Open Interval

Q: What is the difference between an open interval and a closed interval?

An open interval $(a, b)$ excludes both endpoints, using strict inequalities $a < x < b$. A closed interval $[a, b]$ includes both endpoints, using $a \le x \le b$. On a number line, open intervals use hollow circles at the endpoints while closed intervals use filled-in circles.

Q: Can an open interval extend to infinity?

Yes. The notation $(a, \infty)$ means all real numbers greater than $a$, and $(-\infty, b)$ means all real numbers less than $b$. Infinity is always paired with a parenthesis, never a bracket, because infinity is not a real number and cannot be included as an endpoint.

Q: Why do parentheses and brackets matter in interval notation?

Parentheses $(\,)$ indicate that an endpoint is excluded (strict inequality), while brackets $[\,]$ indicate that an endpoint is included ($\le$ or $\ge$). Confusing the two changes the set of numbers entirely. For instance, $(0, 5]$ includes 5 but not 0, whereas $[0, 5)$ includes 0 but not 5.

Open Interval

An interval that does not contain its endpoints.

Number line showing open interval (−2, 3) with open circles at −2 and 3, and a line segment between them.

Key Formula

(a, b) = \{x \in \mathbb{R} \mid a < x < b\}

Where:

$a$ = The left (lower) endpoint, not included in the interval
$b$ = The right (upper) endpoint, not included in the interval
$x$ = Any real number strictly between a and b

Worked Example

Problem: Determine whether the values 2, 3.5, and 5 belong to the open interval (2, 5).

Step 1: Write the defining inequality for the open interval (2, 5).

(2, 5) = \{x \mid 2 < x < 5\}

Step 2: Test x = 2. The condition requires x to be strictly greater than 2. Since 2 is not greater than 2, it fails.

2 < 2 \quad \text{is FALSE}

Step 3: Test x = 3.5. Check that 2 < 3.5 < 5. Both inequalities hold.

2 < 3.5 < 5 \quad \text{is TRUE}

Step 4: Test x = 5. The condition requires x to be strictly less than 5. Since 5 is not less than 5, it fails.

5 < 5 \quad \text{is FALSE}

Answer: Only 3.5 belongs to the open interval (2, 5). The endpoints 2 and 5 are excluded.

Another Example

This example shows how to represent an open interval on a number line and involves negative numbers, unlike the first example which focused on membership testing.

Problem: Graph the open interval (−3, 1) on a number line and express it as an inequality.

Step 1: Convert the interval notation to an inequality.

(-3, 1) \implies -3 < x < 1

Step 2: Identify the endpoints: −3 on the left and 1 on the right. Because the interval is open, draw open (hollow) circles at both −3 and 1.

Step 3: Shade the entire region on the number line between the two open circles. Every point in the shaded region satisfies the inequality.

Step 4: Verify a point inside: x = 0. Check −3 < 0 < 1. Both parts are true, so 0 is in the interval. Verify a boundary point: x = −3. Check −3 < −3, which is false, confirming the endpoint is excluded.

-3 < 0 < 1 \quad \checkmark

Answer: The open interval (−3, 1) is graphed with open circles at −3 and 1 and a shaded line segment between them. As an inequality:

-3 < x < 1

Frequently Asked Questions

What is the difference between an open interval and a closed interval?

An open interval

(a, b)

excludes both endpoints, using strict inequalities

a < x < b

. A closed interval

[a, b]

includes both endpoints, using

a \le x \le b

. On a number line, open intervals use hollow circles at the endpoints while closed intervals use filled-in circles.

Can an open interval extend to infinity?

Yes. The notation

(a, \infty)

means all real numbers greater than

a

, and

(-\infty, b)

means all real numbers less than

b

. Infinity is always paired with a parenthesis, never a bracket, because infinity is not a real number and cannot be included as an endpoint.

Why do parentheses and brackets matter in interval notation?

Parentheses

(\,)

indicate that an endpoint is excluded (strict inequality), while brackets

[\,]

indicate that an endpoint is included (

\le

\ge

). Confusing the two changes the set of numbers entirely. For instance,

(0, 5]

includes 5 but not 0, whereas

[0, 5)

includes 0 but not 5.

Open Interval vs. Closed Interval

	Open Interval	Closed Interval
Notation	$(a, b)$ — parentheses	$[a, b]$ — square brackets
Endpoints	Excluded (not in the set)	Included (in the set)
Inequality form	$a < x < b$ (strict)	$a \le x \le b$ (non-strict)
Number line graph	Open (hollow) circles at endpoints	Filled (solid) circles at endpoints
Contains its boundary?	No	Yes
Example	$(1, 4)$ : 1 and 4 are out	$[1, 4]$ : 1 and 4 are in

Why It Matters

Open intervals appear throughout algebra and calculus whenever you need to describe a range of values that excludes boundary points. For example, the domain of

f(x) = \frac{1}{x}

on a restricted region or the set where a function is increasing might be expressed as an open interval. Understanding the distinction between open and closed intervals is also essential for solving inequalities and reading solution sets correctly.

Common Mistakes

Mistake: Confusing parentheses with brackets and accidentally including the endpoints.

Correction: Remember: parentheses

(\,)

always mean "endpoint excluded" and brackets

[\,]

always mean "endpoint included." If you see

(3, 7)

, neither 3 nor 7 is part of the set. A helpful mnemonic: parentheses curve away from the number, as if pushing it out.

Mistake: Using a bracket with infinity, such as writing

[2, \infty]

Correction: Infinity is not a real number, so it can never be reached or included. Always use a parenthesis next to

\infty

-\infty

(2, \infty)

Related Terms

Interval — General concept that open intervals are a type of
Interval Notation — The notation system using parentheses and brackets
Closed Interval — Interval that includes both endpoints
Half-Open/Half-Closed Interval — Interval that includes one endpoint but not the other
Inequality — Open intervals correspond to strict inequalities
Set-Builder Notation — Alternative way to express the same set of values
Number Line — Visual tool for graphing open intervals