A region in the coordinate plane is defined by x ≥ 0, y ≥ 0, and x + y ≤ 10. Which statement must be true for every point (x, y) in this…

Every point in the region has $y \geq 0$ and $x + y \leq 10$. Since $y$ is nonnegative, $x = (x + y) - y \leq x + y \leq 10$. So $x \leq 10$ for every point, choice B. Choice A ($x = 10$) is true only at some points, such as $(10, 0)$, but not at $(0, 0)$ or $(1, 1)$. Choice C ($y \geq 5$) fails at $(1, 1)$. Choice D ($x + y = 10$) describes only the boundary line, while the region also includes interior points such as $(1, 1)$ with $x + y = 2$. A statement must hold for every point, so one counterexample eliminates a choice.