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If x is positive, which of the following

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If x is positive, which of the following

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If x is positive, which of the following could be correct ordering of $$\frac{1}{x},\ 2x\ ,\ \ \ \ and\ \ x^2\ ?$$ $$(I)\ \ \ \ x^2<2x<\frac{1}{x}$$ $$(II)\ \ \ x^2<\frac{1}{x}<2x$$ (III) 2x
(A) none
(B) I only
(C) III only
(D) I and II
(E) I, II and III

The OA is the option D.

How can I know the correct answer? Experts, I need your help here.

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GMAT/MBA Expert

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Hello Vjesus12.

First of all, I want to comment that the third option is not displayed. The third option is: $$(III)\ \ \ 2x
Let's try with x=1/2. Then $$\frac{1}{x}=\frac{1}{\frac{1}{2}}=2\ \ \ \ \ \ \ \ \ 2x=2\cdot\frac{1}{2}=1\ \ \ \ \ \ \ \ \ \ x^2=\left(\frac{1}{2}\right)^2=\frac{1}{4}.$$ Therefore, the option (I) is possible.

If x=8/9 then $$\frac{1}{x}=\frac{1}{\frac{8}{9}}=\frac{9}{8}=1.125\ \ \ \ \ \ \ \ \ 2x=2\cdot\frac{8}{9}=\frac{16}{9}=1.777\ \ \ \ \ \ \ \ \ \ x^2=\left(\frac{8}{9}\right)^2=\frac{64}{81}=0.79.$$ Therefore, the option can be possible too.

When x is between 1 and 2, the order is 1/x < x^2 < 2x and when x is greater than 2, the order is 1/x < 2x < x^2, but these two options are no in the list.

So, the option (III) can not be possible.

So, the correct answer is D=(I) and (II).

I hope this explanation may help you.

Feel free to ask me again if you have any doubt.

Regards.

_________________
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