To solve this problem, we need to graph the set [tex]\(\{x \mid -5 < x \leq -1\}\)[/tex] and write it in interval notation.
### Step 1: Understand the Inequality
The given inequality [tex]\(-5 < x \leq -1\)[/tex] describes all values of [tex]\(x\)[/tex] that are greater than [tex]\(-5\)[/tex] and less than or equal to [tex]\(-1\)[/tex].
### Step 2: Graph the Inequality on a Number Line
1. Identify the endpoints:
- The left endpoint is [tex]\(-5\)[/tex].
- The right endpoint is [tex]\(-1\)[/tex].
2. Determine the type of endpoints:
- For the endpoint [tex]\(-5\)[/tex], the inequality is [tex]\(x > -5\)[/tex], meaning [tex]\(-5\)[/tex] is not included in the set. We represent this on the number line with an open circle at [tex]\(-5\)[/tex].
- For the endpoint [tex]\(-1\)[/tex], the inequality is [tex]\(x \leq -1\)[/tex], meaning [tex]\(-1\)[/tex] is included in the set. We represent this on the number line with a closed circle at [tex]\(-1\)[/tex].
3. Draw the number line:
- Place an open circle at [tex]\(-5\)[/tex].
- Place a closed circle at [tex]\(-1\)[/tex].
- Shade the region between [tex]\(-5\)[/tex] and [tex]\(-1\)[/tex], indicating all values of [tex]\(x\)[/tex] within this interval.
Here is a visual representation of the graph:
```
<----|----|----|----|----|----|----|----|-->
-7 -6 -5 -4 -3 -2 -1 0 1 2 3
o========================•
```
- The open circle [tex]\(`o`\)[/tex] at [tex]\(-5\)[/tex] indicates that [tex]\(-5\)[/tex] is not included.
- The closed circle [tex]\(`•`\)[/tex] at [tex]\(-1\)[/tex] indicates that [tex]\(-1\)[/tex] is included.
- The shaded region between [tex]\(-5\)[/tex] and [tex]\(-1\)[/tex] shows all values in the set.
### Step 3: Write the Set Using Interval Notation
In interval notation, we express the set as:
[tex]\[
(-5, -1]
\][/tex]
- The open parenthesis [tex]\((\)[/tex] at [tex]\(-5\)[/tex] indicates that [tex]\(-5\)[/tex] is not included in the interval.
- The closed bracket [tex]\([\)[/tex] at [tex]\(-1\)[/tex] indicates that [tex]\(-1\)[/tex] is included in the interval.
Thus, the set [tex]\(\{x \mid -5 < x \leq -1\}\)[/tex] is represented in interval notation as [tex]\((-5, -1]\)[/tex].
