Difference between revisions of "2014 AMC 10A Problems/Problem 7"
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<math>\textbf{(II)}</math> states that <math>x-y<a-b \implies -3 - (-4) < 1 - 4 \implies 1 < -3</math>.Therefore, <math>\textbf{(II)}</math> is false. | <math>\textbf{(II)}</math> states that <math>x-y<a-b \implies -3 - (-4) < 1 - 4 \implies 1 < -3</math>.Therefore, <math>\textbf{(II)}</math> is false. | ||
+ | <math>\newline</math> | ||
<math>\textbf{(III)}</math> states that <math>xy<ab \implies (-3) \cdot (-4) < 1 \cdot 4 \implies 12 < 4</math>. Therefore, <math>\textbf{(III)}</math> is false. | <math>\textbf{(III)}</math> states that <math>xy<ab \implies (-3) \cdot (-4) < 1 \cdot 4 \implies 12 < 4</math>. Therefore, <math>\textbf{(III)}</math> is false. | ||
+ | <math>\newline</math> | ||
<math>\textbf{(IV)}</math> states that <math>\frac{x}{y}<\frac{a}{b} \implies \frac{-3}{-4} < \frac{1}{4} \implies 0.75 < 0.25</math>. Therefore, <math>\textbf{(IV)}</math> is false. | <math>\textbf{(IV)}</math> states that <math>\frac{x}{y}<\frac{a}{b} \implies \frac{-3}{-4} < \frac{1}{4} \implies 0.75 < 0.25</math>. Therefore, <math>\textbf{(IV)}</math> is false. | ||
− | + | <math>\newline</math> | |
One of our four inequalities is true, hence, our answer is <math>\boxed{\textbf{(B) 1}}</math> | One of our four inequalities is true, hence, our answer is <math>\boxed{\textbf{(B) 1}}</math> | ||
Revision as of 11:26, 7 September 2021
Contents
Problem
Nonzero real numbers ,
,
, and
satisfy
and
. How many of the following inequalities must be true?
Solution
Let us denote where
and
where
. We can write that
.
It is important to note that counterexample fully disproves a claim. Let's try substituting
.
states that
.Therefore,
is false.
states that
. Therefore,
is false.
states that
. Therefore,
is false.
One of our four inequalities is true, hence, our answer is
~MathFun1000 (Majority of Solution)
Video Solution
~savannahsolver
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.