The inequality you're given is:

$|x + 4| < 9$

To solve this, we break it down into two separate inequalities since the absolute value $|A| < B$ can be rewritten as:

$- B < A < B$

In this case, $A = x + 4$ and $B = 9$. Therefore, we can write:

$-9 < x + 4 < 9$

Now, solve for $x$ by subtracting 4 from all parts of the inequality:

$-9 - 4 < x < 9 - 4$

$-13 < x < 5$

So, the solution to the inequality is:

$x \in (-13, 5)$

Would you like more details or have any questions?

Here are five related questions that expand on this concept:

1. How would the solution change if the inequality were $|x + 4| \leq 9$?
2. What if the inequality was $|x - 4| < 9$?
3. How do you solve $|x + 4| > 9$?
4. What is the geometric interpretation of the inequality $|x + 4| < 9$?
5. How would you solve $|2x + 4| < 9$?

Tip: Remember, absolute value inequalities can be split into two cases, which helps solve them easily!