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# If x is a non-zero number, is x^5<x^7?

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Director
Joined: 18 Feb 2019
Posts: 602
Location: India
GMAT 1: 460 Q42 V13
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If x is a non-zero number, is x^5<x^7?  [#permalink]

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24 Mar 2019, 02:27
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Difficulty:

55% (hard)

Question Stats:

58% (02:01) correct 42% (01:52) wrong based on 19 sessions

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If x is a non-zero number, is x^5<x^7?

I. x<x^2
II. 1/x>1/x^2
Math Expert
Joined: 02 Aug 2009
Posts: 8281
Re: If x is a non-zero number, is x^5<x^7?  [#permalink]

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24 Mar 2019, 07:00
If x is a non-zero number, is x^5<x^7?

Let us rework on the initial equation $$x^5<x^7... => ..x^7-x^5>0......x^5(x^2-1)>0$$
So, two cases..
A. Both positive .. so $$x^5>0$$, that is $$x>0$$, then $$x^2-1>0$$, that is $$x^2>1...x<-1$$ or $$x>1$$.... so if x>0, then x>1
A. Both negative .. so $$x^5<0$$, that is $$x<0$$, then $$x^2-1<0$$, that is $$x^2<1...-1<x<1$$ ]..... so if x<0, then -1<x<0

I. x<x^2
$$x^2-x>0....x(x-1)>0$$... so if x>0, then x>1 OR if x<0, then x<1....
So, if x>1, it is true, but the value of x<-1, say x=-2 will satisfy $$x<x^2$$ but then $$x^5>x^7$$

II. 1/x>1/x^2
Now $$x^2$$ is always positive as xis not 0, so we can multiply both sides by $$x^2$$..
$$\frac{x^2}{x}>\frac{x^2}{x^2}........x>1$$
Thus $$x^5<x^7$$
Sufficient

B
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Re: If x is a non-zero number, is x^5<x^7?   [#permalink] 24 Mar 2019, 07:00
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