GMAT Question Pack 1 - modulus & inequality

[Mo2men]

Is |xy| > x^2*y^2 ?

(1) 0 < x^2 < 1/4
(2) 0 < y^2 < 1/9

OA: C

[Jay@ManhattanReview]

Is |xy| > x^2*y^2 ?

(1) 0 < x^2 < 1/4
(2) 0 < y^2 < 1/9

OA: C

We have to determine whether |xy| > x^2*y^2.

|xy| > x^2*y^2 can be written as |xy| > (xy)^2.

We see that the LHS = |xy| and the RHS = (xy)^2 are positive irrespective of whether one or both of x and y are negative.

|xy| > x^2*y^2 in the following conditions.

1. None of x and y is 0.
2. The product of |x| and |y| = |xy| < 1.

Let's take each statement one by one.

(1) 0 < x^2 < 1/4

=> 0 < |x| < 1/2, but we do not have any information about y. Insufficient.

(2) 0 < y^2 < 1/9

=> 0 < |y| < 1/2, but we do not have any information about x. Insufficient.

(1) and (2) together

We have 0 < |x| < 1/2 and 0 < |y| < 1/3, thus 0 < |xy| < 1/6 and 0 < (xy)^2 < (1/2*1/3)^2 => 0 < (xy)^2 < (1/6)^2 => 0 < (xy)^2 < 1/36.

We see that |xy| < (xy)^2. The answer is yes. Sufficient.

The correct answer: C

Hope this helps!
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[GMATGuruNY]

Is |xy| > x^2*y^2 ?

(1) 0 < x^2 < 1/4
(2) 0 < y^2 < 1/9

|xy| > x²y² ?
Since neither an absolute value not the square of a value can be negative, both sides are of the inequality above are NONNEGATIVE.
Thus, we can safely square the inequality:
x²y² > x�y�.
Since the resulting inequality is valid only if x and y are NONZERO, we can now safely divide both sides by x²y², which must be POSITIVE:
(x²y²)/(x²y²) > (x�y�)/(x²y²)
1 > x²y².
Question stem, rephrased:
Is x²y² < 1, where x and y are nonzero?

Statement 1:
No information about y².
INSUFFICIENT.

Statement 2:
No information about x².

Statements combined:
Since x² and y² are both positive fractions, x²y² < 1, where x and y are nonzero.
SUFFICIENT.

The correct answer is C.

[Mo2men]

Is |xy| > x^2*y^2 ?

(1) 0 < x^2 < 1/4
(2) 0 < y^2 < 1/9

|xy| > x²y² ?
Since neither an absolute value not the square of a value can be negative, both sides are of the inequality above are NONNEGATIVE.
Thus, we can safely square the inequality:
x²y² > x�y�.
Since the resulting inequality is valid only if x and y are NONZERO, we can now safely divide both sides by x²y², which must be POSITIVE:
(x²y²)/(x²y²) > (x�y�)/(x²y²)
1 > x²y².
Question stem, rephrased:
Is x²y² < 1, where x and y are nonzero?

Statement 1:
No information about y².
INSUFFICIENT.

Statement 2:
No information about x².

Statements combined:
Since x² and y² are both positive fractions, x²y² < 1, where x and y are nonzero.
SUFFICIENT.

The correct answer is C.

Dear GMATGuru,

After combing the two statements, I found them nether x nor y equals to zero. As both sides in the question stem is positive, can I deal with as follows:
|xy| > x^2*y^2

|xy| > |xy|^2.....divide by |xy|

the questions becomes is
|xy| < 1??

Then answer would be C.

Is my reasoning above correct?

Thanks in advance for your help

[GMATGuruNY]

Dear GMATGuru,

After combing the two statements, I found them nether x nor y equals to zero. As both sides in the question stem is positive, can I deal with as follows:
|xy| > x^2*y^2

|xy| > |xy|^2.....divide by |xy|

the questions becomes is
|xy| < 1??

Then answer would be C.

Is my reasoning above correct?

Thanks in advance for your help

Nice work!