Question

Graph the set

$(-\infty ,6]\cap \left(2,10\right)$

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 intersection.

The intersection of S and T is the set $S\cap T$consisting of all elements that are in both S and T.

Step 3. Find the intersection.

$\begin{array}{l}\left(-\infty ,6\right]\cap \left(2,10\right)\\ =\left\{x|x\le 6\right\}\cap \left\{x|2<x<10\right\}\\ =\left\{x\right|x\le 6\text{and}2<x<10\}\\ =\left\{x\right|2<x\le 6\}\\ =\left(2,6\right]\end{array}$

Step 4. Graph the intervals.

We will graph all three intervals:

$\left(-\infty ,6\right]$

It includes all the numbers less than and equal to 6.

$\left(2,10\right)$

It includes all the numbers between 2 and 10.

$\left(-\infty ,6\right]\cap \left(2,10\right)=\left(2,6\right]$

It includes all the numbers between 2 and 6, including 6