Question

Graph the set

$\left(-\infty ,-4\right)\cup \left(4,\infty \right)$

Short answer

The graph of the given set is,

Step-by-step solution

Step 1. Apply the concept of interval and number line.

If $a<b$, then the open interval from a to b consists of all numbers between a and b and is denoted by $\left(a,b\right)$. The closed interval from a to b includes the endpoints and is denoted by $\left(a,b\right)$. If points are included in the interval, they are represented by dark circles and if the points are not included in the interval, they are represented by Hollow circles on the number line.

Using the set-builder form, we can write the intervals as inequality:

$$\begin{gathered} \left(a,b\right)=\left\{x|a<x<b\right\} \\ \left[a,b\right]=\left\{x|a\le x\le b\right\} \end{gathered}$$

Step 2. Apply the concept of the union of sets.

If S and Tare set, then their union $S\cup T$is the set that consists of all elements that are in S or T (or in both).

Step 3. Find the union.

$\begin{array}{l}\left(-\infty ,-4\right)\cup \left(4,\infty \right)\\ =\left\{x|x<-4\right\}\cup \left\{x|x>4\right\}\\ =\left\{x\right|x<-4\text{or}x>4\}\end{array}$

Step 4. Graph the intervals.

We will graph all three intervals:

$\left(-\infty ,-4\right)$

It includes all the numbers less than -4

$\left(4,\infty \right)$

It includes all the numbers greater than 4

$\left(-\infty ,-4\right)\cup \left(4,\infty \right)=\left\{x\right|x<-4\text{or}x>4\}$

It includes all the numbers less than -4 or greater than 4