Types of Intervals: Open, Closed, Half-Open

Interval notation like (2,5) or [-1,∞) is not calculator shorthand. Each notation precisely describes which real numbers are included. The distinction between open and closed intervals is the difference between a correct answer and a point off. OpenStax College Algebra 2e, sections 1.1 and 2.7, defines the rules: a parenthesis means the endpoint is not included; a bracket means the endpoint is included. Infinity always takes a parenthesis because it is not a number that can be reached.

Write an open interval as (a, b). It includes every real number between a and b, but excludes a and b themselves. The inequality form is a < x < b. The set-builder equivalent is { x | a < x < b }. On a number line, draw open circles at a and b and shade the segment between them. Use this interval when you solve a strict inequality such as x + 1 > 0 and x - 3 < 0. OpenStax College Algebra 2e, section 2.7, uses open intervals to represent the solution sets of strict linear inequalities. To write a domain that excludes a boundary value exactly, an open interval is the correct form.

Writing And Graphing Closed Intervals

Write a closed interval as [a, b]. It includes a, b, and every number between them. The inequality form is a ≤ x ≤ b. The set-builder notation is { x | a ≤ x ≤ b }. On the number line, closed circles mark both endpoints and the segment is shaded solid. This is the default interval when you solve a non-strict inequality like x ≥ 2 and x ≤ 7. In OpenStax College Algebra 2e, section 2.7, closed intervals appear when absolute value inequalities use the ≤ symbol.

Length And Midpoint

A bounded closed interval always has a finite length, b - a, and a midpoint at (a + b) / 2. Both endpoints are included, so the midpoint is guaranteed to be inside the set.

A half-open interval includes one endpoint but not the other. There are two forms. [a, b) includes a, excludes b. The inequality is a ≤ x < b. (a, b] includes b, excludes a. The inequality is a < x ≤ b. Each form has exactly one bracket and one parenthesis. Use these when solving compound inequalities where one is strict and the other is non-strict. For example, the domain of f(x) = √(x) / (x - 5) requires x ≥ 0 from the square root and x ≠ 5 from the denominator. Write that domain as [0, 5) ∪ (5, ∞). The half-open interval [0, 5) includes 0 but not 5. OpenStax College Algebra 2e, section 3.2, uses half-open intervals for domain and range of radical and rational functions.

Forms And Rules

An unbounded interval extends to infinity in at least one direction. The forms are (a, ∞), [a, ∞), (-∞, b), and (-∞, b]. Never pair the infinity symbol with a bracket. Write (-∞, 3] not (-∞, 3]. The interval (-∞, ∞) represents all real numbers; it is the only interval unbounded in both directions. Unbounded intervals always have infinite length; you cannot compute b - a and get a finite number. However, you can still compute the midpoint of a bounded sub-interval if one endpoint is finite.

Bounded Versus Unbounded

The key distinction is bounded versus unbounded: a bounded interval has two finite endpoints; an unbounded interval has at least one infinite endpoint. The domain of any polynomial function is the unbounded interval (-∞, ∞). OpenStax College Algebra 2e, section 1.1, defines this classification.

For any bounded interval, whether open, closed, or half-open, the length is the same: the difference between the endpoints. The formula is length = |b - a|. For (2, 7), the length is 5. For [2, 7], the length is also 5. Inclusion of endpoints does not change the distance. The midpoint is (a + b) / 2. For (2, 7), the midpoint is 4.5. That midpoint is inside the open interval because 2 < 4.5 < 7. For [2, 7), the midpoint is still 4.5 and it is inside because 4.5 ≥ 2 and 4.5 < 7. The length and midpoint are properties of the interval's position and span, not of its endpoint inclusion.

The classification of bounded versus unbounded depends solely on whether both endpoints are finite. A bounded interval has two finite numbers as endpoints. (2, 5), [0, 10], ( -3, 4 ], all bounded. An unbounded interval has at least one endpoint at ±∞. (2, ∞), (-∞, 5], (-∞, ∞), all unbounded. This distinction matters because only bounded intervals have a finite length. Unbounded intervals extend forever and cannot be measured. Write the domain of a logarithmic function as an unbounded interval: (0, ∞). The domain of a square root function is often an unbounded interval: [0, ∞). OpenStax College Algebra 2e, section 1.1, defines this classification.

Below is a reference table for all interval types. Each row shows the interval notation, the equivalent inequality, and the number line graph.

• Open (a, b): a < x < b. Graph: open circle at a, open circle at b, shade between.
• Closed [a, b]: a ≤ x ≤ b. Graph: filled circle at a, filled circle at b, shade between.
• Half-open [a, b): a ≤ x < b. Graph: filled circle at a, open circle at b, shade between.
• Half-open (a, b]: a < x ≤ b. Graph: open circle at a, filled circle at b, shade between.
• Unbounded (a, ∞): x > a. Graph: open circle at a, arrow to the right.
• Unbounded [a, ∞): x ≥ a. Graph: filled circle at a, arrow to the right.
• Unbounded (-∞, b): x < b. Graph: arrow from left, open circle at b.
• Unbounded (-∞, b]: x ≤ b. Graph: arrow from left, filled circle at b.
• All reals (-∞, ∞): entire number line shaded.

OpenStax College Algebra 2e, section 1.1, provides the same mapping between notation and inequality. Use this to check any interval you write.

The most frequent error is the infinity bracket error: using a bracket with ∞ or -∞, as in [a, ∞). This is always wrong because infinity is not a number that can be included. The second most common is writing the endpoints in the wrong order, e.g., (5, 2) when a < b. Interval notation must always go from smallest to largest. The third is forgetting to use the union symbol ∪ when combining disjoint intervals. Writing (-∞, 2)(3, ∞) is incorrect; it must be (-∞, 2) ∪ (3, ∞). The fourth is using a double inclusion, writing [a, b] when the inequality is a < x < b. A final common failure is representing the empty set as [ ] or ( ). The correct symbol is ∅ or { }.