Inequalities and modulus

[mallika hunsur]

Hi All,

Could anyone please take a look at this problem and please explain how statement 1 is sufficient..?

Many thanks,
Mallika

[Brent@GMATPrepNow]

If y ≠ 0, is a^x > y?

1) a = -y = |x|
2) a < 1

Target question: Is a^x > y?

Statement 1: a = -y = |x|
We're told that y ≠ 0, which means a ≠ 0 and x ≠ 0.
If x ≠ 0, we can be certain that |x| is POSITIVE, which means a is POSITIVE and -y is POSITIVE
If -y is POSITIVE, then y is NEGATIVE
Aside: x can be either positive of negative, but this doesn't play a role in the answer.

So, all of this means that a^x = (some POSITIVE value)^x
Since a positive number raised to ANY EXPONENT will yield a positive result, we can conclude that a^x must be POSITIVE.
Since we also know that y is NEGATIVE, we can be certain that a^x > y
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: a < 1
Since there's no information about x or y, this statement doesn't FEEL sufficient. So I'm going to TEST some values.
There are several set of values that satisfy statement 2. Here are two:
Case a: a = 1/2, x = 1 and y = -1, in which case a^x > y
Case b: a = 1/2, x = 1 and y = 1, in which case a^x < y
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, you can read my article: https://www.gmatprepnow.com/articles/dat ... lug-values

Answer = A

Cheers,
Brent

[[email protected]]

Hi mallika hunsur,

As Brent has pointed out, this DS question includes a few different Number Properties. If you're not great at handling Number Properties just yet, then that's okay - ALL Number Properties can be proven by TESTing VALUES, so all Number Property-based questions can be solved with this approach.

Here, we're told that Y CANNOT = 0. We're asked if A^X > Y. This is a YES/NO question.

Fact 1: A = -Y = |X|

Let's TEST VALUES....

IF....
X = 1
A = -Y = 1
Y = -1
A = 1
1^1 > -1 and the answer to the question is YES

IF....
X = -1
A = -Y = 1
Y = -1
A = 1
1^(-1) = 1/1 and 1/1 > -1 so the answer to the question is YES

From this work, you should notice that |X| will always be POSITIVE. Thus...

A = -Y = POSITIVE

So A will ALWAYS be positive and Y will ALWAYS be negative. Considering what the question is asking us for.....the 'left side' will always be positive and the 'right side' will always be negative, so the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT

Fact 2: A < 1

This tells us NOTHING about X and Y. There's no way to know if something is greater than Y when we don't know what Y is and we have no basis for comparison.
Fact 2 is INSUFFICIENT

Final Answer: A

GMAT assassins aren't born, they're made,
Rich

[GMATGuruNY]

If y ≠ 0, is a^x > y?

1) a = -y = |x|
2) a < 1

Statement 1: a = -y = |x|
Substituting a=|x| and y=-|x| into a^x > y, the question stem can be rephrased as follows:
Is |x|^x > -|x|?

Since y=-|x| and y≠0, x≠0 and |x| > 0.
Thus, the rephrased question stem implies the following:
Is (positive)^(nonzero power) > negative?
Since the left side must always be POSITIVE, while the right side must always be NEGATIVE, the answer to the question stem is YES.

Statement 2: a < 1
If a=0, x=1, and y=1, then a^x < y.
If a=0, x=1 and y=-1, then a^x > y.
INSUFFICIENT.

The correct answer is A.