Inequalities for positive x
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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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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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As a tutor, I don't simply teach you how I would approach problems.
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Hi ,
This is a big testing value question.
But we are not restricted that x can not be equal to 1 . So if we put x=1 then none is correct.
Please explain and correct me if i am wrong.
Thanks,
This is a big testing value question.
But we are not restricted that x can not be equal to 1 . So if we put x=1 then none is correct.
Please explain and correct me if i am wrong.
Thanks,
- DavidG@VeritasPrep
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The key is the wording of the question: "Which of the following could be the correct ordering?" So the question is asking which could be the correct ordering for some values, not all. As soon as you find a single value that works - even if many others don't work- you know the ordering could be true.So if we put x=1 then none is correct.