We are given the inequality $|x + 6| < 3$.

Step 1: Break the absolute value inequality into two inequalities.

The inequality $|x + 6| < 3$ can be rewritten as: $-3 < x + 6 < 3$

Step 2: Solve the inequalities.

Now, solve for $x$ by subtracting 6 from all parts of the inequality: $-3 - 6 < x + 6 - 6 < 3 - 6$ $-9 < x < -3$

Step 3: Identify the integer values of $x$.

The inequality $-9 < x < -3$ implies that $x$ is an integer between -9 and -3, but not including -9 and -3.

Thus, the possible integer values of $x$ are: $-8, -7, -6, -5, -4$

Step 4: Count the number of integer solutions.

There are 5 integer values that satisfy the inequality.

Final Answer:

The number of different integer values of $x$ that satisfy $|x + 6| < 3$ is $\boxed{5}$.

Would you like any further details on this solution or have any other questions?

Here are 5 related questions:

1. How would the solution change if the inequality were $|x + 6| \leq 3$?
2. What if the inequality was $|x + 6| > 3$?
3. Can you solve inequalities involving absolute values on both sides, such as $|x - 2| < |x + 3|$?
4. How do we handle cases where the absolute value is greater than or equal to a constant?
5. Can you explain how absolute value inequalities relate to distances on a number line?

Tip: When solving absolute value inequalities, always think of them as distance-related problems on the number line!