2003 AMC 12A Problem 6 — Solution | LIVE by Po-Shen Loh

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2003 AMC 12A Problem 6

Below is the professionally curated solution for Problem 6 of the 2003 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 12A solutions, or check the answer key.

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Concepts:absolute value counterexample

Difficulty rating: 1200

6.

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

$x\heartsuit y = y\heartsuit x$ for all $x$ and $y$

$2(x\heartsuit y) = (2x)\heartsuit(2y)$ for all $x$ and $y$

$x\heartsuit 0 = x$ for all $x$

$x\heartsuit x = 0$ for all $x$

$x\heartsuit y \gt 0$ if $x \neq y$

Solution:

Since $x\heartsuit 0=|x-0|=|x|,$ statement (C) claims $|x|=x$ for all $x,$ which fails when $x\lt0.$ For example, $(-1)\heartsuit 0 = 1 \neq -1.$

Every other statement follows directly from properties of the absolute value.

Thus, the correct answer is C.

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