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Manager  Joined: 07 Sep 2010
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If x is positive, which of the following could be correct ordering of  [#permalink]

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Hi Bunnel,

Could you please provide a reasoning to the below text... how did you find the range...Pls help

I am not sure how the OA is D
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Re: If x is positive, which of the following could be correct ordering of  [#permalink]

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imhimanshu wrote:
Hi Bunnel,

Could you please provide a reasoning to the below text... how did you find the range...Pls help

The reasoning is that in these ranges x (2x), 1/x and x^2 are ordered differently:

For $$x>2$$ --> $$x^2$$ has the largest value. Since no option offers this we know that $$x$$ cannot be more that 2;
For $$1<x<2$$ --> $$2x$$ has the largest value, then comes $$x^2$$. Since no option offers this we know that $$x$$ cannot be from this range either;

So, we are left with last range: $$0<x<1$$. In this case $$x^2$$ has the least value. Options, I and II offer this, so we should concentrate on them and test the values of x from 0 to 1.

Hope it's clear.
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GMAT 1: 710 Q47 V40 Re: If x is positive, which of the following could be the correct ordering  [#permalink]

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Bunuel wrote:
Basically we should determine relationship between , and in three areas: 0------1------2-------.

Buneul,

Could you please explain as to how you came to pick these ranges ? The rest of it is perfectly fine.

Thanks
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Re: If x is positive, which of the following could be correct ordering of  [#permalink]

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Hello Bunuel,
Can you show us graphical approach to this question.
I was able to draw the graph for all three equations and intersection points, however, I was not able to negate the third ordering. Would you please help me out.

Thanks
imhimanshu
P.S - How can I post graphs here.
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Re: If x is positive, which of the following could be correct ordering of  [#permalink]

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I picked numbers: 1/2, 1, 3/2, 2, 3

However, it didn occur to me that I must look something like 0.9. Request experts to help me understand the logic behind picking such numbers. Have my actual GMAT in 10 days, any help would be immensely valuable!

Thanks!
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Re: If x is positive, which of the following could be correct ordering of  [#permalink]

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Vavali wrote:
If x is positive, which of the following could be correct ordering of $$\frac{1}{x}$$, $$2x$$, and $$x^2$$?

(I) $$x^2 < 2x < \frac{1}{x}$$
(II) $$x^2 < \frac{1}{x} < 2x$$
(III) $$2x < x^2 < \frac{1}{x}$$

(a) none
(b) I only
(c) III only
(d) I and II
(e) I, II and III

first, let's get rid of at least one x in the given expressions, as x is positive just multiply the expressions by x and we'll get
(I) $$x^3 < 2x^2 < 1$$
as $$2x^2$$ < 1 we must pick a value which is < 1, let's pick 1/2 and it works. COULD BE

(II) $$x^3 < 1< 2x^2$$
Here we can see that $$x^3$$<1 so we must pick a value <1 BUT which will make $$2x^2$$ >1 if possible. Let's pick 0.9 and it works also here. COULD BE

(III) $$2x^2 < x^3 < 1$$
We must pick a value < 1 BUT as we've already seen, if we pick a fraction < 1 we cannot make $$2x^2 < x^3$$, in the above cases it $$2x^2 was > x^3$$ each time we picked a fraction < 1

Hope it helps.
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Re: If x is positive, which of the following could be the correct ordering  [#permalink]

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if i take value 9/10 it satisfies 2nd ordering but if i take 2/3 it doesnt satisfies 2nd ordering. why?
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Re: If x is positive, which of the following could be the correct ordering  [#permalink]

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sananoor wrote:
if i take value 9/10 it satisfies 2nd ordering but if i take 2/3 it doesnt satisfies 2nd ordering. why?

The question says "...could be the correct ordering of..."
So if even 1 positive value of x satisfies the ordering, it is included. Since 9/10 satisfies the second ordering, it is included in the ordering.

A transition point here is 1/sqrt(2). That is why values less than 1/sqrt(2) (such as 2/3) behave differently from values more than 1/sqrt(2) (such as 9/10).

Check here: http://www.veritasprep.com/blog/2013/05 ... on-points/
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Re: If x is positive, which of the following could be correct ordering of  [#permalink]

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Bunuel wrote:
ykaiim wrote:
IMO B.
for 0<x<1, only statement I holds.
Brunuel, if u put x=1/2:
II. II. x^2<1/x<2x >>>>> will not hold true.
x^2 = 1/4, 1/x=2 and 2x=1 then this expression will not hold.
1/4<2<1 [Incorrect]

If x=1/9 then:
x^2=1/81, 1/x=9 and 2x=2/9
1/81<9<2/9 [Incorrect]

Let's check the III option for above values:
III. 2x<x^2<1/x
For x=1/2: 1<1/4<2 [Incorrect]
For x=1/9: 2/9<1/81<9 [Incorrect]

OA IS D.

