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M60-06

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Math Revolution GMAT Instructor
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M60-06  [#permalink]

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New post 11 Jun 2018, 02:47
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A
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D
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Question Stats:

67% (00:53) correct 33% (01:13) wrong based on 3 sessions

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Is \(x^3 - x^2 + x < 0?\)



1) \(x < 0\)

2) \(x^5 + x < 0\)

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Re M60-06  [#permalink]

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New post 11 Jun 2018, 02:47
Official Solution:

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

The first step of the VA (Variable Approach) method is to modify the original condition and the question, and then recheck the question.



Modifying the question:

\(x^3 - x^2 + x < 0\)

⇔ \(x(x^2 - x + 1) < 0\)

⇔ \(x < 0\) since \(x^2 - x + 1 > 0\) always.

So, the question becomes, 'is \(x < 0?\)'.



Condition 1) is certainly sufficient.



Condition 2), \(x^5 + x < 0\), is equivalent to \(x(x^4 + 1) < 0\) or \(x < 0\), since \(x^4 + 1 > 0\) is always true. So, condition 2) is also sufficient.

Therefore, D is the answer.



Answer: D
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Re: M60-06  [#permalink]

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New post 13 Sep 2018, 04:14
Didnt understand this part - ⇔ x<0x<0 since x2−x+1>0x2−x+1>0 always. How is x2−x+1>0 ?

Please can someone explain ?
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Re: M60-06  [#permalink]

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New post 29 Sep 2018, 18:18
Anirudh134 wrote:
Didnt understand this part - ⇔ x<0x<0 since x2−x+1>0x2−x+1>0 always. How is x2−x+1>0 ?

Please can someone explain ?


x(x2-x+1)<0 implies either x or x2-x+1 has to be negative (only 1 HAS to be negative).

If x is positive, then x2-x+1 has to be negative for the equation to be true. For no value of x (x>0), x2-x+1 is <0. So x<0 & for all values of x<0, x2-x+1 is >0. Does it make sense?
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Re: M60-06  [#permalink]

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New post 29 Sep 2018, 18:38
Anirudh134 wrote:
Didnt understand this part - ⇔ x<0x<0 since x2−x+1>0x2−x+1>0 always. How is x2−x+1>0 ?

Please can someone explain ?


Anirudh134

\(X^2 - X + 1 = X^2 - 2X + 1 + X = (X - 1)^2 + X = K ( Say)\)

Now for X >= 0 , above expression always positive.

Next, for X < 0 also it is always positive.

Plug some values and check

X = - 2 then K = 9 - 2= 7

X = -0.5 then K = 2.25 - 0.5 = 1.75

Does this help?
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Re: M60-06 &nbs [#permalink] 29 Sep 2018, 18:38
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