https://file.aiquickdraw.com/mathbot/file/1729226400921u77kwxay.jpg

The problem presented is an absolute value inequality:

\[
\frac{|x + 6|}{8} \leq 1
\]

To solve this, follow these steps:

### Step 1: Eliminate the denominator.
Multiply both sides of the inequality by 8 to get rid of the denominator:

\[
|x + 6| \leq 8
\]

### Step 2: Split into two inequalities.
Since we are dealing with an absolute value, we split this into two separate inequalities:

\[
-8 \leq x + 6 \leq 8
\]

### Step 3: Solve for \(x\).
Now, solve for \(x\) by subtracting 6 from all parts of the inequality:

\[
-8 - 6 \leq x \leq 8 - 6
\]

\[
-14 \leq x \leq 2
\]

### Final Answer:
The solution is:

\[
x \geq -14 \quad \text{and} \quad x \leq 2
\]

So, the solution in interval notation would be:

\[
x \in [-14, 2]
\]

The boxes in the image would be filled with:

- \(x \leq 2\) for the first blank.
- \(x \geq -14\) for the second blank.

Would you like more details or have any further questions?

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Here are 5 related questions for practice:
1. How do you solve an absolute value inequality like \(|2x - 3| \leq 5\)?
2. What happens if the inequality is strict, for example \(\frac{|x + 3|}{2} < 4\)?
3. How do you solve absolute value inequalities when the inequality symbol is flipped, such as \(\frac{|x - 4|}{5} > 3\)?
4. What is the graphical interpretation of the solution to an absolute value inequality?
5. Can you explain how to solve an absolute value inequality with two absolute values, such as \(|x - 1| + |x + 2| \leq 7\)?

**Tip:** When solving absolute value inequalities, always split the inequality into two cases and remove the absolute value bars!

Solve the absolute value inequality: |x + 6| / 8 ≤ 1

Learn how to solve the absolute value inequality |x + 6| / 8 ≤ 1 with a step-by-step solution. This problem focuses on key algebraic concepts such as absolute value properties, splitting inequalities, and solving for x.

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