2003 AMC 12A Problem 6

Attempt Problem 6 of the 2003 AMC 12A below, then check your answer against the professionally curated solution 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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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

x \neq y

x\heartsuit 0=|x|

Test a negative value of

x

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.

Problem 6 in other years