1. The problem states: When 9 is increased by 3x, the result is greater than 36. We need to find the least possible integer value for $x$. 2. Write the inequality based on the problem: $$9 + 3x > 36$$ 3. To isolate $x$, subtract 9 from both sides: $$9 + 3x - \cancel{9} > 36 - \cancel{9}$$ $$3x > 27$$ 4. Now divide both sides by 3 to solve for $x$: $$\frac{3x}{\cancel{3}} > \frac{27}{\cancel{3}}$$ $$x > 9$$ 5. Since $x$ must be greater than 9, the least possible integer value for $x$ is 10. **Final answer:** $$\boxed{10}$$