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| − | == Problem ==
| + | #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?
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| − | <math> \mathrm{(A) \ } x \heartsuit y = y \heartsuit x </math> for all <math>x</math> and <math>y</math>
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| − | <math>\mathrm{(B) \ } 2(x \heartsuit y) = (2x) \heartsuit (2y) </math> for all <math>x</math> and <math>y</math>
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| − | <math>\mathrm{(C) \ } x \heartsuit 0 = x </math> for all <math>x</math>
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| − | <math>\mathrm{(D) \ } x \heartsuit x = 0 </math> for all <math>x</math>
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| − | <math> \mathrm{(E) \ } x \heartsuit y > 0 </math> if <math>x \neq y</math>
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| − | == Solution ==
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| − | Examining statement C:
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| − | <math> x \heartsuit 0 = |x-0| = |x| </math>
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| − | <math>|x| \neq x</math> when <math>x<0</math>, but statement D says that it does for all <math>x</math>.
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| − | Therefore the statement that is not true is "<math>x \heartsuit 0 = x</math> for all <math>x</math>" <math>\Rightarrow C</math>
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| − | 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.
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| − | == See Also ==
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| − | {{AMC10 box|year=2003|ab=A|num-b=5|num-a=7}}
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| − | [[Category:Introductory Algebra Problems]] | |
