Difference between revisions of "2014 AMC 10A Problems/Problem 7"
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Also, with some intuition, we could have plugged <math>0=X</math>, <math>1=A</math>, <math>-3=Y</math>, and <math>-2=B</math> and then plugged these values into the equations to see which ones held. | Also, with some intuition, we could have plugged <math>0=X</math>, <math>1=A</math>, <math>-3=Y</math>, and <math>-2=B</math> and then plugged these values into the equations to see which ones held. | ||
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==See Also== | ==See Also== |
Revision as of 14:26, 26 January 2015
Contents
Problem
Nonzero real numbers ,
,
, and
satisfy
and
. How many of the following inequalities must be true?
$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 3\qquad\textbf{(E)}\ 4$ (Error compiling LaTeX. Unknown error_msg)
Solution
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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