Data Sufficiency

Is x<0?

1) x^3+1<0
2) x^3+2x^2+x+2=0

Max@Math Revolution wrote:Is x<0?

1) x^3+1<0
2) x^3+2x^2+x+2=0

Statement 1:
x³ + 1 < 0
x³ < -1
The resulting inequality will hold true only if x < 0.
Thus, the answer to the question stem is YES.
SUFFICIENT.

Statement 2:
x³ + 2x² + x + 2 = 0
x²(x+2) + (x+2) = 0
(x+2)(x²+1) = 0
The blue factor is equal to 0 if x=-2.
It is not possible for the red factor to be equal to 0.
Since the only viable solution is x=-2, the answer to the question stem is YES.
SUFFICIENT.

The correct answer is D.

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.

Since we have 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each condition on its own first.

Condition 1)
x^3+1<0
=> (x+1)(x^2-x+1) < 0
=> x+1<0 since x^2-x+1 > 0
=> x < -1
In inequality questions, the law "Question is King" tells us that if the solution set of the question includes the solution set of the condition, then the condition is sufficient
Thus, since the solution set of the question, 'x<0', includes that of condition 1), 'x<-1', condition 1) is sufficient.

Condition 2)
x^3+2x^2+x+2=0
=> x^2(x+2)+(x+2)=0
=> (x^2+1)(x+2)=0
=> x+2=0 since x^2+1 > 0
=> x = -2
Thus, x < 0 and condition 2) is sufficient.

Therefore, D is the answer.


If the original condition includes "1 variable", or "2 variables and 1 equation", or "3 variables and 2 equations" etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.