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Open Interval and Closed Interval: Definition and Examples | EDU.COM

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Open Interval and Closed Interval: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Open Interval and Closed IntervalOpen Interval and Closed Interval: Definition and ExamplesTable of ContentsOpen Interval and Closed Interval Definition of Open Interval and Closed Interval

An interval is the collection of all real numbers in a given range. It contains all the real numbers between two given numbers, called endpoints of the interval. In an open interval (a,b)(a, b)(a,b), the endpoints are not included. Open intervals are denoted with parentheses and represent all values where axba axb. When graphed on a number line, open intervals use hollow circles at the endpoints to show they are not part of the interval.

A closed interval [a,b][a, b][a,b] includes both endpoints and all numbers between them. Closed intervals are written with square brackets and represent all values where a≤x≤ba \leq x \leq ba≤x≤b. On a number line, closed intervals use solid circles at the endpoints to show they are included in the interval. There are also half-open intervals where only one endpoint is included, such as [a,b)[a, b)[a,b) or (a,b](a, b](a,b].

Examples of Open Interval and Closed Interval Example 1: Comparing Intervals for Number Inclusion Problem:

There are two intervals [3,7][3, 7][3,7] and (2,6)(2, 6)(2,6). Which interval includes the number 333? Which interval includes 666?

Step-by-step solution:

Step 1, Understand what each interval contains. The closed interval [3,7][3, 7][3,7] includes all real numbers between 333 and 777, including both 333 and 777. The open interval (2,6)(2, 6)(2,6) includes all real numbers greater than 222 but less than 666.

Step 2, Check which interval contains 333. The number 333 is an endpoint of [3,7][3, 7][3,7], so it's included in this interval. For (2,6)(2, 6)(2,6), 333 is between 222 and 666, so it's also included in this interval.

Step 3, Check which interval contains 666. The number 666 is not an endpoint of [3,7][3, 7][3,7], but since 666 is less than 777, it's included in this interval. For (2,6)(2, 6)(2,6), 666 is an endpoint which is not included in an open interval, so 666 is not in (2,6)(2, 6)(2,6).

Step 4, Summarize our findings: Both intervals include the number 333. The closed interval [3,7][3, 7][3,7] includes the number 666, but the open interval (2,6)(2, 6)(2,6) does not include 666.

Example 2: Converting Inequalities to Intervals Problem:

Write the given inequalities as intervals.

i) 0x990 0x99 ii) −5≤x≤8-5 \leq x \leq 8−5≤x≤8 Step-by-step solution:

Step 1, Look at the inequality symbols in the first expression: 0x990 0x99. Since we have strict inequalities (

Step 2, Choose the correct notation for the first inequality. Because endpoints are not included, we use an open interval with parentheses: (0,99)(0,99)(0,99).

Step 3, Look at the inequality symbols in the second expression: −5≤x≤8-5 \leq x \leq 8−5≤x≤8. Since we have non-strict inequalities (≤), the endpoints are included.

Step 4, Choose the correct notation for the second inequality. Because endpoints are included, we use a closed interval with square brackets: [−5,8][-5, 8][−5,8].

Example 3: Representing an Inequality on a Number Line Problem:

Observe the inequality −4x3-4 −4x3 and state if it is an open interval or a closed interval. Represent it on a number line.

Step-by-step solution:

Step 1, Look at the inequality symbols in −4x3-4 −4x3. Since we have strict inequalities (xxx is greater than −4-4−4 but less than 333.

Step 2, Determine the type of interval. Since the endpoints −4-4−4 and 333 are not included (due to the strict inequalities), this is an open interval.

Step 3, Write the interval notation: (−4,3)(-4, 3)(−4,3).

Step 4, Draw the number line. Place hollow circles at −4-4−4 and 333 to show these points are not included. Draw a line connecting these points to show all numbers between them are in the interval.

Representing an Inequality on a Number Line

Comments(2)WWriterEllaNovember 4, 2025I've been trying to explain open and closed intervals to my students. This page's clear defs and examples made it so much easier! Thanks!

NNatureLover25September 17, 2025I’ve been helping my kids with math, and this page made intervals so easy to understand! The examples are great, and the clear explanations really clicked for them. Thanks for this resource!

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