DS Question----Please explain OA is D

[Ajitpal Sian]

Is 2 < x < 4?
1) x² - 5x + 6 < 0
2) 5x² - 25x > 0

A) Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.

B) Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.

C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.

D) Either statement BY ITSELF is sufficient to answer the question.

E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question

[Anurag@Gurome]

Is 2 < x < 4?
1) x² - 5x + 6 < 0
2) 5x² - 25x > 0

(1) x² - 5x + 6 < 0 implies (x - 2)(x - 3) < 0, which means that one of the two terms is positive and the other is negative.
If x - 2 > 0, x - 3 < 0 then x > 2 and x < 3 or 2 < x < 3, which implies if x lies between 2 and 3 then x will always lie between 2 and 4; SUFFICIENT.
If x - 2 < 0, x - 3 > 0 then x < 2 and x > 3, which cannot hold true as there is no number which is less than 2 and greater than 3 at the same time.

(2) 5x² - 25x > 0 implies 5x(x - 5) > 0, which means that either both the terms are positive or both the terms are negative.

Following can be the 2 possibilities:
x > 0; x > 5 implies x > 5
x < 0; x < 5 implies x < 0
Combining we get, x < 0, x > 5, so x can not lie between 2 and 4; Sufficient.

But this is a contradiction, as statement 1 gives the answer to the main question (Is 2 < x < 4?) as "yes" but statement 2 gives the answer as "no", which can not be the case in a good GMAT problem.

[neelgandham]

First up, Please don't reveal the OA. You can use the spoiler tab to conceal the answer(How to use it ? - Look here https://www.beatthegmat.com/new-spoilers-t5302.html)

Is 2 < x < 4?

1) x² - 5x + 6 < 0
x² - 5x + 6 < 0, Implies (x-2)*(x-3) < 0. From the in-equation one can clearly tell that the sign of the expression changes ((- to +) or (+ to -)) at 2 and 3. So, let us check the sign of the expression using these limits

a) x<2
(x-2)*(x-3)>0

b) 2<x<3
(x-2)*(x-3)<0

c)x>3
(x-2)*(x-3)>0

So, we know that for all x, where 2<x<3, the expression (x-2)*(x-3)<0. This is sufficient to answer the question 'YES'


2) 5x² - 25x > 0

5x² - 25x > 0, implies 5x*(x-5)>0. From the in-equation one can clearly tell that the sign of the expression changes ((- to +) or (+ to -)) at 0 and 5. So, let us check the sign of the expression using these limits

a) x<0
5x*(x-5)>0

b) 0<x<5
5x*(x-5)<0

c)x>5
5x*(x-5)>0

So, we know that for all x, where x>5 and x<0, the expression 5x*(x-5)>0. So, x doesn't lie between 2,3.This is sufficient to answer the question 'NO'


As Anurag quoted

This is a contradiction, as statement 1 gives the answer to the main question (Is 2 < x < 4?) as "yes" but statement 2 gives the answer as "no", which can not be the case in a good GMAT problem.

Do you mind posting the source of the question please ?