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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8017
GMAT 1: 760 Q51 V42 GPA: 3.82

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1 00:00

Difficulty:   35% (medium)

Question Stats: 70% (01:30) correct 30% (01:46) wrong based on 10 sessions

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Is $$x^3 - x^2 + x < 0?$$

1) $$x < 0$$

2) $$x^5 + x < 0$$

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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8017
GMAT 1: 760 Q51 V42 GPA: 3.82

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1
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.

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Joined: 01 Apr 2018
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Location: India
WE: Consulting (Consulting)

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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 ?
Intern  B
Joined: 02 Jul 2018
Posts: 4

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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?
Director  D
Joined: 08 Jun 2013
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Location: France
Schools: INSEAD Jan '19
GMAT 1: 200 Q1 V1 GPA: 3.82
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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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It will all make sense. Re: M60-06   [#permalink] 29 Sep 2018, 18:38
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