Consider the interval $(-3, 2]$, where $(-3, 2] = \{x : x \in R, -3 < x \leq 2\}$ represents the set of real numbers between $-3$ and $2$ where $2$ is included in the interval but $-3$ is excluded from it. Represent this as a segment of the real number line. Also, consider the set $[-3, 2) = \{x : x \in R, -3 \leq x < 2\}$, representing the set of real numbers between $-3$ and $2$ where $-3$ is included and $2$ is excluded from the set. Represent this as a segment of the real number line.

> [!TLDR] > A half-open interval marks its excluded endpoint with an open circle and its included endpoint with a solid dot. > [!FORMULA] > $(a,b]=\{x\in\mathbb{R}: a<x\le b\}$ (left excluded, right included); $[a,b)=\{x\in\mathbb{R}: a\le x<b\}$ (left included, right excluded). 1. **$(-3,2]=\{x\in\mathbb{R}: -3<x\le2\}$.** - At $x=-3$: the inequality is strict ($-3<x$), so $-3$ is **excluded** $\Rightarrow$ open (hollow) circle at $-3$. - At $x=2$: the inequality is non-strict ($x\le2$), so $2$ is **included** $\Rightarrow$ solid (filled) dot at $2$. - Draw the segment from $-3$ to $2$ with open circle at $-3$ and filled dot at $2$. 2. **$[-3,2)=\{x\in\mathbb{R}: -3\le x<2\}$.** - At $x=-3$: the inequality is non-strict ($-3\le x$), so $-3$ is **included** $\Rightarrow$ solid (filled) dot at $-3$. …

Consider the interval (-3, 2], where (-3, 2] = \x – Class 11 Applied Mathematics

Worked Examples · Example 9

(-3, 2]

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Concept understanding — Types of Intervals

Types of Intervals

Imagine you have a number line. An interval is simply a connected chunk of that line — all the real numbers that lie between two endpoints. The question is: do you include the endpoints themselves, or not? That choice gives you the different types of intervals.

The Intuition

Suppose you're told "pick any number between 2 and 5". That's vague. Do you mean:

Any number from 2 to 5, including 2 and 5 themselves? (So 2, 5, and everything in between are allowed.)
Any number from 2 to 5, excluding 2 and 5? (So 2.0001 is fine, but 2 itself is not.)
Include one endpoint but not the other? (Like "from 2 up to, but not including, 5".)

Each of these is a different type of interval. The notation and names are designed to make this crystal clear.

The Precise Statement

An interval is a set of real numbers. For two real numbers $a$ and $b$ with $a < b$ , there are four standard types:

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

The round bracket ( or ) means "this endpoint is not included". The square bracket [ or ] means "this endpoint is included".

Name	Notation	Includes $a$ ?	Includes $b$ ?	Example numbers
Open	$(a, b)$	No	No	$2.1, 3, 4.999$ but not $2$ or $5$
Closed	$[a, b]$	Yes	Yes	$2, 5$ , and everything in between
Left-open	$(a, b]$	No	Yes	$2.001$ up to and including $5$
Right-open	$[a, b)$	Yes	No	$2$ up to but not including $5$

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A half-open interval marks its excluded endpoint with an open circle and its included endpoint with a solid dot.

(a,b]=\{x\in\mathbb{R}: a<x\le b\}

$(-3,2]=\{x\in\mathbb{R}: -3<x\le2\}$ .
- At $x=-3$ : the inequality is strict ( $-3<x$ ), so $-3$ is excluded $\Rightarrow$ open (hollow) circle at $-3$ .
- At $x=2$ : the inequality is non-strict ( $x\le2$ ), so $2$ is included $\Rightarrow$ solid (filled) dot at $2$ .
- Draw the segment from $-3$ to $2$ with open circle at $-3$ and filled dot at $2$ .
$[-3,2)=\{x\in\mathbb{R}: -3\le x<2\}$ .
- At $x=-3$ : the inequality is non-strict ( $-3\le x$ ), so $-3$ is included $\Rightarrow$ solid (filled) dot at $-3$ . …

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