Interval Notation Calculator

Infinity never gets a square bracket

[3, ∞] is wrong. Not stylistically — mathematically. A square bracket says the endpoint is a member of the set, and ∞ is not a number, so no set has it as a member.

x > 3 is (3, ∞): square on the finite side if the inequality is not strict, round on the infinite side always. The two ends of an interval are decided by different considerations, and only one of them is about the inequality symbol.

The same applies at the left: (−∞, 5] is correct and [−∞, 5] is not. This tool will not print the wrong one, which is a small thing until you are checking your own work against it.

“Or” and “and” describe different sets

x < −1 or x ≥ 4 is (−∞, −1) ∪ [4, ∞) — two separate pieces, joined by ∪, and a number qualifies by satisfying either one.

x < 1 and x > 5 is ∅. Nothing is both less than 1 and greater than 5, so the set is empty. That is a correct answer, not a failure to solve.

The error worth naming is writing an “or” condition as a chained inequality. 5 < x < 1 is not a way of writing “x < 1 or x > 5” — a chain always means “and”, and that particular chain describes the empty set. The two conditions genuinely cannot be written as one chain, which is exactly why the ∪ notation exists.

Absolute value is a distance, and the direction flips

|x − a| < b says “x is within b of a”. That is one interval, centred on a, and it is why the answer to |x| < 3 is (−3, 3) — everything within 3 of zero.

Flip the inequality and you get the outside instead. |x - 2| >= 5 says “x is at least 5 away from 2”, which is (−∞, −3] ∪ [7, ∞): two rays going in opposite directions, joined by a union.

So the direction of the inequality decides whether the answer is one interval or two, and reading |x − a| as a distance makes that automatic rather than a rule to remember. The commonest slip is keeping a single interval when the sign flips, which produces an answer with no solutions in it at all.

Which bracket, and which circle

The correspondence is exact, and it is worth stating once:

• < or > — the endpoint is excluded, so a round bracket and a hollow circle.
• ≤ or ≥ — the endpoint is included, so a square bracket and a filled circle.
• ±∞ — always a round bracket, and an arrow rather than a circle.

2 < x ≤ 5 has one of each: (2, 5], hollow at 2 and filled at 5. The number line above the answer is drawn from the same interval list that produced those brackets, so a mismatch between them is impossible here.

Ordering matters too. The smaller endpoint goes first, and an interval written the wrong way round — [5, 2] — is empty rather than backwards. Reversing it is not a notation preference; it changes what the expression means.

Set-builder notation, and what the bar means

{x ∈ ℝ | 2 < x ≤ 5} reads “the set of real x such that 2 < x ≤ 5”. The vertical bar is “such that”, and a colon is used identically by many texts.

Set-builder is more general than interval notation because the condition can be anything, not just a range — {x ∈ ℝ | x² < 4} is perfectly well-formed. Interval notation only describes unions of intervals, which is most of what comes up and not all of it.

Which is why interval notation is the preferred answer when it applies: it is shorter, it makes the endpoints obvious, and unions are visible at a glance. x < 2 or x > 2 is a case where both are natural — (−∞, 2) ∪ (2, ∞) in one and {x ∈ ℝ | x < 2 or x > 2} in the other.

When the answer is nothing, or everything

Two answers surprise people, and both are correct as written. The empty set, ∅, when no number satisfies the condition — and (−∞, ∞), when every number does.

x < 1 and x > 5 gives the first. Two conditions that cannot both hold produce an empty overlap, and ∅ is the right way to write that. Leaving the answer blank, or writing “no solution” where a set was asked for, is a different and less precise thing.

The second turns up with conditions like |x| ≥ 0, which every real number satisfies because an absolute value is never negative. (−∞, ∞) is the whole line, and it is worth recognising as an answer rather than as a sign of a mistake.

Sources and methodology

Interval and set-builder notation are conventions rather than results; the reference below is the standard statement of them.

• NIST Digital Library of Mathematical Functions — §1.1, Notation and Special Sets
• How we calculate — methodology

Method. Every figure on this page comes from src/lib/interval-notation.ts, which holds a solution as a sorted list of disjoint intervals with exact rational endpoints. Unions, intersections and complements are computed on that representation rather than pattern-matched from the input, so touching intervals merge correctly ([1,2] ∪ (2,3] is [1,3]) and an empty result comes out empty. That engine is verified on every change against 49 hand-written assertions, including that every conversion round-trips — an interval parsed and re-rendered gives back the same set — and that the number-line marks always match the printed brackets. The count and the per-case breakdown are published on the formula verification page.

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Read the guide

Interval notation is how the answer to an inequality gets written down; solving the inequality comes first. The Graphing Quadratic Inequalities Calculator produces sets of exactly this shape by sign chart, and Which Method Should You Use to Solve a Quadratic? covers the solving step behind them.