A positive number x satisfies the inequality √ x<2x if and only if

1980 AMC 12 Problem 6

Attempt Problem 6 of the 1980 AMC 12 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 1980 AMC 12 solutions, or check the answer key.

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A positive number

x

satisfies the inequality

\sqrt x\lt2x

if and only if

x\gt\frac14

x\gt2

x\gt4

x\lt\frac14

x\lt4

Use the fact that both sides are positive

Divide by

\sqrt x

before squaring or isolating

x

Because

x\gt0,

division by

\sqrt x

preserves the inequality:

1\lt2\sqrt x.

Thus

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

which is equivalent to

x\gt\frac{1}{4}.

Therefore, the correct answer is A.