For which nonzero real numbers x is (|x-|x||)/(x) a positive integer?

(E) for no nonzero real numbers x

(D) for all nonzero real numbers x

1975 AMC 12 Problem 7

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For which nonzero real numbers

x

\frac{|x-|x||}{x}

a positive integer?

for negative

x

only

for positive

x

only

only for

x

an even integer

for all nonzero real numbers

x

for no nonzero real numbers

x

Evaluate the quotient separately for

x\gt0

and

x\lt0

For

x\lt0,

replace

|x|

-x

before evaluating the outer absolute value

x\gt0,

the quotient is

0.

x\lt0,

then

x-|x|=2x\lt0,

|x-|x||=-2x

and the quotient is

-2.

Neither value is a positive integer.

Therefore, the correct answer is E.