Question

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

$\left[\mathbf{2}\mathbf{,}\infty \right)$

Short answer

The inequality form of the given interval $\left[2,\infty \right)=\left\{x\right|x\ge 2\}$.

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$\left[2,\infty \right)$

This includes all the numbers greater than and equal to 2.

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

$\left[2,\infty \right)=\left\{x\right|x\ge 2\}$

Step 3. Graph the interval.

From step 2, we have $\left[2,\infty \right)=\left\{x\right|x\ge 2\}$

This represents all the numbers greater than and equal to 2.

Since 2 is included in the interval, it will be represented by a closed circle.

So, the graph will be: