Difference between revisions of "2003 AMC 12A Problems/Problem 6"

Revision as of 14:20, 23 August 2009

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 $x \neq y$

Examining statement C:

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

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

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

2003 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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 12 Problems and Solutions

@@ Line 17: / Line 17: @@
 <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>.
+<math>|x| \neq x</math> when <math>x<0</math>, but statement C 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>
 == See Also ==
-*[[2003 AMC 12A Problems]]
+{{AMC12 box|year=2003|ab=A|num-b=5|num-a=7}}
-*[[2003 AMC 12A/Problem 5|Previous Problem]]
-*[[2003 AMC 12A/Problem 7|Next Problem]]
 [[Category:Introductory Algebra Problems]]