Algebraic approach is given in my solution. Here is number picking:

I. $$x^2<2x<\frac{1}{x}$$ --> $$x=\frac{1}{2}$$ --> $$x^2=\frac{1}{4}$$, $$2x=1$$, $$\frac{1}{x}=2$$ --> $$\frac{1}{4}<1<2$$. Hence this COULD be the correct ordering.

II. $$x^2<\frac{1}{x}<2x$$ --> $$x=0.9$$ --> $$x^2=0.81$$, $$\frac{1}{x}=1.11$$, $$2x=1.8$$ --> $$0.81<1.11<1.8$$. Hence this COULD be the correct ordering.

III. $$2x<x^2<\frac{1}{x}$$ --> $$x^2$$ to be more than $$2x$$, $$x$$ must be more than 2 (for positive $$x-es$$). But if $$x>2$$, then $$\frac{1}{x}$$ is the least value from these three and can not be more than $$2x$$ and $$x^2$$. So III can not be true.

Thus I and II could be correct ordering and III can not.

how did u choose the numbers for plugging in. i guess thats the trick in inequalities
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Re: If x is positive, which of the following could be correct ordering of  [#permalink]

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vsvikas wrote:
how did u choose the numbers for plugging in. i guess thats the trick in inequalities

Use transition points.
Discussed here: https://www.veritasprep.com/blog/2013/0 ... on-points/
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If x is positive, which of the following could be correct ordering of  [#permalink]

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Can't any one or two numbers(that are same) be used in all the three statements? The problem is plugging number in statement II is not satisfying. Pls explain by putting in same numbers in all statements. (I got that Statment 3 is not satisfying)

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Re: If x is positive, which of the following could be correct ordering of  [#permalink]

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topper97 wrote:
Can't any one or two numbers(that are same) be used in all the three statements? The problem is plugging number in statement II is not satisfying. Pls explain by putting in same numbers in all statements. (I got that Statment 3 is not satisfying)

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You cannot plug in same numbers in each statement and expect to get the answer. The requirement of each statement is different. Hence, we need to find transition points to ensure that we get the answer. This is explained in the link given here: https://gmatclub.com/forum/if-x-is-posi ... l#p1748102
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Re: If x is positive, which of the following could be correct ordering of  [#permalink]

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Hi Bunuel, bb
how can I find more practice questions which are of this type?
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GMAT 1: 670 Q46 V36 GMAT 2: 690 Q47 V38 If x is positive, which of the following could be the correct ordering  [#permalink]

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

I. $$x^2 < 2x < \frac{1}{x}$$

II. $$x^2 < \frac{1}{x} < 2x$$

III. $$2x < x^2 < \frac{1}{x}$$

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

Method: Plug Using Transition Points

x = +
Identify Transition Points:

1/x = 2x => 1 = 2x^2 => x = 1/sqrt(2)
2x = x^2 => x^2 - 2x = 0 => x(x-2) = 0 => x = 0 or 2 (0 is not possible since x = +)
x^2 = 1/x => x^3 = 1 => x = 1
Therefore, transition points = 1/sqrt(2), 1, 2
Need to check numbers between those transition points

Range: Value of x, 1/x, 2x, x^2, Order
< 1/sqrt(2): 1/10, 10, 1/5, 1/100 x^2 < 2x < 1/x => I
1/sqrt(2) to 1: 0.99, 1/0.99, 1.98, ~1 x^2 < 1/x < 2x => II
1 to 2: 3/2, 2/3, 3, 9/4 1/x < x^2 < 2x (Not given)
> 2: 5/2, 2/5, 5, 25/4 1/x < 2x < x^2 (Not given)

Hence, only I & II could be true. ANSWER: D

Method: Using Inequalities (along with wavy line)

Split complex inequality into 2 simpler inequalities to see whether there is an overlap. If there is, that complex inequality could be true, otherwise not.

x = +
1st Inequality (I):
a) x^2 < 2x => x^2 - 2x < 0 => x(x-2) < 0 => 0 < x < 2
b) 2x < 1/x => 2x^2 < 1 => -1/sqrt(2) < x < 1/sqrt(2) => 0 < x < 1/sqrt(2) [since x=+]
Overlap: 0 < x < 1/sqrt(2) => I could be true

