To graph the set [tex]\(\{x \mid -3 < x < 3\}\)[/tex] on a number line and write it in interval notation, follow these steps:
### Step-by-Step Solution:
1. Understand the Inequality:
The inequality [tex]\( -3 < x < 3 \)[/tex] means that [tex]\( x \)[/tex] can take any value between [tex]\(-3\)[/tex] and [tex]\(3\)[/tex], but not including [tex]\(-3\)[/tex] and [tex]\(3\)[/tex] themselves.
2. Graph on the Number Line:
- Draw a horizontal line representing the number line.
- Mark the points [tex]\(-3\)[/tex] and [tex]\(3\)[/tex] on this line.
- Since [tex]\(-3\)[/tex] and [tex]\(3\)[/tex] are not included in the set, we use open circles (or hollow dots) at these points.
- Shade the portion of the number line between [tex]\(-3\)[/tex] and [tex]\(3\)[/tex].
The number line should look like this:
```
-------------------o===================o-------------------
-3 0 3
```
Here, the open circles at [tex]\(-3\)[/tex] and [tex]\(3\)[/tex] indicate that these points are not included in the set, and the shaded region (represented by the line between the two open circles) shows all the numbers between [tex]\(-3\)[/tex] and [tex]\(3\)[/tex].
3. Interval Notation:
To write the set in interval notation, we use parentheses to denote that the endpoints are not included.
- For [tex]\(-3 < x < 3\)[/tex], the interval notation is:
[tex]\[
(-3, 3)
\][/tex]
### Summary:
- The set [tex]\(\{x \mid -3 < x < 3\}\)[/tex] on the number line is graphed with an open circle at [tex]\(-3\)[/tex] and an open circle at [tex]\(3\)[/tex], with the area in between shaded.
- In interval notation, this set is written as [tex]\((-3, 3)\)[/tex].
