Section 8.1 Interval Notation

Let’s look at the example we started this section with: all of the numbers between 2 and 7, including 2 but not including 7. Since we want the numbers between 2 and 7, those are our enpoints. Since we are including 2, it’s going to have a square bracket [. Since we are not incluidng 7, it is going to have a round parentheses ). We always write the smaller endpoint first, so in interval notation, we would have \([2,7)\text{.}\)

Suppose we want to write \(x\gt 2\) in interval notation. We know that 2 is one of our endpoints, but the other side goes on forever. You might find it helpful to draw a number line to determine which side of 2 we want to include. In our case, this is what that would look like:

Since the interval goes on forever in the positive direction, we can think of \(\infty\) as the other endpoint. However, since \(\infty\) isn’t an actual number, it will always get a round parentheses ). Since the endpoint 2 is not included, it will also get a round parentheses (. So, in interval notation, our answer would be \((2,\infty)\text{.}\)

Suppose we are working on a problem, and our answer has to include both the interval \(3 \lt x \le 8\) and the interval \(x \gt 10\text{.}\) First, we write each interval using interval notation, then we will combine them with a union. Using what we learned above, we know that \(3 \lt x \le 8\) is written as \((3,8]\text{,}\) and \(x \gt 10\) is written as \((10,\infty)\text{.}\) Since we want our answer to include both, our final answer is \((3,8]\cup(10,\infty)\text{.}\)

Suppose we wanted our answer to be everything except 3. That means we want to include all the numbers less than 3 and all the numbers greater than 3. That’s two intervals that we’ll combine with a union. All the numbers less than 3 is written as \((-\infty, 3)\text{,}\) and all the numbers greater than 3 is written as \((3,\infty)\text{.}\) Therefore, our final answer is \((-\infty, 3)\cup(3,\infty)\text{.}\)