GMAT Question of the Day - Data Sufficiency - Exponents
Is x - 4 > y?
(1) 3^x - 3^(x-5) > 242 * 3^y
(2) x is a positive integer
GMAT Question of the Day Solution
There are a couple of things that make GMAT Data Sufficiency much easier and that I encourage GMAT tutoring students to think about when approaching a DS question:
1. Planning - this generally breaks down to defining the question.
2. Organization - simplifying algebra and/or creating supporting charts, lists, scenarios to help define the question and potential information that you need to solve the question.
In this question we can't do a ton of planning because we don't have much information to begin with but we can do some organization. Consider re-arranging the question to: x > y + 4? You may find that it is easier to think about with addition rather than subtraction.
Statement (1) You consider simplifying this inequality. With GMAT exponents you have several options that come up again and again. You can make the bases the same, factor, or use special quadratics (a special type of factoring). In this question we can't make the bases the same and we don't have a special quadratic so we should consider factoring. You have a couple of ways to factor - in general it will be easier to remove the smaller of the two items. In this case 3^(x-5) must be smaller than 3^x. So 3^(x-5) is the one to factor out. Why? By factoring out the smaller of the two you will have integers rather than fractions inside the parenthesis. You can try it the other way just to see what happens.
After you factor you'll see that you'll be able to cancel the 242 leaving only base 3 exponents. Now you can drop the bases and set the exponents equal to each other leaving you with x-5 > y or x > y + 5. We can answer yes. Sufficient.
(2) That x is a positive integer is irrelevant to the question. Insufficient.