2003 AMC 10A Problem 6

Problem 6 of 25EasierAlgebra Logic

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

Statement (C) claims

x \heartsuit 0 = x,

but

x \heartsuit 0 = |x - 0| = |x|,

which fails for negative

x.

For example,

-1 \heartsuit 0 = 1 \ne -1.

The remaining statements all follow directly from the properties of absolute value. Thus, the correct answer is C.

Tagged: custom operation · absolute value · counterexample