Objective 1 - Set Notation

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Describe solutions as sets of numbers.

Link to section in online textbook.

First, watch the video below to learn about inequalities. You can use the notes here to follow along with the video and record your thoughts.

We start with a terminology review.

A set is a collection of mathematical objects. We’ll commonly look at sets of numbers like the Natural numbers: $\{ 1, 2, 3, 4, \ldots \}$ .

An interval is a collection of Real numbers. For example, $(1, 2)$ is the set of Real numbers between 1 and 2 (but not including 1 or 2). If we want to include the endpoints of an interval, we use brackets, such as $[1, 2]$ .

We can describe solutions that exist in an interval by using the notation $x \in (a, b)$ . We read this as “ $x$ is an element of $(a, b)$ ” and means that $x$ is some number between $a$ and $b$ . For example, $x \in [1, 2]$ means that $x$ is some number between 1 and 2 (and could be one of the two numbers).

We can also describe intervals using inequalities. For example, to describe the set of $x \in (1, 2)$ using inequalities, we would use $1 < x < 2$ . This is usually a more natural way for students to read “ $x$ is a Real number between 1 and 2.” If we want to include the endpoints of an inequality, we use the symbols $\leq$ or $\geq$ .

Write each set described in Interval notation.


Set described in words	Inequality Notation	Interval Notation

All Real numbers between $a$ and $b$ , but not including $a$ or $b$	$a < x < b$	$\{x \, \| \, x \in \answer {(a, b)} \}$
All Real numbers greater than $a$ , but not including $a$	$x > a$	$\{x \, \| \, x \in \answer {(a, \infty )} \}$
All Real numbers less than $b$ , but not including $b$	$x < b$	$\{x \, \| \, x \in \answer {(-\infty , b)} \}$
All Real numbers greater than $a$ , including $a$	$x \geq a$	$\{x \, \| \, x \in \answer {[a, \infty )} \}$
All Real numbers less than $b$ , including $b$	$x \leq b$	$\{x \, \| \, x \in \answer {(-\infty , b]} \}$
All Real numbers between $a$ and $b$ , including $a$	$a \leq x < b$	$\{x \, \| \, x \in \answer {[a, b)} \}$
All Real numbers between $a$ and $b$ , including $b$	$a < x \leq b$	$\{x \, \| \, x \in \answer {(a, b]} \}$
All Real numbers between $a$ and $b$ , including $a$ and $b$	$a \leq x \leq b$	$\{x \, \| \, x \in \answer {[a, b]} \}$
All Real numbers less than $a$ or greater than $b$	$x < a \text { or } x > b$	$\{x \, \| \, x \in \answer {(-\infty , a)} \cup \answer {(b, \infty )} \}$
All Real numbers	$x \geq a \text { or } x < a$	$\{x \, \| \, x \in \answer {(-\infty , \infty )} \}$

On exams, you will answer questions primarily using interval notation. Solve the linear equation below and choose the interval that contains the solution.

$x + 3 = 5.5$

$x = a, \text { where } a \in [-2, -1]$ $x = a, \text { where } a \in [-1, 0]$ $x = a, \text { where } a \in [0, 1]$ $x = a, \text { where } a \in [1, 2]$ $x = a, \text { where } a \in [2, 3]$