Certainly! Let's first graph the set \(\{x \mid -7 \leq x \leq 5\}\) on a number line, and then we'll express it using interval notation.
### Step-by-Step Solution:
1. Understanding the Set:
The given set is \(\{x \mid -7 \leq x \leq 5\}\), which means we are looking at all values of \(x\) between \(-7\) and \(5\), including the endpoints \(-7\) and \(5\).
2. Graphing on the Number Line:
To graph the set on the number line, we need to mark and shade the portion of the number line from \(-7\) to \(5\).
- Draw a horizontal line to represent the number line.
- Mark the points \(-7\) and \(5\) on this line.
- Shade the region between \(-7\) and \(5\) to indicate all the points \(x\) such that \(-7 \leq x \leq 5\).
- Use a solid dot or circle at \(-7\) and \(5\) to show that these endpoints are included in the set.
Here's how it would look like graphically:
```
-10 -8 -6 -4 -2 0 2 4 6 8 10
|------|------|------|------|------|------|------|------|------|------|
•------------------------------------•
-7 5
```
3. Interval Notation:
Interval notation is a concise way of writing the set of numbers between two endpoints.
Since \(-7\) and \(5\) are included in the set, we use square brackets \([ \ ]) to denote that the endpoints are included.
Therefore, the interval notation for the set \(\{x \mid -7 \leq x \leq 5\}\) is:
[tex]\[
[-7, 5]
\][/tex]
### Summary
- Graph:
On a number line, shade the portion between \(-7\) and \(5\), including the endpoints.
- Interval Notation:
[tex]\[
[-7, 5]
\][/tex]
This concludes the steps to graph the set and write it in interval notation.
