Bunuel wrote:

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

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

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

We'll show two approaches:

The first relies on number properties and is a Logical approach.

x^2*y^2 = (xy)^2.

So we're asked when the square of a number (xy)^2 is smaller than its absolute value of a number (xy).

This happens when the square is a fraction.

Written as an equation, |xy| > (xy)^2 when 0 < (xy)^2 < 1.

Looking at our statements, (1) tells us that x^2 is a fraction. Without knowing the value of y^2, this is insufficient.

Similarly, (2) gives us only the range of y^2 and is insufficient.

Combined, we know that 0 < (xy)^2 < 1, exactly what we need.

(C) is our answer.

If this confuses you, then picking numbers is a very easy way to solve the question.

This is an Alternative approach.

(1) Say x^2 = 1/9 and y = 0. Then |xy| = 0 and x^2*y^2 = 0 so the answer is NO.

We'll try to challenge this

Say x^2 = 1/9 and y = 1. Then |xy| = 1/3 and x^2*y^2 = 1/9. Then the answer is YES.

Insufficient.

(2) is entirely symmetrical - we can choose y = 1/16 and x = 0 and then y = 1/16 and x = 1

Insufficient.

Combined:

Let's say x^2 = y^2 = 1/16. Then x and y are 1/4 or -1/4 meaning that |xy| = 1/16 which is definitely larger than (1/16)*(1/16)

So our first numbers give us a YES.

We now need to challenge this by looking for numbers that give a NO.

Trying this out for a few more pairs, we can see that x^2*y^2 is always a fraction and |xy| is always a larger fraction.

Once again, (C) is our answer.

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