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p>q, Is q negative?

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Source: — Data Sufficiency |

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by Keith@ThePrincetonReview » Fri Apr 13, 2018 7:39 pm
VJesus12 wrote:p>q, Is q negative?

(1) pq + q^2 is positive
(2) (p^2)(q) + (p)(q^2) is positive

The OA is E.

How can I determine that the correct answer is the option E? May someone help me?
This is a Yes/No question with variables in the question stem and in the statements, so you can Plug In values for p and q.

Statement (1) states that pq + q^2 > 0
Plug In p=2 and q=3, so that (2)(3) + (3)^2 > 0, and 6 + 9 > 0.
Since the inequality is true when q=3, q can be positive.
Try to prove the statement insufficient by showing that q can also be negative.
Plug In p=2 and q=-3, so that (2)(-3) + (-3)^2 > 0, and -6 + 9 > 0.
Since the inequality is true when q=-3, q can be negative.
Statement (1) is insufficient, so write down BCE.

Statement (2) states that (p^2)(q) + (p)(q^2) > 0
Factor p out of the expression on the left of the inequality sign, so that p(pq + q^2) > 0.
We know from Statement (1) that pq + q^2 > 0 can be true whether q is positive or negative.
Therefore, p(pq + q^2) > 0 can be true whether q is positive or negative.
Statement (2) is insufficient, so eliminate choice B.

The expression in Statement (2) is simply p times the expression in Statement (1), so the combination of the statements provides no additional information.
The correct answer is choice E.
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by GMATGuruNY » Sat Apr 14, 2018 2:56 am
VJesus12 wrote:p>q, Is q negative?

(1) pq + q^2 is positive
(2) (p^2)(q) + (p)(q^2) is positive
Statement 1:
Case 1: q=1
Substituting q=1 into pq + q² > 0, we get:
p(1) + 1² > 0
p > -1.
Since this case requires that p>-1, and the prompt requires that p>q, it's possible that p=2.
Thus, a valid combination for Statement 1 is p=2 and q=1, in which case the answer to the question stem is NO.

Case 2: q=-1
Substituting q=-1 into pq + q² > 0, we get:
p(-1) + (-1)² > 0
-p > -1.
p < 1.
Since this case requires that p<1, and the prompt requires that p>q, it's possible that p=1/2.
Thus, a valid combination for Statement 1 is p=1/2 and q=-1, in which case the answer to the question stem is YES.

Since the answer is NO in Case 1 but YES in Case 2, INSUFFICIENT.

Statement 2:
Test whether Case 1 (p=2 and q=1) satisfies p²q + pq² > 0:
2²(1) + 2(1²) > 0
6 > 0.
This works.
Thus, a valid combination for Statement 2 is p=2 and q=1, in which case the answer to the question stem is NO..

Test whether Case 2 (p=1/2 and q=-1) satisfies p²q + pq² > 0:
(1/2)²(-1) + (1/2)(-1)² > 0
-1/4 + 1/2 > 0
1/4 > 0.
This works.
Thus, a valid combination for Statement 2 is p=1/2 and q=-1, in which case the answer to the question stem is YES.

Since the answer is NO in Case 1 but YES in Case 2, INSUFFICIENT.

Statements combined:
Cases 1 and 2 satisfy both statements.
Since the answer is NO in Case 1 but YES in Case 2, the two statements combined are INSUFFICIENT.

The correct answer is E.
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