The image shows the compound inequality:

$-5 < x + 2 \quad \text{and} \quad x + 4 \leq 9$

Let's solve each part of the inequality separately:

1. First Inequality: $-5 < x + 2$ Subtract 2 from both sides: $-5 - 2 < x$ Simplifying: $-7 < x \quad \text{or} \quad x > -7$

2. Second Inequality: $x + 4 \leq 9$ Subtract 4 from both sides: $x \leq 9 - 4$ Simplifying: $x \leq 5$

Combine the results:

The solution is: $-7 < x \leq 5$

Final Answer:

The solution in interval notation is: $(-7, 5]$

Would you like further details or explanations on this? Here are some related questions:

1. How do we graph the solution of this inequality on a number line?
2. What happens if both parts of the compound inequality have "or" instead of "and"?
3. How do we solve compound inequalities involving multiplication or division by negative numbers?
4. What are some examples of compound inequalities where the variable appears more than once?
5. How can we check if a specific value of $x$ satisfies a compound inequality?

Tip: Always pay attention to the direction of the inequality sign when adding or subtracting values from both sides of an inequality.