2014 AMC 10A Problem 7

Adding the two inequalities together gets us $$x + y \lt a + b,$$ which shows that (I) is correct. One cannot subtract inequalities, which means that (II) is not necessarily true. Consider $x = 1,$ $y = 1,$ $a = 2,$ and $b = 3$ as a counter-example. This would give us $0 \lt -1.$ (III) is also not always true, since $x$ and $y$ might be negative numbers. Let $x = -3,$ $y = -2,$ $a = 1,$ and $b = 1.$ Then $xy = 6$ and $ab = 1$ which shows that (III) is wrong. The same thing occurs with (IV) . Using the same values as above, we have $\dfrac{x}{y} = 1.5$ and $\dfrac{a}{b} = 1.$ This shows that (I) is the only true statement. Thus, B is the correct answer.