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.