Are x and y both positive?
1) 2x - 2y = 1
2) x/y > 1
Solution:
We are not provided any given information in the question stem, so we can immediately move to the analysis of the two statements.
Statement One Alone:
1) 2x - 2y = 1
The first thing we should do here is to simplify statement one.
2(x - y) = 1
x - y = ½
We can clearly see that x and y can both be positive to yield ½ as their difference (for example, x could be 1.5 and y could be 1)
OR
x and y could both be negative (for example, x could be -1 and y could be -1.5)
OR
x could be positive and y could be negative (for example, x could be ¼ and y could be -¼)
Thus, statement one is insufficient.
Statement Two Alone:
2) x/y > 1
Statement two does not provide enough information to determine whether x and y are either both positive or both negative. Remember that if something is > 1, that something is positive. Also, remember that a negative divided by a negative is positive, and a positive divided by a positive is also positive. We can't get anywhere with statement two alone.
Statements One and Two Together:
It's important to be cognizant of situations in which we are provided an inequality and an equation with the same two variables. In these situations, we can substitute the equation into the inequality. Doing so will allow us to simplify the inequality. In this case we need to first simplify our equation from statement one:
x - y = ½
x = ½ + y
Now we can substitute ½ + y for x into the inequality x/y > 1. Thus, we have:
(½ + y)/y > 1
½/y + y/y > 1
½/y + 1 > 1
½/y > 0
Because ½/y is greater than zero, y MUST also be greater than zero. Lastly, because we know that x = ½ + y, it follows that x MUST also be greater than zero. Thus, both x and y are positive.
Answer C.
Jeffrey Miller
Head of GMAT Instruction
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