Interval notation from inequality calculator

Turning a single-variable inequality into interval notation is a small step that shows up on every algebra worksheet, entrance exam, and analysis prerequisite: students must choose square brackets when an endpoint is included, parentheses when it is excluded, and infinity markers when the solution runs without end. This interval notation from inequality calculator accepts either a one-sided statement such as x less than four or a compound window such as negative two less than or equal to x less than five, then returns the matching interval string plus structured lower and upper fields for code clients.

What interval notation means and how one-sided inequalities map to rays

Unbounded ends use the infinity symbol with a parenthesis, never a square bracket, because infinity is not a real number that can be attained. A one-sided inequality already isolates the variable: x less than a bound is the open left ray from negative infinity up to but not including that bound; x less than or equal to the same bound closes the right end with a square bracket; x greater than a bound is the open right ray starting just after the bound and running to positive infinity; x greater than or equal to the bound closes the left end. This calculator does not solve general linear forms such as two x plus three less than eleven—that is a different search intent covered by a linear inequality solver—and instead starts from the classroom form x compared with a number, which is exactly what students write after isolation and what many homework keys already display. Interval notation packages a set of real numbers as a pair of endpoints with a bracket type at each end. A parenthesis means the endpoint is excluded; a square bracket means it is included. The response echoes the operator and bound, builds a human-readable inequality string using standard less-or-equal and greater-or-equal glyphs when appropriate, and emits the interval string ready to paste into an answer key. Structured fields lower, upper, lower_inclusive, and upper_inclusive let API clients build number-line widgets or set membership tests without re-parsing brackets. A boolean unbounded flag marks every successful one-sided conversion, and empty stays false because a single finite bound always yields a non-empty ray on the real line. Rejected inputs include a missing form, an unknown operator, a non-finite bound, and magnitudes beyond the documented ceiling, all surfaced as invalid_input so billing and goldens stay deterministic.

Compound inequalities, empty sets, and the input contract

The left relation may be less than or less than or equal, meaning the lower number is strictly below x or allowed to equal x; the right relation is likewise less than or less than or equal between x and the upper number. Open-open, open-closed, closed-open, and closed-closed combinations map directly to parentheses and brackets in the interval string. A compound inequality of the form lower related to x related to upper describes a bounded window when the lower endpoint sits strictly below the upper endpoint, or a singleton when both ends are closed on the same number. When the lower endpoint is greater than the upper endpoint, or when the endpoints are equal but at least one end is open, no real x can satisfy both sides at once, and the calculator returns the empty set symbol with empty true so automated graders can distinguish a genuine empty answer from a formatting mistake. The preferred fields are form set to one_sided or compound; for one_sided, op and bound; for compound, lower, lower_op, upper, and upper_op. Aliases such as kind and shape for form, operator and comparison for op, value and endpoint for bound, left and right for the compound ends, and left_op and right_op for the compound relations are accepted so spreadsheets and query strings can reuse the same contract. Pair the free browser run on this page with the priced API when you need identical mathematical results inside an automated pipeline without re-implementing bracket rules yourself.

Where converting inequalities to intervals helps study, teaching, and code

On this page the work runs entirely in your browser; through the API each successful conversion costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split—free interactive exploration and a priced, identical API—matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with linear inequality solvers when you still need to isolate x from ax plus b compared with c, with absolute-value tools when modulus constraints appear, and with other algebra endpoints when you are building a small lab of free, deterministic math helpers focused on clear notation rather than heavy symbolic engines.