Difference between revisions of "2003 AMC 10A Problems/Problem 6"

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== Problem ==
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#REDIRECT[[2003 AMC 12A Problems/Problem 6]]
Define <math>x \heartsuit y</math> to be <math>|x-y|</math> for all real numbers <math>x</math> and <math>y</math>. Which of the following statements is not true?
 
 
 
<math> \mathrm{(A) \ } x \heartsuit y = y \heartsuit x </math> for all <math>x</math> and <math>y</math>
 
 
 
<math>\mathrm{(B) \ } 2(x \heartsuit y) = (2x) \heartsuit (2y) </math> for all <math>x</math> and <math>y</math>
 
 
 
<math>\mathrm{(C) \ } x \heartsuit 0 = x </math> for all <math>x</math>
 
 
 
<math>\mathrm{(D) \ } x \heartsuit x = 0 </math> for all <math>x</math>
 
 
 
<math> \mathrm{(E) \ } x \heartsuit y > 0 </math> if <math>x \neq y</math>
 
 
 
== Solution ==
 
Examining statement C:
 
 
 
<math> x \heartsuit 0 = |x-0| = |x| </math>
 
 
 
<math>|x| \neq x</math> when <math>x<0</math>, but statement D says that it does for all <math>x</math>.
 
 
 
Therefore the statement that is not true is "<math>x \heartsuit 0 = x</math> for all <math>x</math>" <math>\Rightarrow C</math>
 
 
 
Alternatively, consider that the given "heart function" is actually the definition of the distance between two points.  Examining all of the statements, only C is not necessarily true; if c is negative, the distance between <math>c</math> and <math>0</math> is the absolute value of <math>c</math>, not <math>c</math> itself, because distance is always nonnegative.
 
 
 
== See Also ==
 
{{AMC10 box|year=2003|ab=A|num-b=5|num-a=7}}
 
 
 
[[Category:Introductory Algebra Problems]]
 

Latest revision as of 14:44, 30 July 2011

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