Intervals describe continuous ranges of numbers: the domain of √x is [0, ∞), the temperatures at which water is liquid at sea level are (0 °C, 100 °C). Being fluent in the three notations makes algebra and calculus far easier to read.
How to use the Interval Notation Calculator
- Conversion type
- Interval type
- Inequality
- Interval
- a
- Left bracket
- Right bracket
- b
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Pick the direction of the conversion and the kind of interval. The calculator shows the interval, the equivalent inequality and the set-builder description.
Tips
- Rewrite the equation in standard form before reading off the coefficients.
- Check your answer by substituting it back into the original equation.
- Keep exact forms (fractions and square roots) until the final step to avoid rounding errors.
- For inequalities, flip the sign whenever you multiply or divide by a negative number.
Common mistakes to avoid
- Dropping a negative sign when copying a coefficient.
- Forgetting the ± when taking a square root.
- Dividing by an expression that could be zero.
Formula
[a, b] = {x | a ≤ x ≤ b} (a, b) = {x | a < x < b} [a, ∞) = {x | x ≥ a}Worked example
Inputs — Conversion type: Interval → inequality; Interval type: Two-sided (bounded); a: 0; Left bracket: ( open; Right bracket: ] closed; b: 100.
Results — Result = 0 < x ≤ 100; Set-builder form = (0, 100] = {x | 0 < x ≤ 100}; Equivalent form = (0, 100].
Key terms
- Set-builder notation
- {x | condition}, the set of x satisfying a condition.
- Ray
- An interval that extends to infinity in one direction.
Frequently asked questions
What does a half-open interval look like?
One bracket of each kind, such as [2, 7), which includes 2 but not 7.
How do I write the union of two intervals?
With ∪, for example (−∞, 1) ∪ (3, ∞).
Can an interval contain a single number?
[a, a] = {a} is a degenerate closed interval.
What is the empty set?
∅ – an interval such as (3, 3) with nothing in it.
What formula does the Interval Notation Calculator use?
[a, b] = {x | a ≤ x ≤ b} (a, b) = {x | a < x < b} [a, ∞) = {x | x ≥ a}.