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

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by Max@Math Revolution » Sun Jan 14, 2018 11:23 pm
$$Is\ x^3-x^2+x<0?$$

$$\left(1\right)\ x\ <\ 0$$
$$\left(2\right)\ x^5+x<0$$
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Source: — Data Sufficiency |
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by Max@Math Revolution » Tue Jan 16, 2018 11:54 pm
=>
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:

$$x^3-x^2+x<0$$
$$x\left(x^2-x+1\right)<0$$
$$x<0$$ 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 $$x\left(x^4+1\right)<0\ or\ x<0$$ 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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