Question

Express the inequality in interval notation, and then graph the corresponding interval.

$-1<x\le 5$

Short answer

The required interval notation for the given inequality is$\left(-1,5\right]$ 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 inequality is:

$-1<x\le 5$

This inequality contains all real values which are greater than$-1$ and less than or equal to $5$. Here,$-1$ is not included but$5$ is included. So, the required interval notation for the given inequality is:

$\left(-1,5\right]$

Step 3. Graph the interval.

Since$5$ is included and$-1$ is not included, therefore there is a closed circle at$5$ and an open circle at $-1$.

Thus, the required interval notation for the given inequality is$\left(-1,5\right]$ and the above graph is the required graph.