Sure, let's break this down step by step.
1. Understanding the given set notation:
The given set is:
[tex]\[
\{ x \mid -6 < x < 3 \}
\][/tex]
This means we are considering all real numbers [tex]\(x\)[/tex] that are greater than [tex]\(-6\)[/tex] and less than [tex]\(3\)[/tex].
2. Converting to interval notation:
In interval notation, the open interval [tex]\( (-6, 3) \)[/tex] indicates that [tex]\(-6\)[/tex] and [tex]\(3\)[/tex] are not included in the set. The parentheses denote that the endpoints are excluded.
Therefore, the interval notation is:
[tex]\[
(-6, 3)
\][/tex]
3. Graphing the interval on a number line:
- Draw a horizontal number line.
- Place open circles (indicating that the points are not included) at [tex]\(-6\)[/tex] and [tex]\(3\)[/tex].
- Shade the region between [tex]\(-6\)[/tex] and [tex]\(3\)[/tex] to represent all numbers greater than [tex]\(-6\)[/tex] and less than [tex]\(3\)[/tex].
Here is a sketch of the number line representation:
```
-7 -6 0 3 4
|-- (--========--)------|
```
In the diagram above:
- The parentheses `(` and `)` represent the open circles at [tex]\(-6\)[/tex] and [tex]\(3\)[/tex].
- The shaded region `========` between [tex]\(-6\)[/tex] and [tex]\(3\)[/tex] shows all the [tex]\(x\)[/tex] values that satisfy [tex]\(-6 < x < 3\)[/tex].
Through these steps, we have successfully graphed the set and written the interval notation for the given set:
[tex]\[
(-6, 3)
\][/tex]
