Is x > y ?
(1) ax > ay
(2) a^2x > a^2y
Q/A-B can someone explain
inequality
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Unless I'm missing something, there is a problem with the question as you posted it. As written, the statements do not give us enough information, even when combined so the official answer should not be what you indicated.
Statement 1
Simplify (1) by dividing both sides by a. If a is positive, the inequality is maintained, and (1) becomes x>y. On the other hand, if a is negative, dividing by a will flip the sign, and (1) becomes x<y.
Because we can get conflicting results, statement (1) is NOT SUFFICIENT
Statement 2
We can think of a^2x as (a^2)^x. Likewise, a^2y = (a^2)^y. Thus, (2) can be rewritten as (a^2)^x > (a^2)^y. Now, just think of a^2 as an unknown positive value. If this value is greater than 1, for example if a^2 = 4, then (a^2)^x > (a^2)^y would mean that x>y. However, if this value is less than 1, for example if a^2=1/4, then (a^2)^x > (a^2)^y would mean that x<y (because as the exponent of a proper fraction increases, the result only gets smaller; for example, x=1, y=2 would work).
Because we can get conflicting results, statement (2) is NOT SUFFICIENT
Merge Statements
The quick way: we determined from (1) that x>y if a is positive, but x<y if a is negative. So for the statements together to be sufficient, (2) should give us a way to determine the sign of a. Since (2) gives us info on a^2, there is no way to isolate the sign of a. Thus the statements together are not sufficient..
The other quick way: we determined from (2) that x>y if a^2>1 (meaning if a>1 or <-1), but x<y if a^2<1 (meaning a is between -1 and 1). So for the statements together to be sufficient, (1) should give us a way to determine whether a falls between -1 and 1. Since (1) doesn't put any restriction on the value of a, we cannot determine this. Thus the statements together are not sufficient.
We can prove that the statements are not sufficient together by plugging in values that agree with both statements but still yield conflicting answers.
If a=2, x=2, y=1 both statements are respected, and x>y
If a=-1/2, x=1, y=2 both statements are respected, and x<y
We have conflicting results. The answer is E
If you correctly copied the question, beware of its source.
Hope this helps. Read my signature below for more
-Patrick
Statement 1
Simplify (1) by dividing both sides by a. If a is positive, the inequality is maintained, and (1) becomes x>y. On the other hand, if a is negative, dividing by a will flip the sign, and (1) becomes x<y.
Because we can get conflicting results, statement (1) is NOT SUFFICIENT
Statement 2
We can think of a^2x as (a^2)^x. Likewise, a^2y = (a^2)^y. Thus, (2) can be rewritten as (a^2)^x > (a^2)^y. Now, just think of a^2 as an unknown positive value. If this value is greater than 1, for example if a^2 = 4, then (a^2)^x > (a^2)^y would mean that x>y. However, if this value is less than 1, for example if a^2=1/4, then (a^2)^x > (a^2)^y would mean that x<y (because as the exponent of a proper fraction increases, the result only gets smaller; for example, x=1, y=2 would work).
Because we can get conflicting results, statement (2) is NOT SUFFICIENT
Merge Statements
The quick way: we determined from (1) that x>y if a is positive, but x<y if a is negative. So for the statements together to be sufficient, (2) should give us a way to determine the sign of a. Since (2) gives us info on a^2, there is no way to isolate the sign of a. Thus the statements together are not sufficient..
The other quick way: we determined from (2) that x>y if a^2>1 (meaning if a>1 or <-1), but x<y if a^2<1 (meaning a is between -1 and 1). So for the statements together to be sufficient, (1) should give us a way to determine whether a falls between -1 and 1. Since (1) doesn't put any restriction on the value of a, we cannot determine this. Thus the statements together are not sufficient.
We can prove that the statements are not sufficient together by plugging in values that agree with both statements but still yield conflicting answers.
If a=2, x=2, y=1 both statements are respected, and x>y
If a=-1/2, x=1, y=2 both statements are respected, and x<y
We have conflicting results. The answer is E
If you correctly copied the question, beware of its source.
Hope this helps. Read my signature below for more
-Patrick
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Based on the official answer (B), I believe the question should be worded as I have it above.Ankitaverma wrote:Is x > y ?
