https://file.aiquickdraw.com/mathbot/file/1725387016657yjfhm0k6.jpg

Let's solve the compound inequality given in the image:

\[
-9 < 4x + 3 \leq 11
\]

### Step 1: Solve the first part of the inequality.
\[
-9 < 4x + 3
\]
Subtract 3 from both sides:
\[
-9 - 3 < 4x
\]
\[
-12 < 4x
\]
Divide by 4:
\[
-3 < x
\]
This can also be written as:
\[
x > -3
\]

### Step 2: Solve the second part of the inequality.
\[
4x + 3 \leq 11
\]
Subtract 3 from both sides:
\[
4x \leq 8
\]
Divide by 4:
\[
x \leq 2
\]

### Step 3: Combine the two inequalities.
We now have:
\[
x > -3 \quad \text{and} \quad x \leq 2
\]
This is written as:
\[
-3 < x \leq 2
\]

### Conclusion:
The solution corresponds to the inequality where \( x \) is greater than \(-3\) and less than or equal to \( 2 \).

**The correct answer is option A: \( x > -3 \) and \( x \leq 2 \).**

Would you like more details or have any questions?

Here are five related questions for further practice:
1. How would you solve the inequality \( 2x + 5 \leq 15 \) and \( x - 3 > -1 \)?
2. What steps would you follow to solve the compound inequality \( 1 < 3x - 2 \leq 7 \)?
3. How do you express the solution set of \( -5 < 2x - 1 < 9 \) using interval notation?
4. Can a compound inequality have a solution where no values of \( x \) satisfy both inequalities? Explain with an example.
5. What does it mean if the solution to a compound inequality is all real numbers?

**Tip:** Always remember to consider whether to use "and" or "or" when combining inequalities—this affects whether you're looking for the intersection or union of the solution sets.

Learn how to solve the compound inequality -9 < 4x + 3 ≤ 11 with a detailed step-by-step solution. This problem covers essential algebraic concepts, including solving and combining linear inequalities. Suitable for students in Grades 8-10.

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