Convert any inequality to interval notation instantly — see the bracket and parenthesis rules, endpoint logic, and number line visualization. Free, no sign-up required.
Why this page is built for inequality-to-interval conversion
Interval Notation Converter
Enter any inequality — solved or unsolved — and get the interval notation with bracket rules, endpoint logic, and a number line.
Supported Input Styles
Quick rules
Use parentheses because the endpoint is excluded.
Use brackets because the endpoint is included.
Infinity always uses parentheses because it is never included.
Math Keyboard
Tap brackets, infinity, unions, and inequality signs for fast interval-notation input.
Result
The inequality converts to interval notation by reading the included and excluded endpoints from the final solution set.
Interval
(3, +∞)
Endpoints
Open at 3 · No upper bound
Number Line
Number Line
Drag to pan. Use the controls or mouse wheel to zoom the view.
Interval Notation
Left endpoint
Left endpoint: 3
Symbol was: > (strict)
Bracket type: ( parenthesis — endpoint excluded
Right endpoint
Right endpoint: +∞
Infinity always uses: ) parenthesis — never included
> or <
parenthesis ( )
endpoint excluded
≥ or ≤
bracket [ ]
endpoint included
∞
always parenthesis
never included
Conversion Rules
Use this quick reference when you want to confirm the bracket pattern before you convert a new problem. Each row shows the inequality form, the interval notation, the bracket type on each side, and the meaning.
| Inequality | Interval Notation | Left bracket | Right bracket | Meaning | Try it |
|---|---|---|---|---|---|
| x > a | (a, +∞) | ( excluded | ) infinity | All values greater than a | |
| x ≥ a | [a, +∞) | [ included | ) infinity | All values greater than or equal to a | |
| x < a | (-∞, a) | ( infinity | ) excluded | All values less than a | |
| x ≤ a | (-∞, a] | ( infinity | ] included | All values less than or equal to a | |
| a < x < b | (a, b) | ( excluded | ) excluded | Values strictly between a and b | |
| a ≤ x ≤ b | [a, b] | [ included | ] included | Values between a and b, endpoints included | |
| a < x ≤ b | (a, b] | ( excluded | ] included | Left excluded, right included | |
| a ≤ x < b | [a, b) | [ included | ) excluded | Left included, right excluded | |
| x < a or x > b | (-∞, a) ∪ (b, +∞) | — | — | Two separate rays, both endpoints excluded | |
| x ≤ a or x ≥ b | (-∞, a] ∪ [b, +∞) | — | — | Two separate rays, both endpoints included | |
| x ≠ a | (-∞, a) ∪ (a, +∞) | — | — | All reals except one point | |
| All reals | (-∞, +∞) | ( infinity | ) infinity | No restriction |
Zone 4
The fastest way to use this calculator is to type the inequality exactly as it appears — x > 3, x ≤ -2, -1 < x ≤ 5, or x < -3 or x ≥ 4. If the inequality is already solved, the page converts it to interval notation immediately. If the inequality still contains an unsolved expression such as 2x + 1 > 7, the page solves it first and then converts the result.
After you enter a problem, the Interval Notation tab shows the result with the bracket and parenthesis choice explained for each endpoint. The rule is consistent: strict symbols (> and <) always produce a parenthesis because the endpoint is excluded, inclusive symbols (≥ and ≤) always produce a bracket because the endpoint is included, and infinity always uses a parenthesis because it is never a reachable value. The Steps tab walks through those decisions one at a time.
The Conversion Rules table below the calculator is useful when you want to check the pattern for a specific inequality type without entering a new problem. Each row shows the inequality form, the interval notation, and which bracket goes on each side. Click any Try it button to load that example directly into the calculator.
If you also need reverse conversion or set operations, move up to the interval notation calculator. If you want to read the same answer as a graph first, open the number line inequality calculator.
Enter an inequality such as x > 3, -1 < x ≤ 5, or 2x + 1 > 7.
Read the Interval Notation tab first — it shows the result and explains the bracket choice for each endpoint.
Use the Steps tab to see the conversion logic: boundary identification, bracket type selection, and final notation.
Check the Number Line and All Notations tabs to confirm the same answer in visual and symbolic form.
Interval notation is a compact way to write the solution set of an inequality. Instead of writing x > 3, you write (3, +∞). Instead of writing -1 < x ≤ 5, you write (-1, 5]. The notation carries the same information as the inequality — the boundary values and whether they are included — but in a shorter, more standardized form.
The conversion from inequality to interval notation follows three consistent rules. First, identify the boundary value or values. Second, choose the bracket type: a parenthesis for a strict symbol (> or <) because the boundary is excluded, and a bracket for an inclusive symbol (≥ or ≤) because the boundary is included. Third, use ±∞ for any unbounded direction, always with a parenthesis because infinity is never a reachable value.
Compound inequalities follow the same rules applied to each part. An AND inequality such as -1 < x ≤ 5 produces a single bounded interval (-1, 5] with one bracket decision on each side. An OR inequality such as x < -3 or x ≥ 4 produces a union of two intervals (-∞, -3) ∪ [4, +∞) with each part converted independently and joined with the union symbol ∪.
For a longer concept guide, see interval notation explained. For multi-boundary practice after this page, the compound inequality calculator is the natural next step.
> or < means the boundary is not included. Use ( or ) on that side.
x > 3 → (3, ...
≥ or ≤ means the boundary is included. Use [ or ] on that side.
x ≥ 3 → [3, ...
+∞ and -∞ are never reachable values. Always use ) or (.
x > 3 → (3, +∞)
Converting an inequality to interval notation is a three-step process: find the boundary, choose the bracket, write the interval. The only decision that requires attention is the bracket type — and that decision depends entirely on the inequality symbol.
For x > a: write (a, +∞) — parenthesis on the left because > excludes a, parenthesis on the right because +∞ is never included.
For x ≥ a: write [a, +∞) — bracket on the left because ≥ includes a.
For x < a: write (-∞, a) — parenthesis on both sides.
For x ≤ a: write (-∞, a] — bracket on the right because ≤ includes a.
For a < x < b: write (a, b) — both boundaries excluded.
For a ≤ x ≤ b: write [a, b] — both boundaries included.
For a < x ≤ b: write (a, b] — left excluded, right included.
For a ≤ x < b: write [a, b) — left included, right excluded.
Apply the bracket rule to each side independently.
Convert each part to interval notation separately, then join them with ∪.
For x < a or x > b: (-∞, a) ∪ (b, +∞)
For x ≤ a or x ≥ b: (-∞, a] ∪ [b, +∞)
If the inequality is not yet solved — for example 2x + 1 > 7 — solve for x first, then apply the conversion rules to the solved form. The conversion rules apply only to solved inequalities where x stands alone.
A parenthesis means the endpoint is excluded from the solution. A bracket means the endpoint is included. Parentheses correspond to strict symbols (> and <). Brackets correspond to inclusive symbols (≥ and ≤).
Infinity is not a real number — it is a direction, not a reachable value. Since you can never actually reach infinity, it is always excluded, and a parenthesis is always used.
Apply the bracket rule to each side independently. The left side uses < (strict), so write a parenthesis: (-1. The right side uses ≤ (inclusive), so write a bracket: 5]. Result: (-1, 5].
Convert each part separately and join with ∪. x < -3 becomes (-∞, -3). x ≥ 4 becomes [4, +∞). Result: (-∞, -3) ∪ [4, +∞).
Yes. The calculator solves the inequality first and then converts it to interval notation. You will see both the solving steps and the conversion steps.
This page converts inequalities to interval notation only — the single direction most students need. The interval notation calculator also converts interval notation back to inequalities and handles union and intersection operations.
Calculator Types
Convert any inequality to interval notation with bracket rules, endpoint logic, and number line output.
Convert in both directions and handle union or intersection operations in one workspace.
Graph any inequality on a number line with open and closed endpoint detection, shading direction, and interval notation output.
Solve compound inequalities with interval intersection, union logic, and graph output.
Solve one-variable linear inequalities with steps, interval notation, and a clean number-line graph.
Keep Exploring
Interval Notation Calculator
Convert inequalities to interval notation and back with steps, number-line visuals, bracket rules, and interval set operations.
Compound Inequality Calculator
Solve compound inequalities with interval intersection, union logic, and graph output.
Number Line Inequality Calculator
Graph any inequality on a number line with open and closed endpoint detection, shading direction, and interval notation output.