2003 AMC 12A Problem 6

Problem 6 of 25EasierAlgebra

Define

x\heartsuit y

to be

|x - y|

for all real numbers

x

and

y.

Which of the following statements is not true?

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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.

Concepts: absolute value · counterexample

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