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is x > 0?

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by Brent@GMATPrepNow » Fri Jan 18, 2019 5:37 am
DivyaD wrote:Is x > 0 ?
(1) x - y > -8
(2) x + y = -8
Target question: Is x > 0 ?

Statement 1: x - y > -8
This statement doesn't feel sufficient, so I'll TEST some values.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 1 and y = 0. In this case, the answer to the target question is YES, x IS greater than 0
Case b: x = -1 and y = 0. In this case, the answer to the target question is NO, x is NOT greater than 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x + y = -8
Let's test some values again.
There are several values of x and y that satisfy statement 2. Here are two:
Case a: x = 1 and y = -9. In this case, the answer to the target question is YES, x IS greater than 0
Case b: x = -1 and y = -7. In this case, the answer to the target question is NO, x is NOT greater than 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Take Statement 2, x + y = -8, and solve for y to get: y = -8 - x
Now take Statement 1, x - y > -8, and replace y with -8 - x to get: x - (-8 - x ) > -8
Simplify: 2x + 8 > -8
Subtract 8 from both sides: 2x > -16
Divide both sides by 2 to get: x > -8
This means x COULD equal 3, in which case the answer to the target question is YES, x IS greater than 0
Or x COULD equal -2, in which case the answer to the target question is NO, x is NOT greater than 0
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent
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by fskilnik@GMATH » Fri Jan 18, 2019 4:59 pm
DivyaD wrote:Is x > 0 ?
(1) x - y > -8
(2) x + y = -8
$$x\,\,\mathop > \limits^? \,\,0$$
$$\left( {1 + 2} \right)\,\,\,\, - 8\,\,\mathop = \limits^{\left( 2 \right)} \,\,x + y\,\, = \,\,\left( {x - y} \right) + 2y\,\,\mathop > \limits^{\left( 1 \right)} \,\, - 8 + 2y\,\,\,\,\,\, \Rightarrow \,\,\,\,\,y < 0$$
$$\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( { - 7, - 1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1, - 9} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$


We follow the notations and rationale taught in the GMATH method.

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by fskilnik@GMATH » Fri Jan 18, 2019 5:07 pm
Brent@GMATPrepNow wrote:
DivyaD wrote:Is x > 0 ?
(1) x - y > -8
(2) x + y = -8
Target question: Is x > 0 ?

Statement 1: x - y > -8
This statement doesn't feel sufficient, so I'll TEST some values.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 1 and y = 0. In this case, the answer to the target question is YES, x IS greater than 0
Case b: x = -1 and y = 0. In this case, the answer to the target question is NO, x is NOT greater than 0
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x + y = -8
Let's test some values again.
There are several values of x and y that satisfy statement 2. Here are two:
Case a: x = 1 and y = -9. In this case, the answer to the target question is YES, x IS greater than 0
Case b: x = -1 and y = -7. In this case, the answer to the target question is NO, x is NOT greater than 0
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Take Statement 2, x + y = -8, and solve for y to get: y = -8 - x
Now take Statement 1, x - y > -8, and replace y with -8 - x to get: x - (-8 - x ) > -8
Simplify: 2x + 8 > -8
Subtract 8 from both sides: 2x > -16
Divide both sides by 2 to get: x > -8
This means x COULD equal 3, in which case the answer to the target question is YES, x IS greater than 0
Or x COULD equal -2, in which case the answer to the target question is NO, x is NOT greater than 0
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent
Hi, Brent!

The nice bifurcation you have presented for the statement (2) (repeated above) is also a possible bifurcation for the statement (1),
hence you could have closed your solution at that moment: (E) was already proved!

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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