Inequality to Interval Notation Calculator

Convert one-variable inequalities into accurate interval notation with the correct open or closed endpoints.

Convert an inequality

Enter a simple or compound inequality using x, comparison signs, and numeric boundaries.

Inequality

About inequality and interval notation

An inequality describes a range of values by comparing a variable with one or more boundaries. Interval notation expresses that same range compactly with ordered endpoints. This calculator translates inequalities involving x and numeric boundaries into interval notation. It accepts one-sided statements such as x greater than 3, reversed statements such as 3 less than x, and compound statements such as negative 2 less than x less than or equal to 4. Spaces are optional, and either typed comparison pairs or the single-character less-than-or-equal and greater-than-or-equal symbols may be used. Parentheses and square brackets communicate whether an endpoint belongs to the solution. A strict inequality excludes its boundary, so x greater than 3 becomes (3, infinity). A non-strict inequality includes its boundary, so x greater than or equal to 3 becomes [3, infinity). For a bounded interval, each side is considered separately. Negative 2 less than x less than or equal to 4 therefore uses a parenthesis at negative 2 and a bracket at 4, producing (-2, 4]. Infinity and negative infinity are directions rather than attainable numbers. They always receive parentheses, never square brackets. The finite endpoints must also appear from least to greatest in interval notation, even when the inequality is written in reverse. For example, 8 greater than x greater than or equal to 1 describes values beginning at 1 and continuing up to, but not including, 8. Its interval is [1, 8). Interval notation is useful in algebra, precalculus, calculus, statistics, and domain analysis. It makes unions and intersections easier to scan, and it displays inclusion decisions without repeatedly writing a variable. Graphically, a parenthesis corresponds to an open endpoint circle and a bracket corresponds to a filled endpoint circle. The points and shading on a number line should agree with the converted interval. The converter focuses on a single continuous interval with numeric endpoints. Expressions joined by or may produce a union of intervals and require separate conversion of each branch. Likewise, algebraic expressions should first be solved until x is isolated. If multiplying or dividing an inequality by a negative number, reverse the comparison sign before entering the final inequality. This ensures that the interval represents the actual solution set. To verify a conversion, test the boundary and one interior value. A bracket means the boundary must satisfy the original statement; a parenthesis means it must not. Then confirm that an ordinary point inside the interval satisfies every comparison. This quick check catches reversed endpoints and incorrect inclusion marks while reinforcing the connection among inequalities, number-line graphs, and interval notation.

Inequality conversion examples

Compare each inequality with its equivalent interval and endpoint rule.

Inequality	Interval	Endpoint rule
x > 5	(5, infinity)	Five is excluded by the strict sign.
x <= 7	(-infinity, 7]	Seven is included by the non-strict sign.
-2 < x <= 4	(-2, 4]	The left endpoint is open and the right endpoint is closed.
8 > x >= 1	[1, 8)	Endpoints are reordered from least to greatest.

How to convert an inequality to interval notation

Isolate x and enter the completed one-variable inequality.
Choose Convert to Interval to analyze both comparison signs.
Read the endpoints from least to greatest in the result.
Check that strict signs use parentheses and inclusive signs use brackets.

Inequality to interval notation FAQ

When should I use a bracket?

Use a square bracket when the finite endpoint is included by a less-than-or-equal or greater-than-or-equal sign. The bracket represents a filled point on a number line.

When should I use a parenthesis?

Use a parenthesis when a strict comparison excludes the finite endpoint. Infinity also always uses a parenthesis because it is not an attainable endpoint.

Can the inequality be written backward?

Yes, statements such as 8 greater than x are supported. The calculator places interval endpoints in increasing order regardless of the original direction.

What does a compound inequality mean?

A compound inequality places x between two boundaries and applies one comparison at each end. Both comparisons must be true for a value to belong to the interval.

Does the calculator solve algebraic inequalities?

It converts an inequality after x has been isolated; it does not simplify arbitrary algebraic expressions. Solve the expression first and remember to reverse the sign after multiplying or dividing by a negative number.