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

## 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 C says that it does for all $x$.

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

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 (Problems • Answer Key • Resources) Preceded byProblem 5 Followed byProblem 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 10 Problems and Solutions
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