2nd Inequality (II):
a) x^2 < 1/x => x^3 < 1 => x < 1
b) 1/x < 2x => 1 < 2x^2 => x < -1/sqrt(2) or x > 1/sqrt(2) => x > 1/sqrt(2) [since x=+]
Overlap: 1/sqrt(2) < x < 1 => II could be true

3rd Inequality (III):
a) 2x < x^2 => 0 < x^2 - 2x => 0 < x(x-2) => x < 0 or x > 2 => x > 2 [since x=+]
b) x^2 < 1/x => x^3 < 1 => x < 1
Overlap: None => III could not be true

This Q has been explained in depth here: https://gmatclub.com/forum/inequalities ... l#p1582961
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Re: If x is positive, which of the following could be correct ordering of  [#permalink]

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Vavali wrote:
If x is positive, which of the following could be correct ordering of $$\frac{1}{x}$$, $$2x$$, and $$x^2$$?

I. $$x^2 < 2x < \frac{1}{x}$$

II. $$x^2 < \frac{1}{x} < 2x$$

III. $$2x < x^2 < \frac{1}{x}$$

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

First, if x = 1/2, then 1/x = 2, 2x = 1, and x^2 = 1/4; i.e. x^2 < 2x < 1/x. Thus, Roman numeral I is possible. We eliminate answer choice A.

Since we eliminated the answer choice “none” and since 1/x is greater than x^2 in every Roman numeral; it must be true that x < 1.

For Roman numeral II, in order for 1/x < 2x to hold, we must have 2x^2 > 1, which is equivalent to x^2 > 1/2. This implies that x > 1/√2 or x < -1/√2. The latter is not possible because x is positive. Since √2 is roughly 1.41, we see that if we let x = 1/1.4 = 5/7, then we have 1/x = 7/5, 2x = 10/7, and x^2 = 25/49. In this case, we have x^2 < 1/x < 2x and thus, Roman numeral II is also possible.

Finally, for Roman numeral III to be true, we must have x^2 > 2x, or equivalently, x^2 - 2x > 0. Factoring the left hand side, we get x(x - 2) > 0. In order for the product of x and (x - 2) to be positive, either both of them must be positive or both of them must be negative. If both x and (x - 2) are positive, then x > 2; but then x^2 > 1/x. If both of them are negative, then x < 0 but this contradicts the fact that x is positive. Therefore, Roman numeral III is not possible.

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GMAT 1: 690 Q50 V33 Re: If x is positive, which of the following could be correct ordering of  [#permalink]

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Bunuel how to decide which ranges to be checked ?

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

I. $$x^2<2x<\frac{1}{x}$$

II. $$x^2<\frac{1}{x}<2x$$

III. $$2x<x^2<\frac{1}{x}$$

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

First note that we are asked "which of the following COULD be the correct ordering" not MUST be.
Basically we should determine relationship between $$x$$, $$\frac{1}{x}$$ and $$x^2$$ in three areas: $$0<1<2<$$.

$$x>2$$

$$1<x<2$$

$$0<x<1$$

When $$x>2$$ --> $$x^2$$ is the greatest and no option is offering this, so we know that x<2.
If $$1<x<2$$ --> $$2x$$ is greatest then comes $$x^2$$ and no option is offering this.

So, we are left with $$0<x<1$$:
In this case $$x^2$$ is least value, so we are left with:

I. $$x^2<2x<\frac{1}{x}$$ --> can $$2x<\frac{1}{x}$$? Can $$\frac{2x^2-1}{x}<0$$, the expression $$2x^2-1$$ can be negative or positive for $$0<x<1$$. (You can check it either algebraically or by picking numbers)

II. $$x^2<\frac{1}{x}<2x$$ --> can $$\frac{1}{x}<2x$$? The same here $$\frac{2x^2-1}{x}>0$$, the expression $$2x^2-1$$ can be negative or positive for $$0<x<1$$. (You can check it either algebraically or by picking numbers)

Second condition: $$x^2<\frac{1}{x}<2x$$

The question is which of the following COULD be the correct ordering not MUST be.

Put $$0.9$$ --> $$x^2=0.81$$, $$\frac{1}{x}=1.11$$, $$2x=1.8$$ --> $$0.81<1.11<1.8$$. Hence this COULD be the correct ordering.

Hope it's clear.
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_________________ Re: integers and inequalities   [#permalink] 09 Oct 2019, 04:13

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