(1) ax > ay
(2) a²x > a²y
Target question: Is x > y ?
Statement 1: ax > ay
Some students will divide both sides by a and incorrectly conclude that x > y.
However, before we divide by a variable, we must ensure that the variable is EITHER positive OR negative, because if we divide by a negative value, we must reverse the direction of the inequality, and if we divide by a positive value, the direction of the inequality stays the same.
To see what I mean, consider these values of a, x and y that satisfy the given condition:
Case a: a = 1, x = 3 and y = 2, in which case x > y
Case b: a = -1, x = 2 and y = 3, in which case x < y
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: a²x > a²y
First recognize that a² > 0 for ANY value of a
Also recognize that a ≠0, since the inequality would not hold true of a = 0.
This means that a² MUST BE POSITIVE
Since we can be certain that a² is positive, we can divide both sides of the inequality by a² to get x > y
In other words, it must be true that x > y
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
Answer = B
Cheers,
Brent
- Patrick_GMATFix
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Brent@GMATPrepNow wrote:I believe the question should be worded as I have it above.
Makes sense.
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To find: x > y
Statement 1:
ax > ay
if a > 0 ==> x > y
if a < 0 ==> x < y
INSUFFICIENT
Statement 2:
a^2x > a^2y
It doesn't matter now whether what sign does "a" has.. as its (a)^2
So, we can strike-out a^2 from both sides
x > y
SUFFICIENT
Answer [spoiler]{B}[/spoiler]
Statement 1:
ax > ay
if a > 0 ==> x > y
if a < 0 ==> x < y
INSUFFICIENT
Statement 2:
a^2x > a^2y
It doesn't matter now whether what sign does "a" has.. as its (a)^2
So, we can strike-out a^2 from both sides
x > y
SUFFICIENT
Answer [spoiler]{B}[/spoiler]
R A H U L
- sanju09
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Your work in bold is misinterpretation of the situation presented. [spoiler]B[/spoiler] is the correct answer.Patrick_GMATFix wrote:Unless I'm missing something, there is a problem with the question as you posted it. As written, the statements do not give us enough information, even when combined so the official answer should not be what you indicated.
Statement 1
Simplify (1) by dividing both sides by a. If a is positive, the inequality is maintained, and (1) becomes x>y. On the other hand, if a is negative, dividing by a will flip the sign, and (1) becomes x<y.
Because we can get conflicting results, statement (1) is NOT SUFFICIENT
Statement 2
We can think of a^2x as (a^2)^x. Likewise, a^2y = (a^2)^y. Thus, (2) can be rewritten as (a^2)^x > (a^2)^y. Now, just think of a^2 as an unknown positive value. If this value is greater than 1, for example if a^2 = 4, then (a^2)^x > (a^2)^y would mean that x>y. However, if this value is less than 1, for example if a^2=1/4, then (a^2)^x > (a^2)^y would mean that x<y (because as the exponent of a proper fraction increases, the result only gets smaller; for example, x=1, y=2 would work).
Because we can get conflicting results, statement (2) is NOT SUFFICIENT
Merge Statements
The quick way: we determined from (1) that x>y if a is positive, but x<y if a is negative. So for the statements together to be sufficient, (2) should give us a way to determine the sign of a. Since (2) gives us info on a^2, there is no way to isolate the sign of a. Thus the statements together are not sufficient..
The other quick way: we determined from (2) that x>y if a^2>1 (meaning if a>1 or <-1), but x<y if a^2<1 (meaning a is between -1 and 1). So for the statements together to be sufficient, (1) should give us a way to determine whether a falls between -1 and 1. Since (1) doesn't put any restriction on the value of a, we cannot determine this. Thus the statements together are not sufficient.
We can prove that the statements are not sufficient together by plugging in values that agree with both statements but still yield conflicting answers.
If a=2, x=2, y=1 both statements are respected, and x>y
If a=-1/2, x=1, y=2 both statements are respected, and x<y
We have conflicting results. The answer is E
If you correctly copied the question, beware of its source.
Hope this helps. Read my signature below for more
-Patrick
The mind is everything. What you think you become. -Lord Buddha
Sanjeev K Saxena
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The Princeton Review - Manya Abroad
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Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com