Interval Notation Calculator

Interval notation describes a contiguous set of real numbers lying between two endpoints. Instead of writing algebraic inequalities like a < x < b, interval notation uses parentheses and brackets. Parentheses, such as (a, b), indicate that the endpoints are excluded from the set, representing strict inequalities like less than or greater than. Brackets, such as [a, b], denote that the endpoints are included, corresponding to inequalities like less than or equal to. When dealing with unbounded ranges, the infinity symbol (infinity or -infinity) is paired with a parenthesis since infinity is a concept rather than a finite number. The calculator evaluates minimum and maximum boundaries using straightforward mathematical logic, determining whether the lower and upper bounds require inclusive or exclusive markers based on the input operators.

Let us walk through converting the compound inequality 1 less than or equal to x less than 5 into interval notation. First, identify the lower bound, which is 1, and the upper bound, which is 5. Next, examine the inequality operators attached to each value. The symbol less than or equal to associated with 1 indicates an inclusive boundary, meaning we use a square bracket [ on the left side. Conversely, the strict less than symbol associated with 5 indicates an exclusive boundary, requiring a round parenthesis ) on the right side. Combining these elements yields the final interval notation expression: [1, 5). This notation clearly communicates that 1 is part of the set, 5 is excluded, and all real numbers in between are included.

Always list your interval bounds from smallest to largest, reading naturally from left to right on the number line. Never use a square bracket alongside infinity or negative infinity, as these directions represent unbounded limits rather than concrete numbers. When combining disjoint sets of numbers, remember to use the union symbol (U) to bridge separate intervals accurately.