If x and y are nonzero integers, is (x-1 + y-1)-1 > [(x-1)(y-1)]-1 ?

(1) x = 2y

(2) x + y > 0

This DS question is from MGMAT question bank.
I am struggling to understand the solution of this problem for 2) statement.

1) why did we not simplify the question by canceling xy on both sides?
2) If I pick numbers for 2nd statement, its always NO, that satisfies the answer. So why would this be insufficient?
The answer is A

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Solution:

First, let's simplify the question:



(1) SUFFICIENT: If we plug x = 2y into our simplified question we get the following:


Since 2y2 must be positive we can divide both sides of the inequality by 2y2 which leaves us with the following:

If we investigate this carefully, we find that if y is an nonzero integer, 1/3y is never greater than 1. Try y = 2 and y = -2, In both cases 1/3y is less than 1.

(2) INSUFFICIENT: Let’s plug in values to investigate this statement. According to this statement, the x and y values we choose must have a positive sum. Let’s choose a set of values that will yield a positive xy and a set of values that will yield a negative xy.



This not does yield a definitive yes or no answer so statement (2) is not sufficient.

The correct answer is A.

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Appreciate your help. I was unable to paste the calculations/simplification here.
Thank you

Refer to this topic if-x-and-y-are-nonzero-integers-t7687.html

You cant cancel out like that in any inequality if you don't know the sign of that term

because we don't know the sign of xy, we cant cancel it out. For example
-2/3 > -2,
cancelling out -2 will reduce the inequality to
1/3 > 1, This is obviously wrong.

In general, if you know the sign of the term, then you can cancel out positive terms. For negative terms, the inequality sign has to be reversed.
Repeating the above example,
-2/3 > -2,
cancelling out -2 will reduce the inequality to
1/3 < 1,


The same rule applies to multiplications in inequalities.


Hope its clear

Thank you pellucide.

Yes, because multiplying or dividing an inequality by a negative number requires one to flip the sign, you cannot multiply or divide both sides of an inequality unless you are certain that you are multiplying or dividing by a positive number.

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Jamie Nelson
ManhattanGMAT Instructor