Last updated October 4, 2026 · Report an error
This is the reverse of converting an inequality into interval notation: starting from an interval like (-3, 5], it reconstructs the equivalent inequality, letting you read the compact bracket notation back into plain inequality symbols. This direction is just as commonly needed as the forward conversion — a textbook or exam might present a solution set in interval notation and ask you to describe it as an inequality, or you might need to double-check that a given interval was written correctly in the first place. That second part matters more than it might seem: interval notation has strict rules about which bracket type is valid where, and a closed bracket ("[" or "]") placed next to an infinity symbol is actually invalid notation, since infinity can never be "included" the way a square bracket implies. This tool converts interval notation into its equivalent inequality, and validates that the notation itself follows the correct rules, flagging invalid bracket/infinity combinations rather than silently accepting them.
How it works
A square bracket on either end of the interval translates to a "≤" or "≥" relation at that boundary (since a square bracket means the boundary value is included in the solution set), while a round parenthesis translates to a strict "<" or ">" relation (meaning the boundary is excluded). Infinite bounds are required to use a parenthesis specifically, since infinity is not a number that can actually be reached or included — the tool checks for this and flags a closed bracket next to an infinity symbol as invalid notation rather than silently converting it.
- Enter lower bracket.
- Enter lower bound type.
- Enter lower bound value (if a number).
- Enter upper bound type.
- Enter upper bound value (if a number).
- Enter upper bracket.
- Click Calculate to see your results.
Examples
A closed interval
[2, 7] means 2 ≤ x ≤ 7 — both boundary values are included, since both ends use square brackets.
An unbounded interval
(-∞, 4) means x < 4 — the parenthesis on the infinite end is required, and the parenthesis on the 4 means 4 itself is excluded.
A half-open interval
(0, 10] means 0 < x ≤ 10 — the left boundary (0) is excluded by the parenthesis, while the right boundary (10) is included by the square bracket.
Who should use it
- Algebra and precalculus coursework on interval notation.
- Double-checking that a given interval's notation is actually valid.
- Converting a textbook-provided interval back into an inequality for a word problem.
Industry applications
- Mathematics education
Advantages
- Validates bracket/infinity combinations rather than assuming the input is well-formed.
- Classifies the interval as open, closed, half-open, or all real numbers.
- Useful for both converting and double-checking notation.
Limitations
- Only handles a single interval, not a union of multiple disjoint intervals.
Common mistakes to avoid
- Writing a closed bracket "[" or "]" next to an infinite bound.
- Reversing which bracket corresponds to which inequality symbol when converting back to an inequality.
- Assuming an interval's notation is automatically valid without checking the infinity/bracket rule.
Best practices
- When unsure whether to use a round or square bracket, ask whether the boundary value itself should be counted as part of the solution.
- Always use a parenthesis, never a square bracket, next to an infinity symbol.
- Cross-check a converted inequality against the original interval by testing a boundary value in each.
Tips
- This tool pairs naturally with the Inequality to Interval Notation Calculator — use one to convert forward, and this one to convert back and verify.
- If a result flags invalid notation, check whether a square bracket was mistakenly placed next to an infinity symbol.