Definition1.7.3Interval Notation

Interval notation tries to just say the numbers where the interval starts and stops. For example, in Figure 1.7.1, the interval starts at \(18\text{.}\) To the right, the interval extends forever and has no end, so we use the \(\infty\) symbol (meaning "infinity"). This particular interval is denoted:

\begin{equation*} [18,\infty) \end{equation*}

Why use “\([\)” on one side and “\()\)” on the other? The square bracket tells us that \(18\) is part of the interval and the round parenthesis tells us that \(\infty\) is not part of the interval. 1 

In general there are four types of infinite intervals. Take note of the different uses of round parentheses and square brackets.

a number line with a mark at a; the portion of the number line from a to the right has a thick line overlaying it; there is a left parenthesis at a and an arrow on the right pointing right
a number line with a mark at a; the portion of the number line from a to the right has a thick line overlaying it; there is a left bracket at a and an arrow on the right pointing right
Figure1.7.4An open, infinite interval denoted by \((a,\infty)\) means all numbers \(a\) or larger, not including \(a\text{.}\)
Figure1.7.5A closed, infinite interval denoted by \([a,\infty)\) means all numbers \(a\) or larger, including \(a\text{.}\)
a number line with a mark at a; the portion of the number line from a to the left has a thick line overlaying it; there is a right parenthesis at a and an arrow on the left pointing left
a number line with a mark at a; the portion of the number line from a to the left has a thick line overlaying it; there is a right bracket at a and an arrow on the left pointing left
Figure1.7.6An open, infinite interval denoted by \((-\infty,a)\) means all numbers \(a\) or smaller, not including \(a\text{.}\)
Figure1.7.7A closed, infinite interval denoted by \((-\infty,a]\) means all numbers \(a\) or smaller, including \(a\text{.}\)
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