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Desperately Need Help with Inequality Strategy

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Desperately Need Help with Inequality Strategy

by leighton » Wed Apr 13, 2011 9:04 am
Hi all,

I've been a member of the great community for some time now, but this is the first time I'm posting anything.

I desperately need help with an approach on how to handle questions dealing with inequalities, especially in Data Sufficiency (DS). I feel like I have a good handle on the concept on a high level, but I've taken 4 MGMAT CATs and gotten only one out of twelve questions right! I particularly feel weak in DS. When I get a DS inequality question, I don't quite know what to do. First I try to simplify the question stem as much as possible, but with regards to the statements, I never know whether to try to answer the question conceptually (making algebraic manipulations on the inequalities etc.), or test numbers, and if I test numbers, which ones. I know the numbers must meet the criteria of the question stem/statements, but I can never see what numbers to test to get a YES and NO to prove insufficiency, for example. Through my studies, I know that positive, negative, zero, and fractions are always a good idea to check.

I know that there's no cut and dry method to solving these types of questions, but have you guys found any strategies/methods that has helped you to correctly solve them? Among questions dealing with inequalities, are there certain types that lend themselves better to testing numbers vs. a conceptual approach/manipulations? From the CATs I've taken so far and what others have posted on this great site, the GMAT LOVES to test the concept of inequalities, particularly in DS questions and in order to obtain a respectable Quant score, one has do well with these questions.

My GMAT date is quickly approaching and I'm in dire straits. If I cannot improve on this major weakness, I feel like my Quant performance is really going to suffer. So far on my CATs my Quant score has ranged from 40-45 (I am by no means a math person, so this has been a challenge), and my goal for GMAT day is to score between 45-47; getting a better handle on these types of question would put me in a better position to achieve my goal.

Thank you so much in advance for your wisdom and expert advice!
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by Brian@VeritasPrep » Wed Apr 13, 2011 10:00 am
Hey Leighton,

Great question - and, yes, the GMAT does love to test inequalities in Data Sufficiency.

A few guiding principles that may help...

Remember that one of the biggest thought processes that DS questions test is your ability to play devil's advocate - to look for exceptions to the obvious and to avoid "groupthink". So think of the ways in which inequalities can act counter to "conventional wisdom":

-When a negative number is present, you'll need to flip the sign of the inequality if you multiply or divide by it.

-More importantly, when a variable is present, you need to consider whether it could be negative. So if you have something like a/b > 5, you'd better consider whether b could be negative before you multiply it out of the denominator. This is crucial - negative numbers react counter to conventional wisdom with inequalities, so every time you see an inequality you should trigger yourself to look for the possibility of a negative number.

-When you see that a variable is greater than (or less than) a number (say, x > 5), your natural inclination is to think of the next counting number (x is 6 or higher). But, again, your job is to think about what else is possible outside of the obvious. What if x were just a negligible shade over that number (say, x could be 5.00000000001)? When an inequality gives you the parameters for the range of a number, make sure you use that entire range when testing numbers. For example, if a statement tells you that x^2 > 24, x could still be less than 5. 4.9^2 is going to be above 24.

-Remember that > 0 means "positive" and < 0 means "negative". So if a question is posed with an inequality surrounding the number 0, it's often quite helpful to translate that mentally and/or in your scratchwork to "positive" or "negative" so that you see it more conceptually and can apply what you know about positive/negative number properties.

-Also remember that "equal to" is NOT "greater than" or "less than". So if a question asks "is x > 2", and you want to find an exception to the fairly clear possibility that x is greater than 2, see if you can get it to equal 2...you don't need it to be "less than" if "equal to" gives you the answer "no".

-Similarly, I think most would agree that the "default" inequality is either > or <. > or < are special-cases to denote that "either greater than OR equal to" will work. So if you see that underline with an inequality, it's almost always going to matter. In that event, make certain that you test both cases "greater than" and "equal to".


Overall, I think the dominant strategy for DS is to play devil's advocate. You mentioned below about trying to find a YES and a NO - that's perfect. Once you get an "obvious" answer of one of those, your goal should immediately become to find the opposite answer, and the inequality provides you with a range of possibilities. Say, for example, that the statement gives you: x^2 > 49. Your job now is to figure out how to use those possible values of x to your advantage to get a different answer. x = 8 may be the most obvious one, but remember that x could be:

-a shade above 7 (so try 7.000000001 - or just try 7 and ask yourself what would happen if you nudged it just a hair)
-something way, way above 7 (so try 1,000 or something giant)
-something less than -7 (so -8 or -10 are decent numbers to test that)
-something just a hair less than -7 (-7.0000001)
-something insanely negative (-10,000)

And you probably don't need to try all those, as you should develop a hunch by looking at the question as to which of those possibilities is going to likely change the game. Remember that your job is often to find that "other" possibility that many (or most) test-takers won't take the time to find, and just by keeping that in mind you should push yourself to use the allowable values to your advantage.
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by VivianKerr » Wed Apr 13, 2011 10:11 am
Hi Leighton!

You'll find a LOT of resources for Inequalities here:

https://www.beatthegmat.com/mba/category ... equalities

I would aim to read through the step-by-step explanations for the challenging Data Sufficiency questions. It sounds like your skills are high, but you just need more practice.

You can check out this article for my own process regarding DS:

https://grockit.com/blog/gmat/2010/12/18 ... fficiency/

And a few interesting posts:

https://www.manhattangmat.com/forums/if- ... t2997.html

https://www.beatthegmat.com/mba/2010/06/ ... equalities

Remember that for multiple inequalities, you can add them together just like equalities, as long as the symbol is going the same way, but you cannot subtract two inequalities.
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