Let's solve the compound inequality $-9 < 4x + 3 \leq 11$ step by step.

Step 1: Solve the first part of the inequality $-9 < 4x + 3$.

1. Subtract 3 from both sides: $-9 - 3 < 4x$ $-12 < 4x$

2. Divide both sides by 4: $\frac{-12}{4} < x$ $-3 < x$ This can be rewritten as $x > -3$.

Step 2: Solve the second part of the inequality $4x + 3 \leq 11$.

1. Subtract 3 from both sides: $4x \leq 11 - 3$ $4x \leq 8$

2. Divide both sides by 4: $x \leq \frac{8}{4}$ $x \leq 2$

Step 3: Combine the results.

From the first part, we have $x > -3$, and from the second part, we have $x \leq 2$.

The combined solution is $-3 < x \leq 2$, or equivalently $x > -3$ and $x \leq 2$.

Conclusion:

The correct answer is Option A: $x > -3$ and $x \leq 2$.

Would you like further details or have any questions? Here are some related questions you might find useful:

1. How do you solve inequalities with more than two parts?
2. What are the key differences between "and" and "or" in compound inequalities?
3. How do you represent solutions to inequalities on a number line?
4. How does solving inequalities differ from solving equations?
5. What are some common mistakes to avoid when solving compound inequalities?

Tip: When solving compound inequalities, always pay attention to the direction of the inequality signs when multiplying or dividing by negative numbers.