Is (y - 3x)/(y - 2x) > 1?

Statement 1 : \(\frac{(y - 2x)}{x} < 0\)

\(\frac{y}{x} - 2 < 0\)

\(\frac{y}{x} < 2\)

Lets rephrase the given equation: \(\frac{(y - 3x)}{(y - 2x)} > 1\)

\((\frac{y}{x} - 3)/(\frac{y}{x} - 2) > 1\)

In order to get the fraction greater than 1 we need same sign for numerator and denominator and the numerator > denominator

From Statement 1 :\(\frac{y}{x}< 2\), Both the num and denom are negative and num>denom Sufficient

Statement 2 : \(y - 2x > 0\)

\(\frac{y}{x} - 2 >0\)

\(\frac{y}{x} > 2\)

This statement will not work if \(2 < \frac{y}{x} <3\) Not Sufficient.

Looking at the statements, my first thought is that this isn't really an inequalities/algebra problem - it's probably more like a positive/negative problem. However, before I start looking at positives and negatives, I'm going to simplify the math in the question. Always do this if there's anything you can simplify!

Is (y-3x)/(y-2x) > 1?

The top and bottom of the fraction are similar, so I'll rewrite them to be alike:

Is (y-2x-x)/(y-2x) > 1?

Is (y-2x)/(y-2x) - (x)/(y-2x) > 1?

Is 1 - (x)/(y-2x) > 1?

Is -(x)/(y-2x) > 0?

Is x/(y-2x) < 0?

In other words, "is x/(y-2x) negative?"

Now let's check out the statements:

Statement 1 already tells us the answer! If a fraction is negative, it stays negative when you flip it. The number properties reason for this is that a negative fraction has a top and bottom with different signs - one is positive, and one is negative. If you switch the top and bottom, this will still be the case, and it doesn't matter which is which - the fraction still comes out negative. This statement is sufficient.

Statement 2 says that y-2x is negative. So, the bottom of the fraction is negative. The answer to the question will depend on whether the top is also negative.

x could be negative as well. For instance, if y = -100 and x = -1, then x and y-2x are both negative.

But x could also be positive. If y = 1 and x = 100, then x is positive and y-2x is negative.

So, this statement is insufficient, and the answer is A.

What level of que is this ?

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