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

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Senior Manager
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Joined: 18 Feb 2019
Posts: 281
Location: India
GMAT 1: 490 Q47 V13
GPA: 3.6
If x is a non-zero number, is x^5<x^7?  [#permalink]

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New post 24 Mar 2019, 02:27
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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
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Joined: 02 Aug 2009
Posts: 7567
Re: If x is a non-zero number, is x^5<x^7?  [#permalink]

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New post 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

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

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