Problem: A positive number xxx satisfies the inequality x<2x\sqrt{x}<2 xx​<2x if and only if
Answer Choices:
A. x>14x>\dfrac{1}{4}x>41​
B. x>2x>2x>2
C. x>4x>4x>4
D. x<14x<\dfrac{1}{4}x<41​
E. x<4x<4x<4
Solution:
Since xxx is positive the following are equivalent:
x<2xx<4x21<4x14<xx>14.\begin{aligned} \sqrt{x} & <2 x \\ x & <4 x^{2} \\ 1 & <4 x \\ \dfrac{1}{4} & <x \\ x & >\dfrac{1}{4} . \end{aligned} x​x141​x​<2x<4x2<4x<x>41​.​