$(-\infty, 3) \text{ or } (5, \infty)$
Choose two other ways that the solution can be represented.
The given set is \((-\infty, 3) \cup (5, \infty)\). This interval notation represents the collection of all real numbers \(x\) that are strictly less than \(3\) OR strictly greater than \(5\). In other words, it includes all real numbers except those in the closed interval \([3, 5]\).
A common way to represent this set is by using set-builder notation. Set-builder notation uses a variable and a condition that defines the set. For the set of all real numbers \(x\) such that \(x < 3\) or \(x > 5\), we can write:
\[\{x \mid x < 3 \text{ or } x > 5\}\]
This directly mirrors the definition provided by the union of the two intervals.
Another way to represent this set is to describe it as the set of all real numbers excluding the closed interval \([3, 5]\). This is known as set complement notation, where we subtract the subset \([3, 5]\) from the set of all real numbers, denoted as \(\mathbb{R}\). Thus, we can write:
\[\mathbb{R} \setminus [3, 5]\]
This notation is very useful in identifying which real numbers are not included in the set.
We have identified two alternative representations for the interval \((-\infty, 3) \cup (5, \infty)\):
These represent the same set of real numbers as the original interval notation. The final answer is \(\{x \mid x < 3 \text{ or } x > 5\}\) and \(\mathbb{R} \setminus [3, 5]\).
In statistics or data analysis, you might be looking for "outlier" ranges. If a standard range for a process is defined between 3 and 5, any data point falling outside this range—represented by \((-\infty, 3) \cup (5, \infty)\)—could be flagged as an anomaly. Representing this set in different ways, such as \(\mathbb{R} \setminus [3, 5]\), helps researchers quickly identify the values that deviate from the normal operating conditions.