ordering different x values

Hi april24,

There's an excellent discussion of this question here:

https://www.beatthegmat.com/if-x-is-posi ... 70254.html

GMAT assassins aren't born, they're made,
Rich

If x is positive, which of the following could be the correct ordering of 1/x, 2x and x²?
I. x² < 2x < 1/x
II. x² < 1/x < 2x
III. 2x < x² < 1/x

(A) None
(B) I only
(C) III only
(D) I and II only
(E) I II and III

Let's start by PLUGGING IN some positive values of x and see what we get.

x = 1/2
1/x = 2
2x = 1
x² = 1/4
So, we get x² < 2x < 1/x
This matches statement I.

x = 3/4
1/x = 4/3
2x = 3/2
x² = 9/16
So, we get x² < 1/x < 2x
This matches statement II

x = 3
1/x = 1/3
2x = 6
x² = 9
So, we get 1/x < 2x < x²
NO MATCHES

At this point, the correct answer is either D or E.
If you're pressed for time, you might have to guess.

Alternatively, you can use some algebra to examine statement III (2x < x² < 1/x)
Notice that there are 2 inequalities here (2x < x² and x² < 1/x)
Take 2x < x² and divide both sides by x to get 2 < x
Take x² < 1/x and multiply both sides by x to get x^3 < 1, which means x < 1
Hmmm, so x is greater than 2 AND less than 1. This is IMPOSSIBLE, so statement III cannot be true.

Answer = D

Cheers,
Brent

Hi april24,

TESTing VALUES (the approach Brent used) is perfect for this question, so I won't rehash any of that here. Instead, I'll offer some insight into these types of questions.

Most Roman Numeral questions are based on Number Properties (the little rules that define how numbers "work", interact, etc.); if you know these rules, then you can move through Number Property questions rather quickly.

Here, we're told that X is POSITIVE, but that doesn't necessarily mean that X is an integer. We have to be THOROUGH to get the correct answer.

X could be ANYTHING POSITIVE, including fractions, 1, other integers, etc. so we have to do enough work to prove which Roman Numerals are possible and which is not.

The number properties that I see:
1) When X = 1, X^2 and 1/X are equal; since we're dealing with inequalities, using 1 is NOT an option
2) When X = a positive fraction, X^2 makes the number SMALLER; 1/X makes the number BIGGER
3) When X = an Integer > 1, X^2 makes the number BIGGER, 1/X makes the number SMALLER
4) When X = anything, 2X makes the number TWICE as BIG

With these rules and TESTing some values, you can easily get the correct answer.

GMAT assassins aren't born, they're made,
Rich