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Interval notation is a compact shorthand for writing the solution to an inequality — instead of writing out "-3 < x ≤ 5" in words and symbols, interval notation compresses the same information into (-3, 5]. The catch is that the notation depends entirely on getting the right bracket type for each boundary: a round parenthesis for a boundary that's excluded (strict "<" or ">"), and a square bracket for a boundary that's included ("≤" or "≥") — mixing these up is one of the most common small errors in algebra and precalculus coursework, since the two bracket types look similar but mean genuinely different solution sets. Interval notation also has a special rule for unbounded sides: infinity is never actually "reached," so any side of an interval extending to positive or negative infinity always uses a round parenthesis, even if the corresponding inequality symbol elsewhere in the same compound inequality is a "≤" or "≥". This tool converts a compound inequality directly into interval notation, choosing the correct bracket for each end and showing the reasoning behind each choice.
How it works
Each side of the inequality is examined separately: a "≤" or "≥" relation (meaning the boundary value itself is included in the solution) gets a square bracket, a strict "<" or ">" relation (meaning the boundary is excluded) gets a round parenthesis, and an unbounded side always gets a parenthesis next to the infinity symbol (∞ or −∞), regardless of what relation symbol is used at the finite end.
- Enter lower bound relation.
- Enter lower bound.
- Enter upper bound relation.
- Enter upper bound.
- Click Calculate to see your results.
Examples
A half-open interval
-3 < x ≤ 5 becomes (-3, 5] — a round parenthesis on the excluded left boundary, and a square bracket on the included right boundary.
A one-sided (unbounded) inequality
x ≥ 2 becomes [2, ∞) — the included boundary at 2 gets a square bracket, while infinity always gets a parenthesis.
A fully closed interval
-1 ≤ x ≤ 4 becomes [-1, 4] — both boundaries use square brackets since both relations are "≤"/"≥", meaning both endpoint values are included in the solution.
Who should use it
- Algebra and precalculus coursework on interval notation.
- Quickly converting a solved inequality into interval notation for a final answer.
- Double-checking bracket choices on a homework problem.
Industry applications
- Mathematics education
Advantages
- Correctly handles one-sided (unbounded) inequalities.
- Explains the bracket-choice reasoning alongside the result.
- Reduces the risk of a common bracket-mixup error.
Limitations
- Only handles a single compound inequality with one variable, not unions of multiple disjoint intervals.
Common mistakes to avoid
- Using a square bracket next to an infinite bound, which is never valid notation.
- Mixing up which bracket goes with a strict versus non-strict inequality symbol.
- Assuming both ends of an interval must use the same bracket type, when a half-open interval legitimately mixes both.
Best practices
- Double check each side of the inequality independently — the lower and upper bounds can use different bracket types.
- Always pair infinity with a round parenthesis, never a square bracket, regardless of the inequality's relation symbol.
- Read the resulting interval back as an inequality to confirm it matches your original problem before finalizing an answer.
Tips
- Reading interval notation left to right always matches reading the inequality left to right — the leftmost number is the smallest value in the interval.
- When in doubt about a bracket, ask "does this exact boundary value satisfy the original inequality?" — if yes, use a square bracket; if no, use a parenthesis.