Question

Nonzero real numbers $x,$ $y,$ $a,$ and $b$ satisfy $x < a$ and $y < b.$ How many of the following inequalities must be true?

(I) $x + y \lt a + b$

(II) $x - y \lt a - b$

(III) $xy \lt ab$

(IV) $\dfrac{x}{y} \lt \dfrac{a}{b}$

$0$

$1$

$2$

$3$

$4$

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.