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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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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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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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13 Jul 2014, 05:32



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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, 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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Updated on: 09 Jul 2016, 00:44
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.
Originally posted by rohit8865 on 13 Mar 2016, 06:50.
Last edited by rohit8865 on 09 Jul 2016, 00:44, edited 2 times in total.



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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 3669 times ]



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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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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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12 Dec 2017, 12:17
Hi All, We're asked if X is NEGATIVE. This is a YES/NO question. We can answer it by TESTing VALUES or using Number Properties. 1) At least one of X and X^2 is greater than X^3. With this Fact, we know that X can either be.... ANY negative value (since X^2 would be positive and X^3 would be negative) and the answer to the question would be YES. ANY positive fraction (re: 0 < X < 1)  since squaring/cubing a positive fraction makes it smaller  and the answer to the question would be NO. Fact 1 is INSUFFICIENT 2) At least one of X^2 and X^3 is greater than X. With this Fact, we know that X can either be.... ANY negative value (since X^2 would be positive and X would be negative) and the answer to the question would be YES. ANY value greater than 1 (since squaring/cubing those values makes the result BIGGER than X) and the answer to the question would be NO. Fact 2 is INSUFFICIENT Combined, there's only one group of values that 'fit' both Facts: NEGATIVES. Thus, the answer to the question is ALWAYS YES. Final Answer: GMAT assassins aren't born, they're made, Rich
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