2003 AMC 12A Problem 6

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.