If X is positive, which of the following could be the correct ordering of 1/x, 2x, and x^2
|. x^2<2x<1/x
||.x^2<1/x<2x
|||. 2x<x^2<1/x
A) None
B) | only
C) |||only
D) | and || only
E) |,||, and |||
OAD
What should be the approach of this kind of question?
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correct ordering of 1/x, 2x, and x^2
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GMATsid2016
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GMATGuruNY
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Determine the CRITICAL POINTS by setting the expressions equal to each other: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
c. III
d. I and II
e. I, II, and III
1/x = 2x
2x² = 1
x² = 1/2
x = √(1/2) = 1/√2 ≈ 1/1.4 ≈ 10/14 ≈ 5/7.
1/x = x²
x^3 = 1
x = 1.
2x = x²
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 left and right of each critical point, the value of one expression must be GREATER than the value of another.
To determine which of I, II and II could be true, plug in values to the left and right of each critical point.
Start with the range that many test-takers will fail to consider: 5/7 < x < 1.
5/7 < x < 1:
If x = 3/4, then:
1/x = 4/3.
x² = 9/16.
2x = 3/2.
Since x² < 1/x < 2x, we know that II could be true.
Eliminate A, B and C.
In statement III, 2x<x², which implies that 2<x.
But if x>2, then 1/x cannot be the greatest of the three values.
Thus, III is not possible.
Eliminate E.
The correct answer is D.
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Brent@GMATPrepNow
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Let's start by plugging in some positive values of x and see what we get.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
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
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Hi GMATsid2016,
Each of the attached explanations can be used to find the correct answer, so rather than provide another explanation, I thought that I'd 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
Each of the attached explanations can be used to find the correct answer, so rather than provide another explanation, I thought that I'd 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
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GMATsid2016
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Determine the CRITICAL POINTS by setting the expressions equal to each other:
1/x = 2x
2x² = 1
x² = 1/2
x = √(1/2) = 1/√2 ≈ 1/1.4 ≈ 10/14 ≈ 5/7.
1/x = x²
x^3 = 1
x = 1.
2x = x²
x=2
(We can divide by x because x>0.)
Hello GMATGuruNY,
Can you please explain the above part?
Thanks,
Sid
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GMATsid2016
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Hello Brent,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
Don't you think that plugging values will take so much of time?
Also, the value plugged is randomly or is there any specific reason?
Please advise.
Thanks,
Sid
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DavidG@VeritasPrep
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If you're decisive with the numbers you pick, you'll be surprised at how little time picking numbers actually takes. As for the strategy for what kind of numbers to pick, at times you'll just think about broad categories (negative integer/positive integer/fraction.)GMATsid2016 wrote:Hello Brent,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
Don't you think that plugging values will take so much of time?
Also, the value plugged is randomly or is there any specific reason?
Please advise.
Thanks,
Sid
Or you might get a little more fine-grained and picked something between 0 and 1/2 something between 1/2 and 1 and then something greater than 1. (This is what Brent did.)
The tip-off here that Brent's more fine-grained approach is the best way to proceed is that we're only working with positive values in this problem. If you could just say x = 1/4 and x = 2 and be done with it, it's not a very challenging question. When anything feels too easy, that's often a clue to be a tad more rigorous, which in this case, would mean trying something between 1/2 and 1.
