How do you write the following in interval notation: $x > 3$ and $x \leqslant 7$?

Hint: We will first mention the types of intervals and then the types of brackets. Then mention the types of brackets we will be using in this problem. You use a bracket to signify an inequality with an or equal to operator.

Complete step by step answer:
We will start off by mentioning the types of intervals and brackets.
1. Open interval is a set of real numbers that does not include its endpoints. Closed interval is a set of real numbers that includes both of its endpoints and a half-bounded interval is a set for which one endpoint is a real number and the other is not.
2. The round brackets $(\,)$ are used when the endpoints of the interval are not included. 3
3. The square brackets $[\,]$ are used when the endpoints of the interval are also included. 4. Combinations of round brackets $(\,)$ and square brackets $[\,]$ are also used according to the requirements.
In interval notation, you use a parenthesis to signify an inequality without an or equal to operator.

Here, we can write $x > 3$ and $x \leqslant 7$ as $(3,7]$.

Additional Information: A real interval is a set of real numbers with the property that any number that lies between two numbers included in the set is also included in the set. To indicate that an endpoint of a set is not included in the set, the square bracket enclosing the endpoint can be replaced with a parenthesis. An open interval does not include its endpoints, and is enclosed in parenthesis. A closed interval included its endpoints, and is enclosed in square brackets.

Note: While solving such types of questions, make sure that you mention all the terms in the starting of the solution. Define the terms like open, close brackets as well as open, close intervals. Also, remember that a set for which both endpoints are real numbers.