If x is positive which of the following could be correct

[Saurabh Malpani]

Hi All,

Source of the question is: GMAT Prep Test 1

If x is positive which of the following could be correct ordering of 1/x, 2x, x^2.
1) x^2 < 2x < 1/x

2) x^2 < 1/x < 2x

3) 2x < x^2 < 1/x

Choices are:

1) None
2) 1 Only
3) 3 Only
4) I and 2 only
5) 1, 2, 3

Please explain your choices!!!

Thanks
Saurabh Malpani

[GMAT 5/18]

Saurabh,

My answer would be 4) 1 & 2 only.

If x = 0.1, then x^2 < 2x < 1/x (so 1 is possible)

If x = 0.9, then x^2 < 1/x < 2x (so 2 is possible)

I don't think there is a positive value for x to make 3 possible.

Please let me know what the answer is and if my explanation makes sense. I just used trial and error to get to the answer here.

Thanks!

[Saurabh Malpani]

Hi GMAT 5/18,

Is their any arithmetic apporach to it..?

I mean during the exam under time constraint I choose the very obvious x^2 < 2x < 1/x . I din't even bother to plug in .9. I plugged in .5 got the answer proved other thwo wrong for Integer values and moved on.

Did you get this problem on your GPrep1?

What apporach did you follow?

Thanks
Saurabh Malpani

Saurabh,

My answer would be 4) 1 & 2 only.

If x = 0.1, then x^2 < 2x < 1/x (so 1 is possible)

If x = 0.9, then x^2 < 1/x < 2x (so 2 is possible)

I don't think there is a positive value for x to make 3 possible.

Please let me know what the answer is and if my explanation makes sense. I just used trial and error to get to the answer here.

Thanks!

[GMAT 5/18]

Saurabh,

This is part of the reason why I run out of time when I take exams.

For this question, I timed myself and it took me 2:49, which is a little more than you should spend per question. But the way I look at it is, if it costs me 49s over to get a question correct, then I might as well spend it.

Anyway, I don't know of an arithmetic approach, although I am sure there is one - maybe one of the MGMAT wizards will come through shortly and help out.

I didn't get this question in my GMAT Prep exam.

I see what you are saying about just trying 0.5 and moving on. I think perhaps under the time pressures of the test I might have done this also, but when I did it earlier today, I noticed that there would be a difference in the answers when trying a decimal close to 0 and a decimal close to 1.

Sorry I couldn't be of more help.

[Guest]

Each one of these gives you two inequalities. You know that x is positive, so you don't need to worry about the sign changing direction.

(1) x^2<2x<1/x

This means that x^2<2x so divide by x to get x<2. The second one tells you that 2x<1/x which simplifies to x < 1/sqrt(2). These can obviously both be satisfied at the same time, so (1) works.

(2) x^2<1/x<2x

This means that x^2<1/x which gives x^3<1, or x<1. The second half gives you 1/x<2x or 1<2(x^2) or x>1/sqrt(2). So any number that satisfies 1/sqrt(2)<x<1 will work.

(3) 2x<x^2<1/x. The first part gives 2x<x^2 or x>2. The second half gives x^2<1/x or x^3<1 or x<1. Since the regions x>2 and x<1 do not overlap, (3) can not be satisfied.


The Answer choice is (4), 1 and 2 only.

Cheers,
Jeff

[dbernst]

Jeff,

Great explanation! If you "see" the algebraic solution by manipulating the inequalities, by all means use it. However, this problem can be solved within a reasonable time frame by plugging in numbers. Personally, I plugged in fractions rather than decimals, but decimals work as well.

[neeraj]

Jeff,

Great explanation! If you "see" the algebraic solution by manipulating the inequalities, by all means use it. However, this problem can be solved within a reasonable time frame by plugging in numbers. Personally, I plugged in fractions rather than decimals, but decimals work as well.

If you put x=1 ,then none of the ordering is correct so the answer would be none.

[dbernst]

Neeraj,

You fell for a very common, and very tricky, GMAT trap. The original problem asked which of the following COULD be correct? You, however, read this as which of the following MUST be correct? With "could" wording, look only for scenarios that work in at least one circumstance rather than in all circumstances.

Hope that helps!
-dan

[onceatsea]

Brilliant methodology - it would save a lot of time !

Each one of these gives you two inequalities. You know that x is positive, so you don't need to worry about the sign changing direction.

(1) x^2<2x<1/x

This means that x^2<2x so divide by x to get x<2. The second one tells you that 2x<1/x which simplifies to x < 1/sqrt(2). These can obviously both be satisfied at the same time, so (1) works.

(2) x^2<1/x<2x

This means that x^2<1/x which gives x^3<1, or x<1. The second half gives you 1/x<2x or 1<2(x^2) or x>1/sqrt(2). So any number that satisfies 1/sqrt(2)<x<1 will work.

(3) 2x<x^2<1/x. The first part gives 2x<x^2 or x>2. The second half gives x^2<1/x or x^3<1 or x<1. Since the regions x>2 and x<1 do not overlap, (3) can not be satisfied.


The Answer choice is (4), 1 and 2 only.

Cheers,
Jeff

[Ben Ku]

Glad this was helpful!

[imanemekouar]

Hey guy,
How did you knows that we have to pick up two different numbers that both close to 0. I plug 1/2 and 3 but I did not get both solution.Please clarify.

[RonPurewal]

Hey guy,
How did you knows that we have to pick up two different numbers that both close to 0. I plug 1/2 and 3 but I did not get both solution.Please clarify.

well, it's mostly one of those things that you learn from experience with these types of functions.
the behavior of singleton variables (like "x"), versus the behavior of variables raised to powers (like "x^2"), changes at the boundaries of 0 and 1. therefore -- especially on problem like this one, where it's impossible to truly tell if you've narrowed it down all the way -- it's prudent to plug in numbers that are close to 0, and also numbers that are close to 1.

just remember what happened here for next time. remember, that's the only reason you really do these problems: TAKEAWAYS.

[ryan.m.doyle]

For Jeff's response, it is also worth noting that you only have the ability to manipulate the inequalities here because you know X is positive. Otherwise you cannot multiply/divide inequalities with variables. One can even take the "x is positive" as a hint to use Jeff's approach.

[RonPurewal]

For Jeff's response, it is also worth noting that you only have the ability to manipulate the inequalities here because you know X is positive. Otherwise you cannot multiply/divide inequalities with variables. One can even take the "x is positive" as a hint to use Jeff's approach.

correct.

nice!

[talk2pyus]

Thanks Jeff!! The explanation was really helpful. You gave me a new approach to solve such problems. Thanks again!