Interval Notation Calculator

This interval notation calculator is built for the exact conversion tasks that cause the most confusion in algebra, precalculus, and calculus: converting a solved inequality into interval notation, translating interval notation back into inequality form, deciding whether an endpoint should be open or closed, handling infinity correctly, and combining intervals with union or intersection. Instead of treating interval notation as a small add-on under another solver, this page makes the notation itself the main lesson.

The hero workspace separates three jobs that students often mix together. In inequality-to-interval mode, the page reads a solved inequality, a compound inequality, or a simple not-equal statement and turns it into interval notation, set-builder notation, and a number line. In interval-to-inequality mode, the page reverses that process and shows how each bracket or parenthesis maps back to a strict or inclusive inequality sign. In union and intersection mode, the page treats intervals as sets and computes the combined result directly.

The result area is designed to make notation choices visible. The Result tab summarizes the conversion and the endpoint rules, the Number Line tab makes the set visual, the Steps tab explains the reasoning in plain English, and the All Notations tab lines up inequality notation, interval notation, set-builder notation, and the graph at once. That side-by-side structure is the fastest way to catch bracket mistakes before they become homework mistakes.

Interval notation is a compact way to describe sets of real numbers. Instead of writing a sentence such as all x greater than 3, you can write (3, +∞). Instead of writing -1 < x ≤ 5, you can write (-1, 5]. The notation is short, but it still contains all the information about whether endpoints are included, excluded, bounded, or unbounded.

The notation belongs to the real number line. Each interval represents a continuous stretch of values, a union of continuous stretches, all real numbers, or the empty set. That is why interval notation appears so often in inequality solving, domain and range work, calculus, and set notation. It is not a different answer from inequality notation. It is the same answer written more efficiently.

Students usually struggle with interval notation because the symbols carry meaning in both directions. Parentheses and brackets tell you whether an endpoint is open or closed, while infinity tells you the set never stops in that direction. Once those three ideas are clear, interval notation becomes one of the easiest ways to read a solution set quickly.

An open interval uses parentheses and excludes the endpoint. For example, (a, b) means a < x < b, so neither a nor b is part of the set. A closed interval uses brackets and includes the endpoint. For example, [a, b] means a ≤ x ≤ b, so both endpoints stay in the solution.

Mixed intervals combine those rules. The interval (a, b] means the left endpoint is open and the right endpoint is closed, while [a, b) means the left endpoint is closed and the right endpoint is open. That one-symbol difference is what separates x < 5 from x ≤ 5, so a bracket mistake changes the actual solution set.

A reliable memory rule is this: parentheses mean the point is not included, brackets mean the point is included. On the number line that becomes an open circle for parentheses and a closed circle for brackets. The symbol on the page and the marker on the graph should always match.

To convert an inequality into interval notation, first identify the boundary values. For x > 3, the boundary is 3. For -1 < x ≤ 5, the boundaries are -1 and 5. Those values become the endpoints of the interval.

Next decide whether each endpoint is open or closed. Strict inequalities, < and >, use parentheses because the endpoint is excluded. Inclusive inequalities, ≤ and ≥, use brackets because the endpoint is included. If the solution extends forever to the left or right, use -∞ or +∞ with parentheses.

Finally write the interval from left to right on the number line. A single bounded solution such as -1 < x ≤ 5 becomes (-1, 5]. A right ray such as x ≥ 3 becomes [3, +∞). A split answer such as x < -3 or x > 7 becomes (-∞, -3) ∪ (7, +∞).

To convert interval notation back into an inequality, read each endpoint symbol first. A parenthesis means the endpoint is not included, so it becomes a strict sign. A bracket means the endpoint is included, so it becomes an inclusive sign.

Then read the direction of the interval. A bounded interval such as [-1, 5) becomes -1 ≤ x < 5 because x stays between the two endpoints. An interval with infinity has only one finite boundary, so (3, +∞) becomes x > 3 and (-∞, -2] becomes x ≤ -2.

If the interval uses a union symbol, convert each piece separately and combine them with OR. For example, (-∞, -3) ∪ (7, +∞) becomes x < -3 or x > 7. The union symbol means the solution can live in either interval.

Infinity is not a real number, so interval notation never treats it like a reachable endpoint. That is why both +∞ and -∞ always use parentheses. Even if an inequality is x ≥ 3, the interval [3, +∞) still ends with a parenthesis on infinity.

The same rule works in the other direction. If you see (-∞, 2], the finite endpoint 2 is closed because x can actually equal 2, but the negative infinity side must stay open because the interval continues forever and never reaches a smallest real number.

All real numbers are written as (-∞, +∞). That notation does not mean the set includes infinity itself. It means the set extends without bound in both directions. Students often remember the bracket rule but forget the infinity exception, so it is worth treating as a separate rule every time.

Union uses the symbol ∪ and means OR. If a value belongs to interval A or interval B or both, it belongs to A ∪ B. On the number line, union makes the highlighted region larger because it keeps every point that appears in at least one interval.

Intersection uses the symbol ∩ and means AND. A value belongs to A ∩ B only if it lies in both intervals at the same time. On the number line, that usually makes the highlighted region smaller because only the overlap survives.

When intervals overlap, union can merge them into one larger interval while intersection can isolate the shared middle. When intervals do not overlap at all, union keeps both pieces and intersection becomes the empty set. Those are the same OR and AND ideas students already know from compound inequalities.

Compound inequalities usually translate into either a single bounded interval or a union of two outer rays. An AND inequality such as -1 < x ≤ 5 keeps the overlap of two conditions, so its interval notation is a single bounded interval, (-1, 5].

An OR inequality behaves differently. The statement x < -3 or x > 7 keeps values on either outer side, so the answer is a union: (-∞, -3) ∪ (7, +∞). The union symbol is the interval-notation version of the word OR.

Not-equal statements are another special compound pattern. The inequality x ≠ 2 means every real number except 2, so it becomes (-∞, 2) ∪ (2, +∞). The number line looks continuous except for one open hole at the excluded point.

Set-builder notation describes the same solution set in sentence form. The expression { x | -1 < x ≤ 5 } reads as the set of all x such that -1 < x ≤ 5. Interval notation compresses that same information into symbols by writing (-1, 5].

The advantage of set-builder notation is that the condition stays explicit. The advantage of interval notation is that the number-line structure becomes obvious at a glance. Strong students learn to move between both forms because textbooks, exams, and calculus problems use them interchangeably.

A useful checking strategy is to compare three forms at once: inequality notation, interval notation, and set-builder notation. If one form says the endpoint is included and another form says it is not, the mismatch becomes easy to spot before you move on.