1 implies that x > 2 AND x < 3
2 implies that either x < 0 OR x > 5

This is probably going to sound silly, but I am not great with inequalities. I understand the factoring portion of this problem, but I do not understand why you change the sign on the inequalities. Any help is greatly appreciated. Thanks.

I have a similar problem with tricky inequality problems involving exponents.

I know for simple inequality problems that involve even exponents, there are usually two answers, a positive and negative, so best way to solve the problem is to split the problem into two scenarios - 1 where variable is positive, 1 where variable is negative and inequality sign is reversed.

But for more complex inequalities that involve squares, best way that works for me is to plug in a few numbers and find the range that way. Does someone have any better best practice tips?

IMO, this is kinda a lousy problem because the two statements seem to contradict each other...

gabriel16 wrote:This is probably going to sound silly, but I am not great with inequalities. I understand the factoring portion of this problem, but I do not understand why you change the sign on the inequalities. Any help is greatly appreciated. Thanks.

I'm not sure if I understand your question, so I'm going to answer it literally (i.e., ignoring the actual example from the OP):

If you multiply or divide both sides of an inequality by a negative number, the direction of the inequality symbol changes.

Here's a very basic example:

-2x < 4
=
(-2x)/-2 > 4/-2
=
x > -2

Sorry if I missed your point.

I too am confused, according to the 2nd poster the following is incorrect (how i would think about it)

I) x^2 - 5x + 6 < 0
wouldnt this equal:
(x-3)(x-2) < 0

so therefore

x<3 and x<2? I dont follow how the sign switches, we arent dividing both sides of the equation?

II) 5x^2 - 25x > 0
I think:
5(x-5)(x+0) >0

so therefore
x>5 and x>0

Am I missing something?

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

OA is D.

Let's think it through as we factor.

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

When is the product of two terms less than 0? When one term is positive and the other negative.

So, either:

Case 1

x-2 < 0 and x-3 > 0

x < 2 and x > 3

OR

Case 2

x-2 > 0 and x - 3 < 0

x > 2 and x < 3

When the inequalities face opposite directions, we must satisfy BOTH conditions.

Clearly, Case 1 make no sense - there's no number that's both less than 2 and greater than 3. Accordingly, we can ignore Case 1.

Therefore, we're stuck with Case 2, in which 2 < x < 3. If x is between 2 and 3, is it always going to be between 2 and 4? YES - sufficient.

2) 5x² - 25x > 0

dividing both sides by 5:

x² - 5x > 0

factoring out x:

x(x - 5) > 0

When is the product of two terms positive? When both terms are positive OR both terms are negative.

Case 1

x > 0 AND x - 5 > 0

x > 0 AND x > 5

When the inequality faces the same direction, we must satisfy the more limiting (i.e extreme) case, so:

x > 5

Case 2

x < 0 AND x - 5 < 0

x < 0 and x < 5

When the inequality faces the same direction, we must satisfy the more limiting (i.e extreme) case, so:

x < 0

Putting the two cases together (both are "legal"):

x < 0 or x > 5.

If x is less than 0 or greater than 5, is it EVER between 2 and 4? NO - sufficient.

HOWEVER - here's why this question would never appear as written on the actual GMAT:

on the GMAT, it must always be possible to combine the statements; they will never contradict each other.

In this question, statement (1) gives us a definite YES and statement (2) gives us a definite NO. We will NEVER see this on the actual test.

So, although the answer may seem to be (D), the answer is actually "this is not a real GMAT question, there is no correct answer".

What's the source?

Ahh, got it. Thanks for the help.