Concept: This is a question which is a great example of what the GMAT expects from test takers in terms of combining statements in a DS question on Inequalities.
It is also an excellent question to learn how to prove/disprove the individual statements in a DS question.
Breaking down the question stem, we can rephrase the question as “Are x and y of the same signs?”. The answers to this question could be,
Yes – which means x and y are either both positive or both negative
No – which means one of them could be positive and the other could be negative OR ZERO (many of us ignore ZERO at our own peril).
From statement I alone, x-y > -2. Let us take some values to find out if this information is sufficient to answer the main question.
If x = 10 and y = 5, x-y = 5 which is definitely greater than -2. For these values of x and y, we answer the main question with a Yes.
If x = 10 and y = -5, x-y = 15 which is also greater than -2. For these values of x and y, we answer the main question with a No.
Statement I alone is insufficient to answer the question with a definite YES or NO. Answer options A and D can be eliminated. Possible answer options are B, C or E.
From statement II alone, x-2y < -6. Again, as in the first statement, let us test some values.
If x = 5, y = 10, x – 2y = -15 which is less than -6. These values give a YES to the main question.
If x = -5, y = 10, x – 2y = -25 which is less than -6. These values give a NO to the main question.
Statement II alone is insufficient to answer the question with a definite YES or NO. Answer options B can be eliminated. Possible answer options are C or E.
Now, at this stage, we are required to combine the statements I and II. Here’s where GMAT expects you to combine the statements using an important property of inequalities and that is – Two inequalities can be added if they have the same inequality sign.
The two inequalities are: x – y > -2 and x – 2y < -6.
When we multiply the first inequality with -2, we obtain -2x + 2y < 4. Now, since the inequality sign is the same as the second inequality, we can add these inequalities.
Adding the inequalities, 2y and -2y cancel out each other and we have –x < -2 which can be rewritten as x > 2.
When we multiply the second inequality with -1, we obtain –x + 2y > 6. Since the inequality sign is the same as the first, we can add these inequalities.
Adding them, x and –x cancel out each other and we have y > 4.
This means xy > 0 and we can answer the main question with a Yes. The combination of statements is sufficient, answer option E can be eliminated.
The correct answer option is E.
Remember that inequalities can be added as long as they have the same sign.