1. Let's translate each statement into a mathematical inequality. 2. "The sum of $x$ and $y$ is at most 10" means the sum cannot be greater than 10. This can be written as: $$x + y \leq 10$$ 3. "$y$ is less than twice $x$" means $y$ is strictly less than $2x$: $$y < 2x$$ 4. "$y$ is at least 2 more than \frac{1}{3}x" means $y$ is greater than or equal to $\frac{1}{3}x + 2$: $$y \geq \frac{1}{3}x + 2$$ 5. Summarizing, the system of inequalities is: $$x + y \leq 10$$ $$y < 2x$$ $$y \geq \frac{1}{3}x + 2$$