If the product of X and Y is a positive number, is the sum

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If the product of X and Y is a positive number, is the sum of X and Y a negative number?

(1) X > Y5
(2) X > Y6

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by Brent@GMATPrepNow » Tue Oct 09, 2018 5:13 am

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If the product of x and y is a positive number, is the sum of x and y a negative number?
(1) x > y^5
(2) x > y^6
IMPORTANT CONCEPTS:
An ODD power preserves the sign of the base.
For example, (-5)^3 = -125 and 2^5 = 32
An EVEN power always yields a positive number (as long as the base ≠ 0
For example, (-5)^4 = 625 and 2^6 = 64


Target question: Is the sum of x and y negative?

Given: the product xy is positive
This tells us that EITHER x and y are both positive, OR x and y are both negative

Also, if the product xy is positive, we know that x ≠ 0 and y ≠ 0

Statement 1: x > y^5
There are several values of x and y that satisfy this condition. Here are two:
Case a: x = -1 and y = -2, in which case the sum of x and y IS negative
Case b: x = 10 and y = 1, in which case the sum of x and y is NOT negative
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x > y^6
Since y ≠ 0 we know that y^6 must be positive [since we have an EVEN exponent]
If x > y^6, then we know that x MUST BE POSITIVE
So, there are only 2 possible scenarios to consider:
Scenario #1: x is positive and y is positive
Scenario #2: x is positive and y is negative
HOWEVER, scenario #2 CANNOT OCCUR because it is given that the product xy is positive, and the product cannot be positive in scenario #2.
So, scenario #1 is the only possible scenario, which means x is positive and y is positive, which means the sum of x and y is definitely NOT negative
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

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Brent
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subh2273 wrote:If the product of X and Y is a positive number, is the sum of X and Y a negative number?

(1) X > Y^5
(2) X > Y^6
\[xy > 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( * \right)\,\,\,\,\left\{ \begin{gathered}
\,\,x > 0\,\,{\text{and}}\,\,y > 0\,\,\,\,\left( {{\text{scenario}}\,\,{\text{I}}} \right) \hfill \\
\,\,\,\,OR\,\,\, \hfill \\
\,\,x < 0\,\,{\text{and}}\,\,y < 0\,\,\,\,\left( {{\text{scenario}}\,\,{\text{II}}} \right) \hfill \\
\end{gathered} \right.\]
\[x + y\,\,\mathop < \limits^? \,\,\,0\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\boxed{\,\,\,?\,\,\,:\,\,\,{\text{scenario}}\,{\text{II}}\,\,}\]
\[\left( 1 \right)\,\,\,x > {y^5}\,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {x,y} \right) = \left( {2,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,\left( {x,y} \right) = \left( { - 1, - 2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,x > {y^6}\,\, \geqslant 0\,\,\,\, \Rightarrow \,\,\,\,x > 0\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\]

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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by happyatharvard » Tue Oct 09, 2018 4:10 pm

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in the first case x >y^5 can be satisfied when x, y both are negative or positive. But we want that both should be negative.
in the second case x>y^6 is possible only when both are positive. So we now know that x+y can't be negative. sufficient.