Question

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

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

Short answer

The required inequality for the given interval is$x\le 4$ and the required graph is shown below:

Step-by-step solution

Step 1. Important Definitions.

Open interval: An open interval is in the form of $\left(a,b\right)$, it is defined as $a<x<b$. It means the interval represents all real values between the numbers$a$ and $b$, and both$a,b$ are not included in the interval.

Closed interval: A closed interval is in the form of $\left[a,b\right]$, it is defined as $a\le x\le b$. It means the interval represents all real values between the numbers$a$ and $b$, and both$a,b$ are included in the interval.

If the end points of an interval are included in the interval, then they are represented by closed circles and if the end points of an interval are not included in the interval, then they are represented by open circles.

Step 2. Inequality form of the given interval.

The given interval is:

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

This interval contains open bracket on the left side and closed bracket on the right side. The value on the left side is $\mathbf{-}\mathit{\infty }$, it means all real values which are less than or equal to$4$ are included. So, the required inequality for the given interval is:

$x\le 4$

Step 3. Graph the interval.

Since$4$is included, therefore there is a closed circle at $4$. All the values on the left side of are included in the solution.

Thus, the required inequality for the given interval is$x\le 4$ and the above graph is the required graph.