Inequality to Interval Notation Calculator
Enter your inequality below and get the interval notation, set-builder notation, and a plain-English description of the solution set. Choose one-sided (x > a), two-sided (a < x < b), or compound OR (x < a or x > b) form. Results update as you type.
What is interval notation?
Interval notation is a compact way to describe a set of real numbers on the number line using brackets and parentheses. A square bracket [ or ] means the endpoint is included in the set (a "closed" boundary), while a parenthesis ( or ) means the endpoint is excluded (an "open" boundary). Infinity is always written with a parenthesis because it is not a reachable number. For example, the set of all numbers greater than 3 is written (3, +inf), and the set of numbers between -1 and 5 including both endpoints is [-1, 5].
How to convert an inequality to interval notation
The conversion follows four simple rules. First, identify whether each endpoint is included or excluded: a strict inequality (< or >) means excluded, a non-strict inequality (<= or >=) means included. Second, use a parenthesis for excluded endpoints and a square bracket for included endpoints. Third, always use parentheses at positive and negative infinity. Fourth, for compound OR inequalities (x < a or x > b), convert each piece separately and join them with the union symbol U. For example, x >= -2 and x < 5 written as a two-sided inequality -2 <= x < 5 becomes [-2, 5) in interval notation.
Open, closed and half-open intervals
A bounded interval between two real numbers a and b can take four forms. The open interval (a, b) excludes both a and b. The closed interval [a, b] includes both. The half-open intervals [a, b) and (a, b] include exactly one endpoint. Unbounded intervals have only one finite endpoint and extend to infinity in one direction: (a, +inf) for x > a, [a, +inf) for x >= a, (-inf, a) for x < a, and (-inf, a] for x <= a. The entirely unbounded interval of all real numbers is written (-inf, +inf).
Set-builder notation compared to interval notation
Set-builder notation describes the same solution sets as interval notation but uses a rule instead of endpoint shorthand. The general form is { x | condition }, read as "the set of all x such that the condition is true". For example, { x | 2 <= x < 8 } is the set of all x at least 2 and less than 8, which is the same set as the interval [2, 8). Both notations are standard in pre-calculus and calculus; interval notation is more common in American textbooks for describing domains, ranges and solution sets, while set-builder notation is more common in discrete mathematics and formal logic.
Compound OR inequalities and unions
A compound OR inequality such as x < -3 or x > 5 describes a set made of two separate pieces. Each piece is converted to its own interval and the two are joined with the union symbol U. The example above becomes (-inf, -3) U (5, +inf). Notice that the real numbers from -3 to 5 are NOT in the set - there is a gap. This contrasts with a compound AND inequality such as x > -3 and x < 5, which becomes the single connected interval (-3, 5). The AND case is equivalent to the two-sided form -3 < x < 5.
Standard interval notation reference
| Inequality | Interval notation | Type | Bounded? |
|---|---|---|---|
| x < a | (-inf, a) | Open | No |
| x <= a | (-inf, a] | Half-open | No |
| x > a | (a, +inf) | Open | No |
| x >= a | [a, +inf) | Half-open | No |
| a < x < b | (a, b) | Open | Yes |
| a <= x <= b | [a, b] | Closed | Yes |
| a < x <= b | (a, b] | Half-open | Yes |
| a <= x < b | [a, b) | Half-open | Yes |
| x < a or x > b | (-inf, a) U (b, +inf) | Union | No |
| x <= a or x > b | (-inf, a] U (b, +inf) | Union | No |
| x < a or x >= b | (-inf, a) U [b, +inf) | Union | No |
| x <= a or x >= b | (-inf, a] U [b, +inf) | Union | No |
Every standard form mapping inequality notation to interval notation. "-inf" means negative infinity, "+inf" means positive infinity.
Frequently asked questions
What is the difference between a parenthesis and a bracket in interval notation?
A parenthesis - either ( or ) - means the endpoint at that end is NOT included in the interval. This corresponds to a strict inequality (< or >). A square bracket - either [ or ] - means the endpoint IS included. This corresponds to a non-strict inequality (<= or >=). For example, [3, 7) includes 3 but excludes 7, so it matches 3 <= x < 7.
Why is infinity always written with a parenthesis?
Infinity is not a real number - it is a concept representing an unbounded direction on the number line. Because no real number can equal infinity, it can never be "included" in a set of real numbers. So the endpoint at infinity is always open, always written with a parenthesis. For example, x > 5 is (5, +inf) - never (5, +inf].
How do I convert a compound AND inequality to interval notation?
A compound AND inequality like x > -1 and x <= 4 means both conditions must be true simultaneously. Rewrite it as a two-sided inequality: -1 < x <= 4. Then apply the rules: the left endpoint -1 is excluded (strict inequality), so use a parenthesis; the right endpoint 4 is included (non-strict inequality), so use a bracket. Result: (-1, 4].
What does the union symbol U mean in interval notation?
The union symbol U means "or" - the solution set includes all numbers that belong to EITHER interval. It appears when a compound OR inequality produces two separate pieces. For example, x < -2 or x >= 6 converts to (-inf, -2) U [6, +inf), which includes all numbers to the left of -2 or to the right of and including 6.
Is interval notation the same as set-builder notation?
They describe the same sets but use different formats. Interval notation uses brackets and parentheses around endpoint values, such as [2, 9). Set-builder notation spells out the rule explicitly: { x | 2 <= x < 9 }. Both are correct and interchangeable; interval notation is more compact and is widely used in calculus for domains, ranges and solution sets.
How do I write "all real numbers" in interval notation?
All real numbers corresponds to the inequality -inf < x < +inf. In interval notation this is written (-inf, +inf). Both endpoints use parentheses because infinity is not a real number and cannot be included.
What is a half-open interval?
A half-open interval (also called half-closed) has one endpoint included and one excluded. For example, [3, 8) includes 3 but not 8, corresponding to 3 <= x < 8. The notation shows a bracket on the closed side and a parenthesis on the open side. Half-open intervals arise naturally in many contexts, such as defining a domain that includes a starting value but not an ending value.