1980 AHSME Problems/Problem 6

Problem

A positive number $x$ satisfies the inequality $\sqrt{x} < 2x$ if and only if

$\text{(A)} \ x > \frac{1}{4} \qquad \text{(B)} \ x > 2 \qquad \text{(C)} \ x > 4 \qquad \text{(D)} \ x < \frac{1}{4}\qquad \text{(E)} \ x < 4$

Solution

$\sqrt{x}< 2x \\ x < 4x^2 \\ 0 < x(4x-1) \\ 0 < 4x-1 \\ 1 < 4x \\ x >\frac{1}{4} \\ \boxed{(A)}$

Note: You can also draw a rough sketch.

See also

1980 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
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