Question

Solve the compound inequality.
negative 7 less than 2 x plus 1 less than or equals 7
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solution set is
  
enter your response here. ​(Type your answer in interval​ notation.)
B.
The solution set is the empty set.

Answer 1

To solve the compound inequality \(-7 < 2x + 1 \leq 7\), we will break it down into two parts:

1. Solve the first part: \(-7 < 2x + 1\)
2. Solve the second part: \(2x + 1 \leq 7\)

### Step 1: Solve \(-7 < 2x + 1\)

Subtract 1 from both sides:
\[
-8 < 2x
\]

Now, divide by 2:
\[
-4 < x
\]
or
\[
x > -4
\]

### Step 2: Solve \(2x + 1 \leq 7\)

Subtract 1 from both sides:
\[
2x \leq 6
\]

Now, divide by 2:
\[
x \leq 3
\]

### Combining the Results

Now we can combine both results from Step 1 and Step 2:
\[
-4 < x \leq 3
\]

In interval notation, this is expressed as:
\[
(-4, 3]
\]

### Conclusion

So, the solution set is:
\[
(-4, 3]
\]

**Part 1 Answer:**

A. The solution set is \( (-4, 3] \).