https://file.aiquickdraw.com/mathbot/file/1728994460821uxvn9g6f.jpg

The problem in the image is asking to solve the absolute value inequality:

\[
\frac{|x + 8|}{7} \geq 1
\]

### Step 1: Remove the fraction
To eliminate the fraction, multiply both sides of the inequality by 7:

\[
|x + 8| \geq 7
\]

### Step 2: Solve the absolute value inequality
The general rule for solving inequalities of the form \( |A| \geq B \) is that:

\[
A \geq B \quad \text{or} \quad A \leq -B
\]

In this case, \( A = x + 8 \) and \( B = 7 \). So, we split the inequality into two cases:

1. \( x + 8 \geq 7 \)
2. \( x + 8 \leq -7 \)

### Step 3: Solve each inequality

1. \( x + 8 \geq 7 \)
   \[
   x \geq 7 - 8
   \]
   \[
   x \geq -1
   \]

2. \( x + 8 \leq -7 \)
   \[
   x \leq -7 - 8
   \]
   \[
   x \leq -15
   \]

### Final Answer:
The solution to the inequality is:
\[
x \geq -1 \quad \text{or} \quad x \leq -15
\]

Thus, the answer is:

- \( x \geq -1 \) and \( x \leq -15 \).

Would you like any further details or clarification?

### Here are some related questions to practice:

1. How do you solve absolute value inequalities involving "<"?
2. Can absolute value inequalities have no solution?
3. What is the difference between strict inequality (\( < \) or \( > \)) and inclusive inequality (\( \leq \) or \( \geq \))?
4. How do you graph the solution of an absolute value inequality on a number line?
5. Can absolute value inequalities involve quadratic expressions?

**Tip:** When solving absolute value inequalities, always remember to split the inequality into two cases: one for the positive and one for the negative.

Solve the following absolute value inequality: |x + 8| / 7 ≥ 1.

Learn how to solve the absolute value inequality |x + 8| / 7 ≥ 1 with step-by-step instructions. This problem covers key concepts like absolute value inequalities and solving inequalities with fractions. Ideal for students in Grades 8-10.

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