Is 2<x<4?

q: is 2<x<4? ie does x lie between 2 and 4? Y / N

(i) x^2 - 5x + 6 < 0
=> (x-3)(x-2) < 0
=> x < 3 or x < 2
=> lower bound 2 < x does not hold true. So, answer is NO.

(ii) 5x^2 - 25x > 0
=> 5x(x-5) > 0
=> 5x > 0 or x-5 > 0
=> x > 0 or x > 5
=> upper bound x < 4 does not hold true. So, answer is NO.

Hence, [D]

Stmt I

x^2-5x+6 < 0

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

Case 1: x-2 > 0 and x-3<0
x>2 and x<3

Case 2:
x-2<0 and x-3>0
x<2 and x>3

Not possible

So case 1 is the only case possible which is 2<x<3 therefore 2<x<4

SUFF

Stmt II

5x^2-25x > 0

5x (x-5) >0

Case 1:
5x>0 and x-5>0
x>0 and x>5

This means the more restrictive condition x>5 takes effect

Case 2:

5x<0 and x-5<0
x<0 and x<5

The more restrictive condition x<0 takes effect

x is either negative or if x is positive then x>5

SUFF


My only concern is the 2 statements contradicting each other.


Stuart /Ian/Logitech or others please share your thoughts on if this is ok in a DS problem or may be I am misinterpretting the contardiction piece here.

Regards,
Cramya

hardik.jadeja wrote:Is 2<x<4?
1. (x^2)-5x+6<0
2. 5(x^2)–25x>0

OA is D but I think its is B.

(1) After reverse FOILing:

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

For this to be true, one bracket must be positive and one must be negative.

So, we could have:

x-3 > 0 and x-2 <0
x>3 and x<2

Well hey, that can't happen, so we can ignore it.

Other possibility:

x-3 < 0 and x-2 > 0
x<3 and x>2
2 < x < 3

If x is between 2 and 3, it's definitely between 2 and 4: sufficient.

(2) 5(x^2)–25x>0
x^2 - 5x > 0
x(x-5) > 0

Positive product, so either both terms are positive or both are negative.

Both positive:

x>0 and x-5>0. Therefore, x>5.

Both negative:

x<0 and x-5<0. Therefore, x<0.

Well, if x<0 Or x>5, then x is definitely NOT between 2 and 4. Sufficient.

HOWEVER,

on the actual GMAT, the two statements never contradict. So, while this question is a good test of factoring skill, it's a 100% bogus GMAT question. What's the source? Always posting your source is good!

Thank you Stuart for the clarification. As always an excellent explanation .

Regards,
Cramya

Thanks for the explanation guys..
@Stuart: Source is Barron's GMAT - CAT 1