2014 AMC 10A Problems/Problem 7

Problem

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

$\textbf{(I)} x+y < a+b\qquad$

$\textbf{(II)} x-y < a-b\qquad$

$\textbf{(III)} xy < ab\qquad$

$\textbf{(IV)} \frac{x}{y} < \frac{a}{b}$

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 3\qquad\textbf{(E)}\ 4$ (Error compiling LaTeX. Unknown error_msg)

Solution

Clearly, $\text{(I)}$ must be true (do you see why?)

Consider $x=-2013, a=1, y=-2013, b=1$ . Clearly, we have $x<a$ and $y<b$ . Note that $\text{(II), (III), }$ and $\text{(IV)}$ are false, so our answer is $\boxed{\textbf{(B) 1}}$

2014 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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All AMC 10 Problems and Solutions

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2014 AMC 10A Problems/Problem 7

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