Question

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 $x \neq y$

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.