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Interval — Definition, Types & Examples

Interval

The set of all real numbers between two given numbers. The two numbers on the ends are the endpoints. The endpoints might or might not be included in the interval depending whether the interval is open, closed, or half-open (same as half-closed).

 

 

See also

Interval notation, inclusive, exclusive

Key Formula

Closed: [a,b]={x∈R∣a≤x≤b}\text{Closed: } [a, b] = \{x \in \mathbb{R} \mid a \le x \le b\} Open: (a,b)={x∈R∣a<x<b}\text{Open: } (a, b) = \{x \in \mathbb{R} \mid a < x < b\} Half-open: [a,b)={x∈R∣a≤x<b}\text{Half-open: } [a, b) = \{x \in \mathbb{R} \mid a \le x < b\} Half-open: (a,b]={x∈R∣a<x≤b}\text{Half-open: } (a, b] = \{x \in \mathbb{R} \mid a < x \le b\}
Where:
  • aa = The left (lower) endpoint of the interval
  • bb = The right (upper) endpoint of the interval
  • xx = Any real number belonging to the interval

Worked Example

Problem: A student scores between 70 and 100 on a test. The minimum passing score of 70 is included, and the maximum score of 100 is also included. Write this as an interval and determine whether the score 70, the score 85, and the score 100.5 belong to it.
Step 1: Identify the endpoints. The lower endpoint is 70 and the upper endpoint is 100.
a=70,b=100a = 70, \quad b = 100
Step 2: Both endpoints are included, so use square brackets to write a closed interval.
[70,100]={x∈R∣70≤x≤100}[70, 100] = \{x \in \mathbb{R} \mid 70 \le x \le 100\}
Step 3: Check each value. Is 70 in the interval? Yes, because 70 ≤ 70 ≤ 100. Is 85 in the interval? Yes, because 70 ≤ 85 ≤ 100. Is 100.5 in the interval? No, because 100.5 > 100.
70∈[70,100],85∈[70,100],100.5∉[70,100]70 \in [70, 100], \quad 85 \in [70, 100], \quad 100.5 \notin [70, 100]
Answer: The set of valid scores is the closed interval [70, 100]. The values 70 and 85 belong to it, but 100.5 does not.

Another Example

Problem: Write the solution set of the inequality 2 < x ≤ 9 using interval notation, then state whether the interval is open, closed, or half-open.
Step 1: Read the inequality. The value x must be strictly greater than 2 (not equal), so 2 is excluded. The value x can equal 9, so 9 is included.
2<x≤92 < x \le 9
Step 2: Use a parenthesis for the excluded endpoint and a bracket for the included endpoint.
(2,9](2, 9]
Step 3: Because one endpoint is excluded and the other is included, this is a half-open (half-closed) interval.
Answer: The solution set is the half-open interval (2, 9].

Frequently Asked Questions

What is the difference between parentheses and brackets in interval notation?
A parenthesis, like ( or ), means the endpoint is excluded from the interval. A bracket, like [ or ], means the endpoint is included. For example, (3, 7] contains every real number greater than 3 and up to and including 7, but it does not contain 3 itself.
Can an interval extend to infinity?
Yes. You write intervals that go on forever using the symbols −∞ and ∞. Because infinity is not an actual number you can reach, you always use a parenthesis next to it, never a bracket. For example, [5, ∞) means all real numbers greater than or equal to 5, and (−∞, 0) means all real numbers less than 0.

Open Interval vs. Closed Interval

An open interval (a, b) excludes both endpoints, so neither a nor b belongs to the set. A closed interval [a, b] includes both endpoints, so a and b are members of the set. A half-open interval mixes the two: one endpoint is included and the other is excluded. On a number line, open endpoints are drawn as hollow circles and closed endpoints as filled-in circles.

Why It Matters

Intervals appear throughout algebra, calculus, and statistics whenever you describe a continuous range of values. Domains of functions, solution sets of inequalities, and confidence intervals in statistics are all expressed using interval notation. Understanding intervals lets you communicate precisely about which boundary values are or are not part of a set.

Common Mistakes

Mistake: Using a bracket next to infinity, such as writing [2, ∞] instead of [2, ∞).
Correction: Infinity is not a real number, so it can never be "reached" or included. Always pair ∞ or −∞ with a parenthesis.
Mistake: Confusing interval notation (a, b) with the ordered pair (a, b) used for coordinates.
Correction: Context determines the meaning. If you are describing a set of numbers, (a, b) is an interval. If you are plotting a point on the coordinate plane, (a, b) is an ordered pair. When ambiguity is possible, use set-builder notation or add a clarifying note.

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