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Difference between revisions of "2014 AMC 10A Problems/Problem 7"

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Nonzero real numbers <math>x</math>, <math>y</math>, <math>a</math>, and <math>b</math> satisfy <math>x < a</math> and <math>y < b</math>. How many of the following inequalities must be true?
 
Nonzero real numbers <math>x</math>, <math>y</math>, <math>a</math>, and <math>b</math> satisfy <math>x < a</math> and <math>y < b</math>. How many of the following inequalities must be true?
  
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<math>\textbf{(I)} x+y < a+b\qquad</math>
+
<math>\textbf{(I)}\ x+y < a+b\qquad</math>
  
 
<math>
 
<math>
−
\textbf{(II)} x-y < a-b\qquad</math>
+
\textbf{(II)}\ x-y < a-b\qquad</math>
  
 
<math>
 
<math>
−
\textbf{(III)} xy < ab\qquad</math>
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\textbf{(III)}\ xy < ab\qquad</math>
  
 
<math>
 
<math>
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\textbf{(IV)} \frac{x}{y} < \frac{a}{b}</math>
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\textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}</math>
  
 
<math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 3\qquad\textbf{(E)}\ 4</math>
 
<math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 3\qquad\textbf{(E)}\ 4</math>

Revision as of 11:35, 9 February 2014

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 $\boxed{\textbf{(B) 1}}$

See Also

2014 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 6 		Followed by
Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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