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

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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>
 
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 c and 0 is the absolute value of c, not c itself, because distance is always nonnegative.
+
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 c and 0 is the absolute value of c, not c itself, because distance is always nonnegative.
  
 
== See Also ==
 
== See Also ==

Revision as of 19:54, 19 May 2008

Problem

Define $x \heartsuit y$ to be $|x-y|$ for all real numbers $x$ and $y$. Which of the following statements is not true?

$\mathrm{(A) \ } x \heartsuit y = y \heartsuit x$ for all $x$ and $y$

$\mathrm{(B) \ } 2(x \heartsuit y) = (2x) \heartsuit (2y)$ for all $x$ and $y$

$\mathrm{(C) \ } x \heartsuit 0 = x$ for all $x$

$\mathrm{(D) \ } x \heartsuit x = 0$ for all $x$

$\mathrm{(E) \ } x \heartsuit y > 0$ if $x \neq y$

Solution

Examining statement C:

$x \heartsuit 0 = |x-0| = |x|$

$|x| \neq x$ when $x<0$, but statement D says that it does for all $x$.

Therefore the statement that is not true is "$x \heartsuit 0 = x$ for all $x$" $\Rightarrow C$

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 c and 0 is the absolute value of c, not c itself, because distance is always nonnegative.

See Also

2003 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
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All AMC 10 Problems and Solutions
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