Is (x - 4)(x - 3)(x + 2)(x + 1) 0

Question

Is (x - 4)(x - 3)(x + 2)(x + 1) > 0

(1) 3 > x
(2) x > -1

Bunuel · Accepted Answer

Is (x - 4)(x - 3)(x + 2)(x + 1) > 0 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: x<-2 ; -24 . 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), -14 (5th range). So, the question asks whether x<-2 , -14 (1) 3 > x. Not sufficient. (2) x > -1. Not sufficient. (1)+(2) -1x2-4x-94661.html#p731476 inequalities-trick-91482.html data-suff-inequalities-109078.html range-for-variable-x-in-a-given-inequality-109468.html everything-is-less-than-zero-108884.html graphic-approach-to-problems-with-inequalities-68037.html All DS Inequalities Problems to practice: search.php?search_id=tag&tag_id=184 All PS Inequalities Problems to practice: search.php?search_id=tag&tag_id=189 700+ Inequalities problems: inequality-and-absolute-value-questions-from-my-collection-86939.html

igotthis · Answer

omg! Simple and nice solution that I understood. But, what is wrong with this:

(1) Since X<3, pick X = 2. Get (-2)*(-2)*(3)*(3) >0 ( for that matter any number or fraction less than 3 would give this expression >0) - Suff

(2) X>-1, Let x = 0 similar approach above, so is suff.

Seems to0 simple for this level of difficulty, but why is this wrong, what was  not  considered?!

Bunuel · Answer

You cannot check sufficiency based only on one arbitrary value. Consider x=-1.5 for (1) and x=3.5 for (2).

monk16 · Answer

(1) states x<3

for all x ={0,1,2} -> (x – 4)(x – 3)(x + 2)(x + 1) >0  YES 
 
for x= -2 & -1 - >(x – 4)(x – 3)(x + 2)(x + 1) = 0  NO

(2) states x>-1

x can be 0,1,2,3,4..

for all x ={0,1,2} -> (x – 4)(x – 3)(x + 2)(x + 1) >0  YES

for x= 3 & 4 - >(x – 4)(x – 3)(x + 2)(x + 1) = 0  NO

(1) & (2) combined

3>x>-1

x lies between 3 and -1 -> x can take values 0,1,2 -> (x – 4)(x – 3)(x + 2)(x + 1) >0 YES

Answer is C

VenoMfTw · Answer

Let us put critical points on the number line as shown in fig: 1.jpg from St 1: Not suff as we get both +ve or -ve or even 0 From st 2 : Not suff Combined : -1 < x < 3. x is positive as seen in the number line. PS: Sorry for such a bad diagram.

GMATinsight · Answer

Question : Is (x – 4)(x – 3)(x + 2)(x + 1) > 0 ?

Statement 1 : 3 > x

@x = 2.5, (x – 4)(x – 3)(x + 2)(x + 1) = (-1.5)(-0.5)(4.5)(3.5) i.e.  Greater  than Zero
@x = -1.5, (x – 4)(x – 3)(x + 2)(x + 1) = (-5.5)(-4.5)(0.5)(-0.5) i.e.  Less  than Zero
 NOT SUFFICIENT

Statement 2 : x > -1

@x = 2.5, (x – 4)(x – 3)(x + 2)(x + 1) = (-1.5)(-0.5)(4.5)(3.5) i.e.  Greater  than Zero
@x = 3.5, (x – 4)(x – 3)(x + 2)(x + 1) = (-0.5)(0.5)(4.5)(3.5) i.e.  Less  than Zero
 NOT SUFFICIENT

Combining the two statements  
we get, 3 > x > -1
@x = -0.5, (x – 4)(x – 3)(x + 2)(x + 1) = (-4.5)(-3.5)(1.5)(0.5) i.e.  Greater  than Zero
@x = 2.5, (x – 4)(x – 3)(x + 2)(x + 1) = (-1.5)(-0.5)(4.5)(3.5) i.e.  Greater  than Zero

In the entire range the function is Positive therefore,
 SUFFICIENT

Answer: option C

ENGRTOMBA2018 · Answer

Drawing the wavy curve for the given inequality (x – 4)(x – 3)(x + 2)(x + 1)> 0 , we get (refer to the attached picture) that this will be true ONLY IF x<-2 or -14. Statements 1 and 2 are insufficient on their own to answer given inequality >0 Combining , we get -1

Bunuel · Answer

800score Official Solution:

The expression (x – 4)(x – 3)(x + 2)(x + 1) is composed of four factors. It will equal 0 if at least one of the factors is 0. It will be positive if all the four factors are positive, if all the four factors are negative, or if two of them are negative and the other two are positive. Otherwise the epxression will be negative.

Statement (1), 3 > x, implies that the factors (x – 4) and (x – 3) are negative. The signs of the other two factors, (x + 2) and (x + 1), are not defined. E.g. they both can be positive if x = 1. Or one of them can equal 0 if x = -1 or x = -2. Or (x + 2) can be positive and (x + 1) can be negative if x = -1.5, etc. Therefore the original expression can be positive, negative or 0 and we can NOT give a definite answer to the original question. Statement (1) by itself is NOT sufficient.

Statement (2), x > -1, implies that the factors (x + 2) and (x + 1) are positive. The signs of the other two factors, (x – 4) and (x – 3) , are not defined. E.g. they both can be positive if x = 5. Or one of them can equal 0 if x = 3 or x = 4. Or (x – 3) can be positive and (x – 4) can be negative if x = 3.5, etc. Therefore the original expression can be positive, negative or 0 and we can NOT give a definite answer to the original question. Statement (2) by itself is NOT sufficient.

If we use the both statements together, statement (1) implies that factors (x – 4) and (x – 3) are negative. Statement (2) implies that factors (x + 2) and (x + 1) are positive. Therefore the original expression must be positive (2 negative factors × 2 positive factors). The both statements taken together are sufficient to answer the question.  The correct answer is C .

Alternative method:  
You may solve the original inequality first and then compare the solution with the inequality (1), inequality (2) and a system of inequalities (1) and (2) using the number line.

Devesh26 · Answer

For x>-1 you took x=0 so x it satisfies equation take x = 4 or x=3 you get 0>0 which is not possible so x should limit at some point so by combining both we get -1 to 3 so c is answer

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Is (x - 4)(x - 3)(x + 2)(x + 1) > 0

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