inequality | Interval notation |
|---|---|
| −2 < x ≤ 5 | (−2, 5] |
| x ≥ 3 | [3, ∞) |
| x < 1 | (−∞, 1) |
| x < −1 or x ≥ 3 | (−∞, −1) ∪ [3, ∞) |
Rules
- ∞ (infinity) is not a number and is never included, so it always gets a round bracket: (−∞, … or …, ∞).
- “or” joins pieces with the union sign ∪; overlapping pieces merge: (1, 3) ∪ [2, 6] = (1, 6].
- “and” keeps only what both parts share: x > 2 and x ≤ 5 is (2, 5].
- In set-builder notation the same set reads {x ∈ ℝ | −2 < x ≤ 5}.
- Some countries, and the standard ISO 31-11, write an excluded bound with a reversed bracket: ]−2, 5] instead of (−2, 5].
To solve an inequality first, use the inequality solver; to see the interval on a graph, try the graphing calculator.
Length and type
[0, 1) is half-open with length 1; [3, 3] is a single point with length 0; any interval with ∞ is unbounded. For unions the lengths of the separate pieces add up.
Frequently asked questions
How do you write an inequality in interval notation?
Write the smaller bound first, then the larger one. Use [ ] for ≤ or ≥ and ( ) for < or >. Example: −2 < x ≤ 5 becomes (−2, 5].
How is x ≥ 3 written in interval notation?
[3, ∞). The 3 is included, infinity never is.
What does the U mean in interval notation?
It is the union sign ∪ and stands for “or”: (−∞, −1) ∪ [3, ∞) means x < −1 or x ≥ 3.
What is the difference between ( and [ ?
A round bracket excludes the bound, a square bracket includes it. (0, 1] contains 1 but not 0.
How do you write all real numbers in interval notation?
(−∞, ∞), also written ℝ.
Sources and legal basis
As of: · Editorial policy
Related tools
- Inequality SolverInequality solver with steps for linear, quadratic and absolute value inequalities. Solution as interval and inequality, with a check and the sign-flip rule.
- Equation SolverEquation solver with steps: solve linear, quadratic and higher-degree equations for x, with brackets, fractions and a check. Free, no sign-up needed.
- Online graphing calculatorGraphing calculator online: plot up to three functions, zoom and pan, find roots, extrema and intersections, use a slider for a, save as PNG or SVG. Free.
- Quadratic Formula CalculatorQuadratic formula calculator: solve ax² + bx + c = 0 with steps – roots, discriminant, exact radical form, vertex, factored form and complex roots.
- Power Set CalculatorPower set calculator: lists all subsets of a finite set ordered by size, gives the count 2ⁿ and the number of subsets per size, with the working shown.
- Online CalculatorCalculator online, free and instant: type or tap and see the answer as you go – order of operations, brackets, percent, square root, steps and a history list.
- Young modulus calculator: stress, strain and elongation of a rodYoung’s modulus calculator: E = (F/A)·(L₀/ΔL) from force, cross-section, length and elongation – or the elongation of steel, aluminium, copper and more.
- Clock Angle CalculatorClock angle calculator: the angle between the hour and minute hand for any time, with the formula |30h − 5.5m|, the smaller and larger angle and every step.
- CGPA calculator for Nigerian universities (5-point scale)CGPA calculator Nigeria: enter each course with its grade or score and credit units to get your GPA or CGPA on the 5.0 scale, the class of degree and the steps.