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Is x negative? [#permalink]
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17 Feb 2013, 12:16
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Is x negative? (1) At least one of x and x^2 is greater than x^3. (2) At least one of x^2 and x^3 is greater than x.
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Re: Is x negative? At least one of x and x^2 is greater x^3 [#permalink]
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17 Feb 2013, 14:50
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If x > 1, then it is always true that x < x^2 < x^3. If 0 < x < 1, then it is always true that x^3 < x^2 < x. From the above, you can see that neither statement is sufficient alone, since in each case, x can be positive. Notice from the above that if x is positive, x^2 is never the largest of the three expressions x, x^2 and x^3. Since Statement 1 guarantees that x^3 is not the largest of the three expressions, and Statement 2 guarantees that x is not the largest of the three expressions, then using both statements, the only possibility is that x^2 is the largest of the three expressions. Since that can't happen when x is positive, x must be negative, and the answer is C.
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Is x negative? [#permalink]
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12 Jul 2016, 08:25
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Just draw out the number line and write out the order of the three functions in each region:
\(x^3<x<x^2 \) \(x<x^3<x^2\) \(x^3<x^2<x\) \(x<x^2<x^3\) <> 1 0 1
(1) At least one of x and x^2 is greater than x^3.
This can be true in regions 1,2, and 3, x can be positive or negative. INSUFFICIENT
(2) At least one of x^2 and x^3 is greater than x.
This can be true in regions 1,2, and 4, x can be positive or negative. INSUFFICIENT
Taking the two together, this is true in regions 1 and 2, which means x is negative. SUFFICIENT.
Answer: C.



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Re: Is x negative? [#permalink]
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13 Jul 2014, 05:32



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Re: Is x negative? [#permalink]
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04 Feb 2014, 07:28
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PraPon wrote: Is x negative? (1) At least one of x and x^2 is greater than x^3. (2) At least one of x^2 and x^3 is greater than x. I second the approach by IanStewart although I followed a slightly different approach Statement 1 X could be either a fraction or a negative number Statement 2 X could be either a positive or a negative number Statement 1 and 2 together X has to be a negative number C Just my 2c Cheers J



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Re: Is x negative? [#permalink]
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07 May 2016, 13:36
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I documented the behavior of a variable in different regions. Uploading its image as I think it would be helpful. Although, I believe, memorizing how every power of 'x' behaves in each region would be pointless, noticing the patterns such as the ones mentioned below would be useful.  The behavior of odd powers of 'x' in the region " 1 < x < 0" is exactly same as that of even powers in the region "x < 1"  The behavior of even powers of 'x' in the region " 1 < x < 0" is exactly same as that of odd powers in the region "x < 1"
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Behavior of 'X' in different regions.png [ 31.35 KiB  Viewed 2168 times ]



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Is x negative? [#permalink]
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13 Jul 2014, 02:30
Can someone please explain me what "at least one of" means in the statements? I really have no idea how to convert this sentence into an equation.



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Re: Is x negative? [#permalink]
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05 Aug 2014, 23:43
PrashantPonde wrote: Is x negative? (1) At least one of x and x^2 is greater than x^3. (2) At least one of x^2 and x^3 is greater than x. Thanks for the question. C is the answer. (1) x <1 for (1) to happen (2) x<0 or x>1 for (2) to happen Combine (1) and (2), x<0 , so C is the answer. (You can solve it by doing a little bit algebra x^2> x^3 > x^2  x^3>0 > x^2 (1x)> 0, so x<1)
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Re: Is x negative? [#permalink]
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09 Feb 2016, 12:27
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Re: Is x negative? [#permalink]
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09 Feb 2016, 16:53
Can somebody please explain how to solve these inequalities in detail? I am not getting the desired answer after solving the equations.



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Is x negative? [#permalink]
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13 Mar 2016, 06:50
KbSharma wrote: Can somebody please explain how to solve these inequalities in detail? I am not getting the desired answer after solving the equations. for statement(1) consider 2 cases x > x^2 > x^31/2 > 1/4 > 1/8 & 2 > 4 > 8. for statement(2) consider 2 cases x < x^2 < x^31/2 < 1/4 < 1/8 & 2 < 4 < 8. combining both statements we see ve values from all 4 cases satisfying that x is ve.
Last edited by rohit8865 on 09 Jul 2016, 00:44, edited 2 times in total.



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Is x negative? [#permalink]
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09 Jul 2016, 00:23
You can see how x, x^2 and x^3 behaves from the graph attached.. We can then answer the qn accordingly Is x negative? (1) At least one of x and x^2 is greater than x^3. (2) At least one of x^2 and x^3 is greater than x. Lets define the regions : A <1 , 1< B <0 , 0<C<1 , 1<D; Blue line  x ,Red line  x^2 , Green line  x^3 (1) So the region can be either of A , B or C.. It can be either positive or negative (2) So the region can be either of A , B or D .. It can be either positive or negative Each is insufficient. Now combine both (1) and (2)... We get the regions A and B ... which are negative ANSWER: C  Kudos if you find the post helpful
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plots.jpg [ 70.3 KiB  Viewed 1748 times ]











