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DevB: Senior | Next Rank: 100 Posts; Posts: 43; Joined: Sun Apr 06, 2014 10:50 am; Thanked: 1 times; Followed by:1 members

Plug in question

by DevB » Tue Jul 22, 2014 5:16 am

The below question is from GMAT Prep practice exam so anyone planning to give the exam may avoid looking below.

For others, Please help in answering the following:

If x is positive, which of the following could be the correct ordering of 1/x, 2x and x^2 ?

1. x^2 < 2x < 1/x
2. x^2 < 1/x < 2x
3. 2x < x^2 < 1/x

a. None
b. 1 only
c. 3 only
d. 1 and 2 only
e. 1, 2, and 3

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Source: Beat The GMAT — Problem Solving | Tue Jul 22, 2014 5:16 am

abhasjha: Master | Next Rank: 500 Posts; Posts: 379; Joined: Tue Sep 30, 2008 7:17 am; Location: NY; Thanked: 28 times; Followed by:11 members

by abhasjha » Tue Jul 22, 2014 5:20 am

solution by mitch hunt

Determine the critical points by setting the expressions equal to each other:

1/x = 2x
2xÂ² = 1
xÂ² = 1/2
x = âˆš(1/2) = 1/âˆš2 â‰ˆ 1/1.4 â‰ˆ 10/14 â‰ˆ 5/7.

1/x = xÂ²
x^3 = 1
x = 1.

2x = xÂ²
x=2
(We can divide by x because x>0.)

The critical points are x=5/7, x=1, x=2.
These critical points indicate where two of the expressions are equal.
Thus, to the right and left of each critical point, the value of one expression must be greater than the value of another.

To determine which answer choices are possible, plug in one value to the left and one value to the right of each critical point.

x < 5/7:
If x=1/2, then:
1/x = 2.
xÂ² = 1/4.
2x = 1.
Since xÂ² < 2x < 1/x, we know that I could be true.
Eliminate A and C.

5/7 < x < 1:
If x = 3/4, then:
1/x = 4/3.
xÂ² = 9/16.
2x = 3/2.
Since xÂ² < 1/x < 2x, we know that II could be true.
Eliminate B.

In III, the largest value listed is 1/x.
For 1/x to be the largest value, x would have to be a fraction.
Having tried a fraction on each side of the critical point of 5/7, we know that there is no way that III could be true.

The correct answer is D.

https://www.beatthegmat.com/need-explanation-t85192.html

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GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

by Brent@GMATPrepNow » Tue Jul 22, 2014 5:41 am

If x is positive, which of the following could be the correct ordering of 1/x, 2x and xÂ²?
I. xÂ² < 2x < 1/x
II. xÂ² < 1/x < 2x
III. 2x < xÂ² < 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
xÂ² = 1/4
So, we get xÂ² < 2x < 1/x
This matches statement I.

x = 3/4
1/x = 4/3
2x = 3/2
xÂ² = 9/16
So, we get xÂ² < 1/x < 2x
This matches statement II

x = 3
1/x = 1/3
2x = 6
xÂ² = 9
So, we get 1/x < 2x < xÂ²
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 < xÂ² < 1/x)
Notice that there are 2 inequalities here (2x < xÂ² and xÂ² < 1/x)
Take 2x < xÂ² and divide both sides by x to get 2 < x [Aside: We can safely do this because we are told that x is positive]
Take xÂ² < 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

Brent Hanneson - Creator of GMATPrepNow.com