How many different integer values of x satisfy $|x+6|\lt3$?

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!

How many different integer values of x satisfy $|x+6|<3$?

Discover how to solve the absolute value inequality |x+6|<3 with step-by-step explanations. Learn key algebraic concepts like inequalities and absolute values. Suitable for Grades 6-8, this problem covers critical techniques for solving inequalities.

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