Question

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

$\left[-6,-\frac{}{}\right]$

Short answer

The inequality form of the given interval $\left[-6,-\frac{1}{2}\right]=\left\{x\right|-6\le x\le -\frac{1}{2}\}$.

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$\left[-6,-\frac{1}{2}\right]=\left[-6,-0.5\right]$

This includes all the numbers between -6 and -0.5, including both.

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

$\begin{array}{l}=\left[-6,-\frac{1}{2}\right]\\ =\left\{x\right|-6\le x\le -\frac{1}{2}\}\\ =\left\{x\right|-6\le x\le -0.5\}\end{array}$

Step 3. Graph the interval.

From step 2, we have $\left[-6,-\frac{1}{2}\right]=\left\{x\right|-6\le x\le -0.5\}$

This represents all the points between -6 and -0.5, including both.

Since both the endpoints are included in the interval, they will be represented by closed circles.

So, the graph will be: