What Is Interval Notation?
Interval notation is a concise mathematical way of describing a set of real numbers between two endpoints. Instead of writing out an inequality like "all numbers x such that negative three is less than x and x is less than or equal to five," you write (-3, 5]. The parentheses and brackets indicate whether each endpoint is included in the set (bracket) or excluded (parenthesis). This notation is fundamental in algebra, calculus, and analysis.
Interval notation appears throughout mathematics: in expressing the domain and range of functions, describing solution sets of inequalities, specifying intervals of convergence for series, defining continuous regions in calculus, and communicating constraints in optimization. Mastering interval notation is essential for success in algebra and all higher-level mathematics courses.
Understanding Brackets and Parentheses
The choice of bracket or parenthesis at each endpoint conveys critical information. A square bracket [ or ] means the endpoint is included in the set (the inequality uses "less than or equal to" or "greater than or equal to"). A parenthesis ( or ) means the endpoint is excluded (the inequality uses strict "less than" or "greater than"). For example, [2, 7) includes 2 but excludes 7, meaning all x where 2 is less than or equal to x and x is less than 7.
Infinity symbols always use parentheses because infinity is not a real number and cannot be included in a set. You write (-infinity, 5] for "all x less than or equal to 5" and [3, infinity) for "all x greater than or equal to 3." The set of all real numbers is written as (-infinity, infinity), sometimes denoted by the symbol for the set of real numbers.
Types of Intervals
Open intervals use parentheses at both ends: (a, b) means a is strictly less than x and x is strictly less than b. Neither endpoint is included. Closed intervals use brackets at both ends: [a, b] means a is less than or equal to x and x is less than or equal to b. Both endpoints are included. Half-open (or half-closed) intervals use one of each: [a, b) includes a but excludes b, while (a, b] excludes a but includes b.
Bounded intervals have two finite endpoints. Unbounded intervals extend to infinity in one or both directions: (-infinity, b], [a, infinity), or (-infinity, infinity). A degenerate interval [a, a] contains exactly one point and is equivalent to the set containing just a. The empty interval occurs when the lower bound exceeds the upper bound.
Converting from Inequality Notation
To convert an inequality to interval notation, identify the lower bound, upper bound, and whether each is included. The inequality -3 is less than x and x is less than or equal to 5 becomes (-3, 5] — parenthesis at -3 because it is strict inequality, bracket at 5 because x can equal 5. For single inequalities like x is greater than or equal to 2, the interval is [2, infinity). For x is less than -1, it is (-infinity, -1).
Compound inequalities involving "or" produce unions of intervals. For example, "x is less than -2 or x is greater than 3" becomes (-infinity, -2) union (3, infinity). The union symbol represents combining two or more separate intervals. This occurs when a solution set is not contiguous, as with absolute value inequalities like |x| > 3.
Set-Builder Notation
Set-builder notation describes a set by stating the property its members must satisfy. It takes the form {x in the real numbers | condition on x}. For the interval (-3, 5], the set-builder notation is {x in the real numbers | -3 is less than x and x is less than or equal to 5}. The vertical bar is read as "such that." This notation is more flexible than interval notation because it can describe any definable set, not just intervals.
Set-builder notation is particularly useful for describing sets that are not simple intervals, such as {x in the real numbers | x is not equal to 0} or {x in the integers | x is greater than 0}. For simple intervals, however, interval notation is preferred for its brevity and clarity.
Number Line Representation
On a number line, intervals are represented by shaded regions between the endpoints. A closed (filled) circle at an endpoint means that point is included (corresponding to a bracket in interval notation). An open (hollow) circle means the point is excluded (corresponding to a parenthesis). Arrows extending to the left or right indicate unbounded intervals extending to negative or positive infinity.
The number line representation is the most intuitive way to visualize intervals. When graphing the solution to an inequality, you shade the portion of the number line satisfying the condition and use appropriate circles at the boundaries. This visual representation directly translates to interval notation: the left end determines the left symbol, the right end determines the right symbol.
Common Mistakes to Avoid
Using brackets with infinity: Always use parentheses with negative infinity and positive infinity because infinity is not a number and cannot be included in a set. Write (-infinity, 5], not [-infinity, 5]. Reversing endpoints: The smaller number always comes first. Write [-3, 5], not [5, -3]. Confusing open and closed: Remember that parentheses exclude the endpoint and brackets include it. The notation (3, 7] means 3 is NOT in the set but 7 IS.
Forgetting unions: When a solution set has gaps, you must use the union of two intervals. "x is less than 1 or x is greater than 4" is (-infinity, 1) union (4, infinity), not (1, 4) or any single interval. Single-point confusion: The set containing just the number 5 is written {5} in set notation, not [5, 5] (though they are technically equivalent). A single point is not typically expressed in interval notation.