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!