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Is \((x + 2)(x + 1)(x - 3)(x - 4) > 0\)? --> the "roots" are -2, -1, 3, and 4 (equate each multiple to zero to get the roots and list them in ascending order), this gives us 5 ranges:
Now, test some extreme value: for example if \(x\) is very large number then all multiples ((x + 2), (x + 1), (x - 3), and (x - 4)) will be positive which gives the positive result for the whole expression, so when \(x>4\) the expression is positive.
Now the trick: as in the 5th range expression is positive then in 4th it'll be negative, in 3rd positive, in 2nd negative, and finally in 1st it'll be positive again: + - + - +. So, the ranges when the expression is positive are: \(x<-2\) (1st range), \(-1<x<3\) (3rd range) and \(x>4\) (5th range).
So, the question asks whether \(x<-2\), \(-1<x<3\) or \(x>4\)
(1) 3 > x. Not sufficient. (2) x > -1. Not sufficient.
What Bunuel did may be the right way to solve this problem, but the easier way is to take some values and check
S1: Here X can be anything number less than 3 to 0 on the positive side and 0 to infinity on the negative side.
Lets check with 2,1,0..all of them give positive numbers. The fractions in between 0 & 3 also give positive numbers. Now -1 & -2 give 0 and -3,-4 etc give -ve numbers. So insufficient
S2: we have checked with -1, -2 etc. But lets check with fractions between -1 & 0 for eg -1/2 [which is greater than -1] gives a positive number. Let us also check for numbers equal to or greater than 3.... 3,4 give negative numbers and above four it is positive. So insufficient.
Combining 1 & 2, for the checks done above, we can easily say that 3>X>-1 and hence Answer C.
Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
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