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The problem asks to solve the following compound inequalities for \( m \):

\[
\frac{m + 52}{6} > 11 \quad \text{or} \quad \frac{51m - 86}{7} < -4
\]

Let's solve each inequality step by step.

### First Inequality: 
\[
\frac{m + 52}{6} > 11
\]

1. Multiply both sides by 6 to eliminate the denominator:
   \[
   m + 52 > 66
   \]

2. Subtract 52 from both sides:
   \[
   m > 14
   \]

### Second Inequality:
\[
\frac{51m - 86}{7} < -4
\]

1. Multiply both sides by 7 to eliminate the denominator:
   \[
   51m - 86 < -28
   \]

2. Add 86 to both sides:
   \[
   51m < 58
   \]

3. Divide both sides by 51:
   \[
   m < \frac{58}{51} \quad \text{or approximately} \quad m < 1.137
   \]

### Final Compound Inequality:
\[
m > 14 \quad \text{or} \quad m < \frac{58}{51}
\]

This is the solution for the compound inequality. If you have any questions or want additional details, feel free to ask!

Here are 5 related questions to further practice:
1. How do you solve an inequality with fractions?
2. What are the steps to solve a compound inequality?
3. How do you graph a compound inequality on a number line?
4. What is the difference between "and" and "or" in compound inequalities?
5. How do you interpret the solution set for a compound inequality?

**Tip:** Always check the boundary points to ensure they satisfy the original inequality!

Solve for m: (m + 52) / 6 > 11 or (51m - 86) / 7 < -4. Write your answer as a compound inequality with integers, proper fractions, or fractions in simplest form.

This page explains how to solve compound inequalities with fractions, including step-by-step solutions for (m + 52) / 6 > 11 or (51m - 86) / 7 < -4. Learn key algebra concepts and solve inequalities involving fractions. Suitable for Grades 8-10.

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