To solve the inequality \( |x + 9| \leq 7 \), we need to break it down into two cases, based on the properties of absolute value inequalities.

### Step 1: Set up the compound inequality
We know that:
\[
|A| \leq B \quad \text{means} \quad -B \leq A \leq B
\]
For this problem, let \( A = x + 9 \) and \( B = 7 \), so we can rewrite the inequality as:
\[
-7 \leq x + 9 \leq 7
\]

### Step 2: Solve the compound inequality
Now, solve for \( x \) by subtracting 9 from all three parts of the inequality:
\[
-7 - 9 \leq x + 9 - 9 \leq 7 - 9
\]
\[
-16 \leq x \leq -2
\]

### Final Solution
The solution to the inequality is:
\[
x \in [-16, -2]
\]

This means that \( x \) can take any value between \( -16 \) and \( -2 \), inclusive.

Do you need any further clarification or details?

Here are some related questions:
1. What is the difference between strict and non-strict inequalities?
2. How would the solution change if the inequality was \( |x + 9| < 7 \)?
3. Can we solve \( |x + 9| \geq 7 \) similarly? How would the solution change?
4. How do we solve absolute value equations, such as \( |x + 9| = 7 \)?
5. What are the properties of absolute value functions in graphing?

**Tip:** Always remember that absolute value inequalities lead to compound inequalities, which need to be solved in two parts!

Learn how to solve the absolute value inequality ∣x+9∣≤7 with a step-by-step breakdown. This problem covers key algebra concepts such as absolute value inequalities and compound inequalities. Suitable for students in Grades 8-10.

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