To solve the inequality \( |x + 7| < 9 \), we need to break it down into two cases because the absolute value inequality \( |A| < B \) implies that \( -B < A < B \).

### Step 1: Split the inequality

\[
-9 < x + 7 < 9
\]

### Step 2: Solve for \( x \)

Now, subtract 7 from all three parts of the inequality:

\[
-9 - 7 < x < 9 - 7
\]

\[
-16 < x < 2
\]

### Solution:

The solution to the inequality is \( -16 < x < 2 \), which means \( x \) can take any value between -16 and 2 (but not including -16 and 2).

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Would you like more details or have any questions?

Here are five related questions to consider:

1. How do we solve inequalities involving \( \leq \) or \( \geq \) with absolute values?
2. What happens when the inequality symbol changes from \( < \) to \( > \)?
3. How do you graph this solution on a number line?
4. Can you convert absolute value inequalities into compound inequalities in general?
5. How does solving absolute value equations differ from inequalities?

**Tip:** Always check your final solution by substituting values back into the original inequality.

Learn how to solve the absolute value inequality |x+7|<9 with a step-by-step solution. This problem covers key algebraic concepts and theorems, including how to break down absolute value inequalities into compound inequalities. Ideal for students in Grades 8-10.

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