To solve the compound inequality:

\[
6 > \frac{y + 9}{3} > 5
\]

we need to break it into two separate inequalities and solve them step by step:

### Step 1: Solve the first part \(6 > \frac{y + 9}{3}\)

1. Multiply both sides by 3:
   \[
   6 \times 3 > y + 9
   \]
   \[
   18 > y + 9
   \]

2. Subtract 9 from both sides:
   \[
   18 - 9 > y
   \]
   \[
   9 > y \quad \text{or} \quad y < 9
   \]

### Step 2: Solve the second part \(\frac{y + 9}{3} > 5\)

1. Multiply both sides by 3:
   \[
   \frac{y + 9}{3} \times 3 > 5 \times 3
   \]
   \[
   y + 9 > 15
   \]

2. Subtract 9 from both sides:
   \[
   y > 15 - 9
   \]
   \[
   y > 6
   \]

### Combine the Two Inequalities:

The solution must satisfy both parts:

\[
6 < y < 9
\]

### Final Answer:
\[
y \in (6, 9)
\]

Would you like more details on any of these steps, or do you have any questions?

### Related Questions:
1. How do you solve compound inequalities involving different operations?
2. How can inequalities be represented on a number line?
3. What happens if there are no possible solutions for a compound inequality?
4. How would the solution change if the inequalities were non-strict (i.e., \(\geq\) or \(\leq\))?
5. Can you solve compound inequalities involving absolute values?

### Tip:
When solving compound inequalities, remember to treat each part of the inequality separately before combining them to find the range of valid values.

This tutorial explains how to solve the compound inequality 6 > (y + 9) / 3 > 5. Learn step-by-step how to break the inequality into parts, apply multiplication and subtraction rules, and find the solution range. Ideal for students in Grades 8-10.

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