To solve the inequality \( 2|x + 9| + 2 \leq 10 \), we will first isolate the absolute value term and then consider the two cases that arise from it.

### Step 1: Isolate the Absolute Value
Subtract \(2\) from both sides:
\[
2|x + 9| \leq 8
\]

Now, divide both sides by \(2\):
\[
|x + 9| \leq 4
\]

### Step 2: Remove the Absolute Value
The inequality \( |x + 9| \leq 4 \) implies that:
\[
-4 \leq x + 9 \leq 4
\]

### Step 3: Solve for \(x\)
Now we can split this compound inequality into two parts and solve each part:

1. **Left Inequality:**
   \[
   -4 \leq x + 9
   \]
   Subtract \(9\) from both sides:
   \[
   x \geq -13
   \]

2. **Right Inequality:**
   \[
   x + 9 \leq 4
   \]
   Subtract \(9\) from both sides:
   \[
   x \leq -5
   \]

### Step 4: Combine the Results
The solution to the inequality is:
\[
-13 \leq x \leq -5
\]

### Solution
In interval notation, the solution set is:
\[
[-13, -5]
\]

This means any \(x\) within this range satisfies the original inequality \( 2|x + 9| + 2 \leq 10 \).

Would you like more details, or do you have any questions about these steps?

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Here are five related questions to deepen your understanding:
1. How would you solve an inequality with a more complex absolute value expression, like \( 3|2x - 1| - 7 \leq 11 \)?
2. What happens if the inequality is strict, such as \( 2|x + 9| + 2 < 10 \)?
3. Can you explain why we need to consider two cases when solving absolute value inequalities?
4. How would the solution change if the inequality were \( 2|x + 9| + 2 \geq 10 \)?
5. What are some practical applications of absolute value inequalities?

**Tip:** Always start by isolating the absolute value term when solving absolute value inequalities. This will simplify the inequality and help you identify the cases to consider.

Discover how to solve the absolute value inequality 2|x + 9| + 2 ≤ 10 with a comprehensive step-by-step solution. This algebra problem explains how to isolate the absolute value term, solve the resulting compound inequality, and express the solution in interval notation. Suitable for students in Grades 8-10.

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