How to note down an interval?
You can note down an interval in three ways. You will most often encounter the first way: it's typical interval notation, where two numbers are are in brackets divided by a semicolon (sometimes a comma is used, which may be misleading and may be confused with a decimal number, so we won't use it in this book).
The first type of notation looks like it has got two endpoints in brackets. The point is on the left side must be less than the one on the right side. For example, the interval (1,5) is written correctly, but the interval (5,1) is wrong.
Of course the brackets may vary, so you can note down four types of intervals, and those are (a, b),[a, b),(a, b], or [a, b]. It depends if you want the endpoint to be part of the interval. If the point is part of the interval, you will use square brackets, i.e. "[" or "]". If the point is not part of the interval, you will use round bracket, thus "(" or ")". I will tell you more about brackets later when discussing finite intervals.
The second type of notation, which is called the set-builder notation, is used primarily for the notation of sets. How it works: first, write the lower endpoint and then the sign of inequality (either "\leq" \rightarrow endpoint belongs to the interval, or "<" \rightarrow endpoint does not belong to the interval). Then note down the variable (it's up to you how you name it, usually it's x) and again you will write the sign of inequality (symbol "\leq" or "<", it depends if the endpoint belongs or doesn't belong to the interval). Finally, write the value of the greater endpoint. The notation 1<x<5 can be read as "variable x is greater than one and, at the same time, less than five."
The third type involves graphing on a real number line. This method is suitable especially in moments when you need to imagine more intervals at once. So, for instance, you can see where they overlap. I will tell you more about graphing under the Finite intervals subheading.
