Question

Express the interval in terms of inequalities, and then graph the interval.

$\left(2,8\right]$

Short answer

The inequality form of the given interval $\left(2,8\right]=\left\{x\right|2<x\le 8\}$.

Step-by-step solution

Step 1. Apply the concept of interval and inequality.

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 $\left[a,b\right]$.

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. Express in terms of inequality.

Given interval is$(2,8)$

This includes all the numbers between 2 and 8, including 8.

Using set-builder form to express the interval as inequality:

$\left(2,8\right]=\left\{x\right|2<x\le 8\}$

Step 3. Graph the interval.

From step 2, we have $\left(2,8\right]=\left\{x\right|2<x\le 8\}$

This represents all the points between 2 and 8, including 8 and excluding 2.

Since 8 is included in the interval, it will be represented by a closed circle and 2 is excluded, it will be represented by an open circle.

So, the graph will be: