What is Interval Notation?
Interval notation is a universally recognized mathematical convention used to describe continuous subsets of real numbers along a 1D continuum. Rather than writing out descriptive algebraic inequalities like \(a \le x < b\), interval notation concisely represents the domain as an ordered pair:
The smaller number is always placed on the left, and the larger number is always placed on the right. Whether an endpoint receives a rounded parenthesis or a square bracket depends strictly on whether that boundary number is included or excluded from the solution set.
How to Use the Inequality to Interval Notation Calculator
Convert any algebraic inequality into standard interval and set-builder notation in three simple steps:
Select Conversion Direction
Choose between Inequality to Interval (e.g. \(-3 \le x < 7\)) or Interval to Inequality (e.g. \([-3, 7)\)).
Enter Expression or Preset
Type single rays (\(x > 4\)), bounded intervals (\(-2 \le x \le 5\)), or union disjunctions (\(x < -1 \text{ or } x \ge 3\)), or select from quick example presets.
Review Interval & Number Line
Immediately view verified bracket vs parenthesis notation, formal set-builder notation, and the accompanying interactive vector SVG number line.
Problems This Interval Converter Solves
Parenthesis vs. Bracket Confusion
Students frequently write \([-\infty, 5]\) instead of \((-\infty, 5]\). Infinity is an unbounded concept, never a reachable number, so it must always take a rounded parenthesis. Our solver enforces correct bracket rules.
Handling Disjoint Unions (\(\cup\)) Accurately
Combining outward-pointing inequalities like \(x \le -2 \text{ or } x > 4\) requires the union symbol \(\cup\). The calculator formats these disjoint intervals into exact textbook mathematical syntax.
Ordering Boundaries Lower to Upper
A common mistake is writing \((5, -2)\) instead of \((-2, 5)\). Interval notation strictly demands that the smaller number appears first. Our validator orders boundary numbers properly.
Domain & Range in Calculus
Essential for defining function domains, intervals of increase/decrease, concavity, and convergence intervals for power series in AP Calculus and university mathematics.
Key Features & Capabilities
Converts seamlessly from Inequality → Interval and from Interval → Inequality.
Generates formal mathematical set-builder notation \(\{x \in \mathbb{R} \mid \dots\}\).
Visualizes boundaries with dynamically rendered open and solid endpoint circles.
Converts on keystroke with zero latency and zero calculate button delay.
Parentheses vs. Brackets: The Golden Rules
Square Brackets: [ and ] (Inclusive)
Used when the endpoint is included in the interval. Corresponds to non-strict inequality relations: \(\le\) (less than or equal to) and \(\ge\) (greater than or equal to).
Example: \(x \ge 3 \implies [3, \infty)\)
Round Parentheses: ( and ) (Exclusive)
Used when the endpoint is excluded. Corresponds to strict relations: \(<\) and \(>\). In addition, positive infinity (\(+\infty\)) and negative infinity (\(-\infty\)) always take parentheses.
Example: \(x < 5 \implies (-\infty, 5)\)
Master Inequality to Interval Conversion Reference
| Inequality Statement | Interval Notation | Set-Builder Notation | Type |
|---|---|---|---|
| \(a < x < b\) | \((a, b)\) | {\(x \mid a < x < b\)} | Open bounded interval |
| \(a \le x \le b\) | \([a, b]\) | {\(x \mid a \le x \le b\)} | Closed bounded interval |
| \(a \le x < b\) | \([a, b)\) | {\(x \mid a \le x < b\)} | Half-open interval |
| \(x > a\) | \((a, \infty)\) | {\(x \mid x > a\)} | Open infinite ray |
| \(x \le b\) | \((-\infty, b]\) | {\(x \mid x \le b\)} | Closed infinite ray |
| \(x < a \text{ or } x \ge b\) | \((-\infty, a) \cup [b, \infty)\) | {\(x \mid x < a \lor x \ge b\)} | Disjoint Union |
| All Real Numbers | \((-\infty, \infty)\) | {\(x \mid x \in \mathbb{R}\)} | Entire real line |