The inequality presented is:

$|x + 4| > 3$

To solve this, we break it into two cases because the absolute value inequality $|A| > B$ can be written as two inequalities:

$x + 4 > 3 \quad \text{or} \quad x + 4 < -3$

Case 1: $x + 4 > 3$

Solve for $x$: $x > 3 - 4$ $x > -1$

Case 2: $x + 4 < -3$

Solve for $x$: $x < -3 - 4$ $x < -7$

Thus, the solution to the inequality is: $x > -1 \quad \text{or} \quad x < -7$

In interval notation, this solution is: $(-\infty, -7) \cup (-1, \infty)$

Would you like any further details or clarifications?

Here are some related questions you might find useful:

1. How do we solve absolute value inequalities where the inequality sign is $\leq$?
2. What is the difference between strict inequalities $>$ and non-strict inequalities $\geq$?
3. Can absolute value inequalities have solutions that are all real numbers?
4. How do we graph the solution set of an absolute value inequality on a number line?
5. What happens if the inequality is $|x + 4| = 3$ instead of $> 3$?

Tip: When solving absolute value inequalities, always remember to split them into two cases—positive and negative—to find all possible solutions.