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Let's solve each of the inequalities shown in the image you uploaded and graph their solutions.

### Problem 1: \( 6(1 - 6p) > 38 - 4p \)
Expand and simplify the inequality:
\[ 6 - 36p > 38 - 4p \]
Add \(36p\) and subtract \(6\) from both sides:
\[ -36p + 36p - 4p < 38 - 6 \]
\[ -4p < 32 \]
Divide by \(-4\) (remember to flip the inequality):
\[ p > -8 \]

### Problem 2: \( -7n + 40 \leq 2(1 - 7n) + 3 \)
Expand and simplify the inequality:
\[ -7n + 40 \leq 2 - 14n + 3 \]
\[ -7n + 40 \leq -14n + 5 \]
Add \(14n\) to both sides:
\[ 7n + 40 \leq 5 \]
Subtract \(40\) from both sides:
\[ 7n \leq -35 \]
Divide by \(7\):
\[ n \leq -5 \]

### Problem 3: \( -5(8 + 5r) \geq -40 + 5r \)
Expand and simplify:
\[ -40 - 25r \geq -40 + 5r \]
Add \(25r\) to both sides:
\[ -40 \geq -40 + 30r \]
Add \(40\) to both sides:
\[ 0 \geq 30r \]
Divide by \(30\):
\[ r \leq 0 \]

### Problem 4: \( 7(3 - 4p) > 21 + 2p \)
Expand and simplify:
\[ 21 - 28p > 21 + 2p \]
Subtract \(21\) and add \(28p\) to both sides:
\[ -28p + 28p + 2p < 0 \]
\[ 2p < 0 \]
Divide by \(2\):
\[ p < 0 \]

### Problem 5: Solve compound inequality and graph the solution
\[ n + 5 \geq 6 \quad \text{and} \quad -3 + n \leq 1 \]
First inequality:
\[ n \geq 1 \]
Second inequality:
\[ n \leq 4 \]
Combined solution:
\[ 1 \leq n \leq 4 \]

### Problem 6: Solve compound inequality and graph the solution
\[ 9 + n \geq -1 \quad \text{and} \quad 6n \leq -36 \]
First inequality:
\[ n \geq -10 \]
Second inequality:
\[ n \leq -6 \]
Combined solution:
\[ -10 \leq n \leq -6 \]

These solutions match the constraints given in each problem. If you have any further questions or need details on the steps, feel free to ask. Here are some follow-up topics you might consider:

1. How do the properties of inequalities affect the steps taken to solve them?
2. What are the effects of multiplying or dividing by a negative number in an inequality?
3. How can you check if a solution to an inequality is correct?
4. Why is it important to consider all terms in an inequality during simplification?
5. What are the implications of having a variable on both sides of an inequality?

**Tip:** When solving inequalities, always remember to flip the inequality sign when multiplying or dividing both sides by a negative number. This is crucial for arriving at the correct solution.

This page walks through solving multiple linear and compound inequalities with clear, step-by-step instructions. Learn how to use the distributive property, solve for variables, and graph solutions on a number line. Appropriate for Grades 8-10.

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