Difference between revisions of "2014 AMC 10A Problems/Problem 7"
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\textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}</math> | \textbf{(IV)}\ \frac{x}{y} < \frac{a}{b}</math> | ||
− | <math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D) | + | <math> \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4</math> |
==Solution== | ==Solution== |
Revision as of 19:12, 23 April 2015
Contents
[hide]Problem
Nonzero real numbers , , , and satisfy and . How many of the following inequalities must be true?
Solution
First, we note that must be true by adding our two original inequalities.
Though one may be inclined to think that must also be true, it is not, for we cannot subtract inequalities.
In order to prove that the other inequalities are false, we only need to provide one counterexample. Let's try substituting
states that Since this is false, must also be false.
states that . This is also false, thus is false.
states that . This is false, so is false.
One of our four inequalities is true, hence, our answer is
Solution 2
Also, with some intuition, we could have plugged , , , and and then plugged these values into the equations to see which ones held.
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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