1. **State the problem:** The value $Y$ varies inversely with $x$, meaning $Y$ and $x$ satisfy the relation $Y = \frac{k}{x}$ for some constant $k$. 2. **Given:** $y = 4$ when $x = 9$. We use this to find $k$. 3. **Find $k$:** Using the formula $y = \frac{k}{x}$, substitute $y=4$ and $x=9$: $$4 = \frac{k}{9}$$ Multiply both sides by 9: $$4 \times 9 = \cancel{9} \times \frac{k}{\cancel{9}}$$ $$36 = k$$ 4. **Find $y$ when $x=6$:** Substitute $k=36$ and $x=6$ into the formula: $$y = \frac{36}{6}$$ Simplify: $$y = 6$$ **Final answer:** When $x=6$, $y=6$.