The problem states that y varies inversely as x, which means the relationship between $y$ and $x$ follows the equation:

$y = \frac{k}{x}$

where $k$ is a constant.

Step 1: Find $k$

We are given that when $x = 6$, $y = 11$. Plugging these values into the equation:

$11 = \frac{k}{6}$

Solving for $k$:

$k = 11 \times 6 = 66$

Step 2: Find $y$ when $x = 9$

Now, using the equation $y = \frac{k}{x}$ with $k = 66$:

$y = \frac{66}{9}$

Simplifying:

$y = \frac{22}{3}$

Final Answer:

$\mathbf{\frac{22}{3}}$

Would you like a more detailed explanation or have any questions?

Here are five related questions you might find useful:

1. What does it mean for a variable to vary inversely?
2. How do you determine the constant $k$ in inverse variation?
3. What are some real-life examples of inverse variation?
4. How is inverse variation different from direct variation?
5. If $y$ varies inversely as $x$ and $x = 4$, how would you find $y$?

Tip:

For inverse variation, as $x$ increases, $y$ decreases proportionally, and vice versa.