If x is such that 1x -3, then:

1968 AMC 12 Problem 16

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16.

x

is such that

\dfrac1x\lt2

and

\dfrac1x\gt-3,

then:

-\dfrac13\lt x\lt\dfrac12

-\dfrac12\lt x\lt3

x\gt\dfrac12

x\gt\dfrac12

-\dfrac13\lt x\lt0

x\gt\dfrac12

x\lt-\dfrac13

Combine the conditions as

-3\lt\frac{1}{x}\lt2

Treat positive and negative

x

separately when taking reciprocals

For

x\gt0,

\frac{1}{x}\lt2

gives

x\gt\frac{1}{2},

while the other inequality is automatic. For

x\lt0,

\frac{1}{x}\gt-3

gives

x\lt-\frac{1}{3},

while the first is automatic. Thus

x\gt\frac{1}{2}

x\lt-\frac{1}{3}.

Therefore, the correct answer is E.