Interval notation, inequality notation, and set-builder notation are three ways to describe
the same set of real numbers. They look different, but they communicate the same mathematical
idea: which values of \(x\) are allowed.
1. Inequality notation
Inequality notation uses symbols such as \(\lt\), \(\le\), \(\gt\), and \(\ge\). For example,
\[
x<-2\quad\text{or}\quad 1
means that \(x\) can be any real number less than \(-2\), or any real number greater than
\(1\) and less than or equal to \(5\).
2. Interval notation
Interval notation uses parentheses and brackets. Parentheses mean the endpoint is not included,
while brackets mean the endpoint is included.
Infinite intervals always use parentheses with infinity because \(\infty\) and \(-\infty\)
are not real numbers and cannot be included as endpoints.
\[
x<-2
\quad\Longleftrightarrow\quad
(-\infty,-2).
\]
3. Set-builder notation
Set-builder notation describes a set by giving a rule. For example,
\[
\{x\in\mathbb{R}\mid x<-2\text{ or }1
is read as “the set of all real numbers \(x\) such that \(x<-2\) or \(1
4. Union and intersection
The symbol \(\cup\) means union. It combines values from either set. In words, union acts like
OR. For example,
\[
(-\infty,-2)\cup(1,5]
\]
means all values in the first interval or in the second interval.
The symbol \(\cap\) means intersection. It keeps only values that are in both sets. In words,
intersection acts like AND. For example,
\[
x>1\quad\text{AND}\quad x\le5
\]
becomes
\[
(1,5].
\]
5. Worked example
Convert:
\[
x<-2\quad\text{or}\quad 1
The first part, \(x<-2\), extends left forever and does not include \(-2\), so it becomes:
\[
(-\infty,-2).
\]
The second part, \(1
\[
(1,5].
\]
Since the word is “or,” we use union:
\[
\boxed{(-\infty,-2)\cup(1,5]}.
\]
The set-builder form is:
\[
\boxed{\{x\in\mathbb{R}\mid x<-2\text{ or }16. Converting interval notation back to inequalities
To convert interval notation back to inequality notation, read each endpoint:
- A left parenthesis \((a,\) means \(x>a\).
- A left bracket \([a,\) means \(x\ge a\).
- A right parenthesis \(\,b)\) means \(x
- A right bracket \(\,b]\) means \(x\le b\).
For example:
\[
[-4,2)
\quad\Longleftrightarrow\quad
-4\le x<2.
\]
7. Number line interpretation
On a number line, open circles match parentheses and strict inequalities. Closed circles
match brackets and non-strict inequalities. A shaded segment shows all values in the interval.
A ray shows values extending forever in one direction.
8. Summary
The conversion rules are:
- Use parentheses for excluded endpoints and brackets for included endpoints.
- Use \(\cup\) when the statement uses OR.
- Use intersection or overlap when the statement uses AND.
- Use \(\{x\in\mathbb{R}\mid \cdots\}\) for set-builder form.
- Always use parentheses with \(\infty\) and \(-\infty\).
