Interval notation helps to clearly express a continuous range of numbers in a simple, concise way. It is handy in many ways, such as to define acceptable weight limits for a machine, indicating a valid range of measurements in scientific experiments, and specifying temperature ranges in which a machine is safe to operate. Use this resource to learn about using interval notation.
Intervals
When the domain or range of a function is restricted to a subset of real numbers, \(\mathbb{R}\), it is called an interval. The endpoints are indicated with \(a\) and \(b\) for the lower and upper limits, respectively.
Intervals can be represented using:
interval notation, which uses brackets \([\,]\) and parentheses \(\left(\,\right)\)
inequality notation, which uses inequality symbols: less than \(<\), greater than \(>\), less than or equal to \(\leq\) and greater than or equal to \(\geq\).
Intervals are always written with the smaller number on the left. For example, \(\left(-3,5\right)\) is correct and \(\left(5,-3\right)\) is not.
Closed intervals
A closed interval is a set of numbers that includes all the numbers between two endpoints, and the endpoints \(a\) and \(b\) themselves.
Consider the interval on the line graph where the endpoints on the real number line are represented by solid circles.
\(a\) and \(b\) are included in the interval. The subset is therefore a closed interval, so brackets \([\,]\) are used. The interval notation would be \([a,b]\) and the inequality notation would be \(a\leq x \leq b\).
Open intervals
If the endpoints are not included in the interval, it is called an open interval.
Consider the same interval with endpoints \(a\) and \(b\) represented by open circles on the line graph.
Curved brackets are used to give the inverval notation \(\left(a,b\right)\). In inequality notation, the interval is \(a<x<b\).
Half-open and half-closed intervals
In some cases, one endpoint is included in the subset and the other is not. This is called a half-open or half-closed interval. Interval notation is written using one bracket and one parenthesis, and inequality notation is written using a mix of inequality symbols.
For some subsets, the range of values may continue indefinitely towards the right i.e. towards infinity (\(\infty\)), or towards the left i.e. towards negative infinity (\(-\infty\)). Infinity is not a numeral, so we should not write the inequality notation as \([b,\infty]\), \([-\infty,a]\) or \(b\leq x\leq\infty\), etc. For the purposes of understanding the concept, this type of notation has been used in this resource, but elsewhere, we would typically write \(x\geq b\) or \(x\leq a\).
The table shows some examples of domains with endpoints \(a\) and \(b\), including their interval notation, inequality notation and line graph.
Interval notation
Inequality notation
Line graph
\([a,b]\)
\(a\leq x\leq b\)
\((a,b)\)
\(a<x<b\)
\([a,b)\)
\(a\leq x<b\)
\((a,b]\)
\(a<x\leq b\)
\([a,\infty)\)
\(x\geq a\)
\((a,\infty)\)
\(x>a\)
\((-\infty,b]\)
\(x\leq b\)
\((-\infty,b)\)
\(x>b\)
Example 1 – using interval and inequality notation
Write \([-2,3)\) in inequality notation and graph it on the real number line.
The left bracket indicates that \(-2\) is included. The right parenthesis indicates that \(3\) is not included. Therefore, the inequality notation is \(-2\leq x\leq 3\).
On the number line, the \(-2\) endpoint uses a solid circle and the \(3\) endpoint uses an open circle.
Write \((-\infty,3]\) in inequality notation and graph it on the real number line.
The \(-\infty\) indicates that the values continue indefinitely towards the left. The right bracket indicates that \(3\) is included in the set. Therefore, the inequality notation is \(x\leq3\).
On the number line, the \(-\infty\) is represented using an arrow towards the left and the \(3\) endpoint uses a solid circle.
Write the interval notation and inequality notation for the following line graph:
The endpoints are \(-5\) and \(6\). \(-5\) uses an open circle, so it is not included in the set. \(6\) uses a solid circle, so it is included in the set.
The interval notation is therefore \((-5,6]\). The inequality notation is \(-5<x\leq6\).
Write the interval notation and inequality notation for the following line graph:
The line graph has an endpoint of \(10\), which extends to the right indefinitely (towards \(\infty\)). \(10\) uses a solid circle, so it is incldued in the set.
The interval notation is therefore \([10,\infty)\) and the inequality notation is \(x\geq10\).
Exercise – using interval and inequality notation
Write the following in interval notation and graph them on a real number line.
\(1\leq x<10\)
\(-6\leq x<-4\)
\(x>5\)
Write the following in interval notation and inequality notation.
\([1,10)\)
\([-6,-4)\)
\((5,\infty)\)
\((-\infty,-5]\) and \(x\leq-5\)
\((-3,0)\) and \(-3<x<0\)
\([-1,4)\) and \(-1\leq x<4\)
Multiple intervals
Functions can have two or more sets of real values. Let's consider a function with one subset with endpoints \(a\) and \(b\), and a second subset with endpoints \(c\) and \(d\).
To indicate the intervals, we need to use the \(\cup\) symbol, which means "in union with". In interval notation, this would be \([a,b]\cup[c,d]\). In inequality notation, we write \(a\leq x\leq b\) with \(c\leq x\leq d\) OR \(\left\{ x:a\leq x\leq b\right\} \cup\left\{ x:c\leq x\leq d\right\} \).
Example – using interval and inequality notation for multiple intervals
Write the following interval and inequality notation.
For the left interval, the left arrow indicates the values extend indefinitely towards \(-\infty\). The solid circle at \(2\) indicates that the subset includes \(2\). This interval would be \((-\infty,2]\) or \(x\leq2\).
For the right interval, the open circle at \(5\) indicates that the subset does not include \(5\). The solid circle at \(12\) indicates that the subset includes \(12\). This interval would be \((5,12]\) or \(5<x\leq12\).
Putting these together with the \(\cup\) symbol, the interval notation would be \((-\infty,2]\cup(5,12]\). The inequality notation would be \(x\leq2\) with \(5<x\leq12\), or \(\left\{ x:x\leq2\right\}\cup\left\{x:5<x\leq12\right\}\).
Exercise
Graph the following on the real number line and write in inequality notation.
\((-\infty,3)\cup(8,13]\)
\(\left[-1,4\right]\cup\left[6,9\right]\)
\((-\infty,3]\cup(6,\infty)\)
\(x<3\) with \(8<x\leq13\) or \(\left\{x:x<3\right\}\cup\left\{x:8<x\leq13\right\}\)
\(-1\leq x\leq4\) with \(6\leq x\leq9\) or \(\left\{x:-1\leq x\leq4\right\}\cup\left\{x:6\leq x\leq9\right\}\)
\(x\leq3\) with \(x>6\) or \(\left\{x:x\leq3\right\}\cup\left\{x:x>6\right\}\)
