We frequently need to describe sets of real numbers that form contiguous segments along the real number line called intervals. For instance, all the real numbers \(x\) that satisfies the inequalities
\begin{equation*}
x \geq -5 \quad \text{AND}\quad x \lt 2
\end{equation*}
forms an interval that includes \(-5\) and all real numbers to right on the number line up to and excluding \(2\text{.}\) Graphically, this can be plotted as we see below.
Note the use of a open circle and closed circle to indicate the behavior at the endpoints of the interval.
Describing these types of sets is so useful that we have special notation for this called interval notation. For example, the interval graphed above is written as \([-5,2)\text{,}\) where we use a bracket next to \(-5\) to indicate that it is included in the set and a parenthesis next to \(2\) to indicate that it is excluded from the set.
The symbol \(\infty\) refers to “infinity”, and is purely a description that these intervals have no upper bound in the case of \(+\infty\) or lower bound in the case of \(-\infty\text{.}\) Also observe that we never include brackets, “\([\)” or “\(]\)”, with infinities because we never “include” it in the interval. Finally, you are free to write \(\infty\) in the case of \(+\infty\text{.}\)
