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Anatomy of an OG Question: How ‘Brute Force’ Can Work Really Fast

By Max Peterson On Nov 6, 2015 In  Quant Algebra Test It General GMAT 

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The following article refers to the Problem Solving question #229 in the OG13, GMAT2015 and GMAT2016 books. You can view the question in its entirety here:

http://gmatclub.com/forum/how-many-of-the-integers-that-satisfy-the-inequality-x-2-x-134194.html

One of the interesting aspects of most GMAT questions is that they can be solved using more than one approach. In that way, you don’t have to be a brilliant mathematician to get to the correct answer - strong critical thinkers, pattern-matchers and ‘workers’ can still get to the correct answer in a reasonable amount of time.

From a pacing standpoint, sometimes the easiest/fastest way to get to the correct answer is to use what’s called ‘brute force’ – no fancy logic or math is required – just the willingness to do a bunch of simple calculations to PROVE what the correct answer actually is.

In this prompt, we’re told that (X+2)(X+3)/(X-2) >= 0. We’re asked for the number of INTEGERS that are LESS than 5 that satisfy this inequality. 

The ‘key’ to recognize that this question is susceptible to ‘brute force’ is that the answer choices are 1 through 5, inclusive. This emphasizes that there are not that many possible values for X - there is at least one solution, but no more than 5 solutions. How hard can it possibly be to find them all? 

To start, we should consider the largest integer that fits the given ‘restrictions.’ In this case, that would be X = 4. 

When X = 4, the fraction becomes….

(6)(7)/(2) = 21

This is clearly greater than or equal to 0, so X = 4 is a solution to the prompt. Now we just have to ‘brute force’ as many additional integers as it takes to discover the number of additional solutions.

Rather than ‘cheat’ you out of experiencing ‘brute force’ for yourself, I’m going to ‘nudge’ you through the steps that have to come next.

Is X = 3 a solution? How long would it actually take you to prove it (if it takes you more than about 10 seconds, then I would be surprised). X can’t equal 2, since that would create a 0 in the denominator of the fraction, but could X = 1 or 0? How about negative integers? Could X = -1 or -2? At what point would you stop ‘brute forcing’ and conclude that you were done? (Hint: look at the ‘sign’ of each of your end calculations).

All in, how long did it actually take you to ‘brute force’ this question? My guess is that it wouldn’t take more than 2 minutes of basic calculations for you to get to the correct answer (AND the calculations were NOT difficult).

This is all meant to show that a flexible thinker (and not one who’s necessarily a ‘genius’) can solve lots of GMAT questions in a relatively short period of time. You too can train to think in these ways. To that end, we’re here to help.

GMAT assassins aren’t born, they’re made,

Rich

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