What Is Interval Notation?

You solve 2x + 3 < 7, get x < 2, and the textbook expects you to write (−∞, 2). That is interval notation: a compact way to describe a continuous set of real numbers using parentheses and brackets instead of inequality statements. Interval notation is a shorthand for writing subsets of real numbers, and it uses a definition, symbols, formal rules, and special cases that trip up students from Algebra 1 through early calculus.

Interval notation is a shorthand for writing subsets of real numbers. Instead of “all real numbers greater than 2 and less than or equal to 5”, you write (2, 5]. The left parenthesis means 2 is not included; the right bracket means 5 is included. The comma separates the two endpoints. That is the core idea. Everything else, infinity, unions, the empty set, extends from that one pattern.

Interval notation uses two kinds of enclosure: parentheses ( ) and brackets [ ]. The meaning is absolute: a parenthesis excludes the endpoint (strict inequality < or >), and a bracket includes the endpoint (non-strict inequality ≤ or ≥). OpenStax College Algebra 2e, section 1.1, defines open endpoints with parentheses and closed endpoints with brackets.

The four basic interval types are:

• Open interval (a, b): both endpoints excluded. Represents a < x < b.
• Closed interval [a, b]: both endpoints included. Represents a ≤ x ≤ b.
• Half-open intervals [a, b) and (a, b]: one endpoint included, the other excluded. Represents a ≤ x < b or a < x ≤ b, respectively.
• Unbounded intervals: use the infinity symbol ∞ or -∞, always with a parenthesis. For example, (−∞, 5] means all real numbers less than or equal to 5.

The infinity symbol is never paired with a bracket because infinity is not a number that can be reached. This is a formal rule in interval notation, not a stylistic choice. OpenStax College Algebra 2e, section 1.1, states this explicitly: unbounded intervals use ∞ or -∞ with a parenthesis.

Interval notation follows three rigid rules. Break any of them and the expression is invalid.

Left Endpoint Must Be Less Than Right Endpoint

The smaller number always comes first. You write (−3, 5), never (5, −3). A reversed interval like (5, 2) is meaningless because no real numbers satisfy 5 < x < 2. The left value names the lower bound; the right names the upper bound. This rule holds for infinite endpoints too: write (−∞, 5), not (5, −∞).

Infinity Gets a Parenthesis Always

The symbols ∞ and -∞ are always paired with a parenthesis. Writing [5, ∞] or [−∞, 3] is an error. The reason is mathematical: infinity is a concept of unboundedness, not a number that can be included in a set. A bracket implies the endpoint is a real number that belongs to the set, which infinity is not. This single mistake accounts for the majority of interval notation errors on homework and exams.

Union for Disjoint Sets

When the solution set includes numbers from two or more separate intervals that do not overlap, you combine them with the union symbol ∪. For example, all real numbers except 2 is written as (−∞, 2) ∪ (2, ∞). The union symbol means “or”: a number belongs to the first interval or the second. Do not write separate intervals without a connector; that is grammatically invalid in interval notation.

These three rules come directly from OpenStax College Algebra 2e, section 2.7, which covers solving linear inequalities and expressing solution sets in interval notation.

Every interval notation expression carries exact meaning. You read it left to right, matching each symbol to its inequality counterpart.

Mapping Symbols to Inequalities

A bracket [ or ] translates to ≤ or ≥. A parenthesis ( or ) translates to < or >. The infinity symbol ∞ means no upper bound; −∞ means no lower bound.The interval (3, ∞) means x > 3.

What the Union Symbol Communicates

The union symbol ∪ tells you the set includes numbers from either interval. For the domain of f(x) = 1/(x² − 4), where the denominator is zero at x = −2 and x = 2, the domain is (−∞, −2) ∪ (−2, 2) ∪ (2, ∞). Each union separates a region where the function is defined. Omitting the union and writing (−∞, −2), (−2, 2), (2, ∞) is incorrect because interval notation requires a connector between separate intervals.

Visualising on the Real Number Line

The real number line is the standard aid. Draw a line, mark the endpoints, use an open circle for an excluded endpoint and a closed circle for an included one, then shade the region between them. For unbounded intervals, extend the shading with an arrow. This visual check prevents the most common errors: wrong order, wrong inclusion, missing union.

