To solve the inequality \(-8 \left| \frac{x}{3} + 1 \right| > -8\), follow these steps:

### Step 1: Divide both sides by -8
Dividing both sides of the inequality by -8 flips the inequality sign (since dividing or multiplying by a negative number reverses the inequality):

\[
\left| \frac{x}{3} + 1 \right| < 1
\]

### Step 2: Remove the absolute value
An absolute value inequality of the form \(\left| A \right| < B\) translates to:

\[
-1 < \frac{x}{3} + 1 < 1
\]

### Step 3: Solve the compound inequality

1. First, subtract 1 from all parts of the inequality:
\[
-2 < \frac{x}{3} < 0
\]

2. Now, multiply through by 3 to eliminate the fraction:
\[
-6 < x < 0
\]

### Final Answer:
The solution to the inequality is:

\[
-6 < x < 0
\]

This means that \(x\) lies between -6 and 0.

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Do you need further clarification or have any other questions?

Here are 5 related questions you might want to explore:

1. How do you solve inequalities involving absolute values with both sides positive?
2. What happens when you multiply or divide inequalities by negative numbers?
3. How can you graph the solution of an absolute value inequality?
4. Can an inequality with absolute value have no solution?
5. What is the difference between strict and non-strict inequalities?

**Tip**: When working with inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number!

Discover how to solve the absolute value inequality -8|x/3 + 1| > -8. This step-by-step guide covers algebraic concepts such as dividing inequalities by negative numbers and handling absolute values. Perfect for students in Grades 8-10.

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