GMAT Inequalities … And Then Some … Part 3:
If you’ve been following this series of posts, by now you’ve figured out that inequalities are not as simple a concept as they appear to be. But, in general, that is the gist of the GMAT exam—what seems to be a simple concept can often be twisted into a crazy difficult quantitative question. The same applies for Critical Reasoning, Sentence Correction … if it seems easy, then you are (likely) falling prey to GMAT deception.
As we’ve found, inequalities with variables and absolute value can prove a tough nut to crack. Same applies for inequality questions that incorporate square roots—or, mainly the reverse understanding of exponents.
Let’s see what the premise of our understanding should be by evaluating this data sufficiency answer:
What is the value of , if is an integer?
Statement 1: <
Statement 2: >
Immediately, in assessing this question I should recognize that it is unlikely I can figure out the value of from one greater than or less than inequality relationship—I am going to need to consider both statements together.
The challenge is that the vast majority of test takers are going to travel down the minefield of trying to figure out how to work with the square root. Yes, a square root is the same as being raised to the ½ power, but what does that do for our assessment of the question? If we FOIL out the question, are we able to make any headway?
Not really. What we need to remember is that “technically” all GMAT questions can be done in under 2 ½ minutes. There is always a simpler way than needing to understand mid-level algebraic calculations, and for this specific questions we want to start by getting rid of the square root by squaring other sides.
When we combine the statements we get:
Ah ha! The question is significantly easier to evaluate. From here, the best option is to try to plug-in numbers to see if there is a pattern:
Starting with , (
When squaring both sides, however, we need to remember that there is also a negative value to assess—which, when is FOILed, we have an as the first value.
So, we need to also consider the negative options of :
And therefore, we have two values. To restate our assessment of this question:
< < , but also < <
It is possible for to be equal to , but also equal to , making the correct answer choice (E)—we don’t have enough information to determine the value of .
When looking at square root problems, the key thing the GMAT is assessing is whether or not you consider that is a “+” or “-” value. Take it one step further with inequalities, and remember you need to flip the sign whenever you divide or multiply by a negative number.
But beyond that, the GMAT is looking to see if test takers have “considered all the options,” a fundamental tenant of the overall exam. Make sure you’ve done your due diligence—when you feel that you’ve worked through all the options for a question, take another look to see what (if) you’ve forgotten anything.