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

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

(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

Let us put critical points on the number line as shown in fig:



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.

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

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 -1<x<3 or x>4.

Statements 1 and 2 are insufficient on their own to answer given inequality >0

Combining , we get -1<x<3 and this does lie in the "ONLY IF" region and thus we can definitely say "yes" . C is the correct 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.

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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