Sometimes when solving mathematical problems, the solution is a set of numbers lying in an interval.

There are several ways to describe these intervals.

• We can write the solution algebraically using set notation.

• We can display the solution graphically on a number line.

• We can use a special notation called interval notation.

A solution can be described algebraically using various symbols.

Example 1

Describe the set of real numbers from \(-5\) to \(3\) that includes \(-5\), but does not include \(3\).

Solution

We read this, “the set of all \(x\) such that \(-5\) is less than or equal to \(x\), which is less than \(3\), and \(x\) is an element of the set of real numbers.”

Example 2

Describe the set of real numbers which does not include \(1\).

Solution

We read this, “the set of all \(x\) such that \(x\) is not equal to \(1\) and \(x\) is an element of the set of real numbers.”

An interval can be described graphically using a number line.

A solid dot means that the endpoint is included in the interval.

An open dot means that the endpoint is excluded from the interval.

The example shown illustrates the solution which could be written in set notation as

Another interval is illustrated on the following number line:

Using set notation, this solution would be written

Note

The use of the word or tells us that there is a union of two solutions. In this case, \(x\) can take on values less than or equal to \(-1\), or \(x\) can take on values greater than or equal to \(1\).

Alternatively, \(x\) cannot take on values between \(-1\) and \(1\).