Intervals on the Real Line

A closed interval $[a, b]$ contains all real numbers $x$ such that $a \leq x \leq b$. Both endpoints $a$ and $b$ are included. On a number line, closed endpoints are drawn as filled (solid) circles. Example: $[2, 5] = \{x \in \mathbb{R} : 2 \leq x \leq 5\}$. The word 'closed' refers to the fact that the interval contains its boundary points.

An open interval $(a, b)$ contains all real numbers $x$ such that $a < x < b$. Neither endpoint is included. Open endpoints are drawn as hollow (empty) circles on the number line. Example: $(2, 5) = \{x \in \mathbb{R} : 2 < x < 5\}$. Although $2$ and $5$ are not members of $(2, 5)$, every number strictly between them is.

Half-open intervals include one endpoint but not the other. There are two varieties: $[a, b)$ satisfies $a \leq x < b$ (left endpoint included, right excluded), and $(a, b]$ satisfies $a < x \leq b$ (left excluded, right included). The square bracket always faces the included endpoint; the round bracket (parenthesis) always faces the excluded endpoint. Example: $[0, 1)$ includes $0$ but not $1$, which is exactly the set used to represent one full cycle in many periodic-function contexts.

When an interval extends without bound in one or both directions, we use the symbol $\infty$ (infinity). Because infinity is not a real number, it is never included, so parentheses are always used next to $\infty$ or $-\infty$. The four unbounded interval types are: $(a, \infty)$, $[a, \infty)$, $(-\infty, b)$, and $(-\infty, b]$. The entire real line is written $(-\infty, \infty)$, which equals $\mathbb{R}$.

Every interval can be described in three equivalent ways: interval notation, set-builder notation, and an inequality. For example, $[3, 7)$ is the same as $\{x \in \mathbb{R} : 3 \leq x < 7\}$, which is the same as the compound inequality $3 \leq x < 7$. Being fluent in all three representations lets you switch between them fluidly, which is vital when solving inequalities, specifying domains, and reading mathematical proofs.

The length (or measure) of a bounded interval with endpoints $a$ and $b$ (where $a < b$) is simply $b - a$, regardless of whether the endpoints are included or not. This is because adding or removing individual points does not change the 'size' of a continuous set. So $[2, 9]$, $(2, 9)$, $[2, 9)$, and $(2, 9]$ all have length $7$.

• a — Left endpoint (dimensionless (real number)): The smaller of the two boundary values; included in the closed interval.
• b — Right endpoint (dimensionless (real number)): The larger of the two boundary values; included in the closed interval.
• x — General element (dimensionless (real number)): A variable representing any real number tested for membership in the interval.
• \mathbb{R} — Set of real numbers (none): The complete number line, containing all rational and irrational numbers.

Problem. Solve the inequality $2x - 3 < 7$ and $x + 1 \geq 0$ simultaneously. Express the solution set in interval notation and represent it on a number line, then find the length of the solution interval.

Answer. The solution set is the half-open interval from −1 to 5, including −1 but excluding 5. The length of this interval is 6.

• Set-Builder Notation — Intervals are defined using set-builder notation; understanding both allows you to move fluently between the two representations of the same set.
• Union and Intersection of Sets — Solving compound inequalities requires intersecting or unioning intervals, a direct application of set operations to interval arithmetic.
• Domain and Range of Functions — Domains and ranges are most naturally expressed as intervals, making interval notation the standard language for describing where functions are defined.
• Solving Linear Inequalities — The solution set of a linear inequality in one variable is always an interval; interval notation is the most concise way to state such solutions.