Interval notation uses exactly five symbols. Each has one job.

Symbol	Name	Meaning	Example
(	Left parenthesis	Excludes the left endpoint	(2, 5) means x > 2
)	Right parenthesis	Excludes the right endpoint	(2, 5) means x < 5
[	Left bracket	Includes the left endpoint	[2, 5] means x ≥ 2
]	Right bracket	Includes the right endpoint	[2, 5] means x ≤ 5
∞	Infinity	Unboundedness; always with parenthesis	(3, ∞) means all numbers greater than 3
∪	Union	Combines two intervals; means “or”	(−∞, 1) ∪ (4, ∞)

The empty set is represented by ∅ or { }, not by [ ] or ( ). This is a standard convention in mathematics. OpenStax College Algebra 2e uses ∅ in its solution sets when an inequality has no real-number solution.

Three edge cases appear regularly in homework and exams. Each has a specific notation.

Empty Set

When an inequality has no real-number solution, the answer is the empty set. Write it as ∅ or { }. For example, the inequality x < 3 and x > 5 has no overlap, so the solution set is ∅. Beginners sometimes write [ ] or ( ), but those are not standard notations in any mathematics textbook. The empty set is not zero and not a point; it is the set with nothing in it.

Single Point

A single number, such as x = 4, can be written in interval notation as [4, 4]. Both endpoints are the same number, and both are included because the condition is x = 4, not a range. You cannot write (4, 4) because that would mean 4 < x < 4, which is empty. If you need to include a single point as part of a larger set, you use union: (−∞, 2] ∪ {3} ∪ [5, ∞). The set-builder notation {x | x = 4} is an alternative, but interval notation is preferred for continuous sets.

All Real Numbers

The set of all real numbers is written as (−∞, ∞). This is identical to ℝ. It means there is no upper or lower bound. Every polynomial function has domain (−∞, ∞). Every square root function with a radicand that is always non-negative also has domain (−∞, ∞). The parentheses at both ends are mandatory because infinity is not a number.

Interval notation, inequality notation, and set-builder notation are three ways to say the same thing. The choice depends on context and audience.

Inequality Notation

Inequality notation uses symbols like <, ≤, >, ≥. For example, −3 ≤ x < 5. It is clearer for a single inequality but becomes unwieldy for compound conditions. Interval notation compresses the same idea into [−3, 5). Textbooks and research papers favor interval notation because it is more compact and easier to scan, especially when describing domains and ranges.

Set-Builder Notation

Set-builder notation uses a variable, a vertical bar (meaning “such that”), and a condition: { x | −3 ≤ x < 5 }. It is more explicit than interval notation and can describe discrete sets, such as {1, 2, 3}, which interval notation cannot. However, it is verbose. Interval notation is the standard for continuous sets because it is faster to write and read.

When to Use Each

Use interval notation when writing the solution to an inequality, the domain of a function, or the range of a function. Use inequality notation when teaching the meaning of the symbols. Use set-builder notation when the set is not a single continuous interval, such as { x | x ≠ 2 }, which interval notation writes as (−∞, 2) ∪ (2, ∞). For discrete sets like {−1, 0, 1}, only set-builder or roster notation works.

The most frequent error in interval notation is using a bracket with infinity. It appears in nearly every first attempt. The fix is simple: check every infinity symbol in your answer and confirm it has a parenthesis. The second most common error is writing intervals in the wrong order, such as (5, 2). The fix is to always place the smaller number on the left.

The third error is omitting the union symbol between disjoint intervals. If your solution is all numbers less than 1 or greater than 4, you must write (−∞, 1) ∪ (4, ∞). The blank space between them is not a valid connector. The final error is confusing the empty set with a zero or a point. The empty set means no real numbers satisfy the condition, not that the answer is 0 or that the answer is a single number.

OpenStax College Algebra 2e, section 2.7, demonstrates these errors in the context of solving absolute value inequalities, showing students how to check their work by testing a number from each interval against the original inequality. That check, picking a number and verifying it makes the inequality true, is the single most reliable way to catch interval notation mistakes.