Skip to content
Calcumatix

Interval Notation Calculator

Turn one-sided or two-sided rules for x into interval notation with the right square mark, round mark, and infinity sign.

Result

[-2, 5)

-2 ≤ x < 5 in interval notation is [-2, 5)

Quick Answer

An interval notation calculator turns an inequality into a standard interval. Use a square bracket when an end value is in the set and a parenthesis when the value is out. The ∞ sign always takes a parenthesis. For -2 ≤ x < 5, include -2 and leave out 5, so write [-2, 5).

What An Interval Notation Calculator Does And Shows

Interval notation shows a set of numbers on one line. The first value is the lower endpoint. The second value is the upper endpoint. A square bracket keeps an end value in the set. A round mark leaves that value out. This tool has two modes. One-sided mode has one finite endpoint and one unbounded side. One-sided mode uses -∞ or ∞ on the open side. Between-values mode has a lower and upper endpoint. Each finite endpoint can be included or excluded. Interval notation names the same set as the inequality and is common in algebra and calculus. Students use the form after solving an inequality, reading a number line, or stating the domain and range of a function in the standard language of sets.

Interval Notation: The Rules Behind Every Endpoint

The notation system uses square brackets for inclusive boundaries (values that satisfy equality) and parentheses for exclusive or unbounded boundaries.

  • x < b gives (-∞, b) and x ≤ b gives (-∞, b].
  • x > a gives (a, ∞) and x ≥ a gives [a, ∞).
  • a < x < b gives (a, b) and a ≤ x ≤ b gives [a, b].

Square brackets [ ] mean the endpoint is included (≤ or ≥).

Parentheses ( ) mean the endpoint is excluded (< or >) or unbounded (±∞).

The ∞ and -∞ symbols represent unbounded directions and always take parentheses.

Interval Notation Steps: Choose, Enter, Convert, Check

Inputs

  • Mode: choose one-sided or between-values.
  • Lower bound: enter the left fixed value when needed.
  • Upper bound: enter the right fixed value when needed.
  • Include switch: choose whether each fixed end is in the set (inclusive vs exclusive).

Steps

  1. Choose the one-sided or between-values mode.
  2. Enter the fixed value or values.
  3. Mark each fixed end as in or out.
  4. Calculate the short range.
  5. Check both end marks against the first rule.

See Interval Notation Applied To A Compound Inequality

Compound inequality -2 ≤ x < 5.

  1. Read the left sign: -2 ≤ x keeps -2 in the set, requiring a square bracket: [-2.
  2. Read the right sign: x < 5 leaves 5 out, requiring a round parenthesis: 5).
  3. Write the endpoints from low to high: [-2, 5).
  4. Verify endpoints: x = -2 satisfies -2 ≤ -2 (included), and x = 5 does not satisfy 5 < 5 (excluded).

Interval notation is [-2, 5). Both end values are exact, so the conversion uses no rounding.

When Interval Notation Fits Your Solution Set Best

Use interval notation after you solve an inequality or read a number line. The form also works for a domain, range, or any unbroken set of valid inputs.

For nearby algebraic, graphing, and set notation tools, see the math calculators hub and look up concepts in the glossary.

Assumptions

  • The low end is less than the high end.
  • Each fixed end has a clear in or out rule.
  • A one-sided set runs without end in one direction.
  • The ∞ sign always uses a round mark.

Limitations

  • The tool converts a finished rule but leaves the solving step to you.
  • Two separate ranges need a union sign.
  • Reversed end values need to be fixed before use.
  • The result does not draw a number line.

In Practice

One frequent error is using a square bracket beside ∞. The ∞ sign is not a point you can reach or include. Use a round mark next to both ∞ and -∞.

Related Calculators

Simplest Radical Form Calculator

Simplify square, cube, or fourth roots by pulling perfect powers from the radicand and showing each factor step for homework checks.

Open the Calculator

Rational Expressions Calculator

Simplify a factored rational expression, cancel shared roots, and keep all values blocked by the first denominator for algebra work.

Open the Calculator

Null Space Calculator

Find all vectors that a 2 by 2 matrix sends to zero, then see the row steps and the basis for the null space during introductory verification.

Open the Calculator

Related Guides

Interval Notation Calculator Questions And Clear Answers

When Do I Use A Bracket Instead Of A Parenthesis?

Use a square bracket when a finite end belongs to the set. Use a round mark when the end is left out by < or >. A sign with equality keeps the end in. The ∞ sign always takes a round mark.

Why Does Infinity Always Use A Parenthesis?

The ∞ sign uses a round mark because no real input can equal infinity. The mark shows that the set keeps going with no last value. A square bracket would wrongly claim that infinity is part of the set.

How Do I Write A One-Sided Inequality As An Interval?

Place the fixed end on the correct side and ∞ on the side with no end. For x ≥ 3, write [3, ∞). The square bracket keeps 3 in the set, and the round mark leaves the far side open.

Can Interval Notation Show Two Separate Solution Sets?

Two ranges can be joined with the union sign to show one split set. This tool handles one unbroken range at a time. Write each range on its own, then place the union sign between the two parts.

Sources

Last updated: . Reviewed for accuracy against the formula shown above.