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If x is positive, which of the following may be true?

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If x is positive, which of the following may be true?  [#permalink]

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New post Updated on: 01 Oct 2019, 15:11
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A
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C
D
E

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If x is positive, which of the following may be true?

I. \(x < x^2 < 1 + x\)
II. \(x < 1 - x < 2x\)
III. \(1 - x < x < 1/x\)

A. 1 Only
B. 1 and 2
C. 1 and 3
D. 2 and 3
E. 1 2 and 3

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Originally posted by TheNightKing on 05 Aug 2019, 13:38.
Last edited by TheNightKing on 01 Oct 2019, 15:11, edited 1 time in total.
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Re: If x is positive, which of the following may be true?  [#permalink]

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New post 07 Aug 2019, 03:01
How can option 2 may be correct? I have understood the point of 1 and 3
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Re: If x is positive, which of the following may be true?  [#permalink]

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New post 07 Aug 2019, 19:24
Hea234ven wrote:
How can option 2 may be correct? I have understood the point of 1 and 3


The answer to your question is that "X" can be a fraction.

For example with #2 when x=4/10, then 4/10<6/10<8/10

Same thing applies to number three but plug in 6/10 for X
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Re: If x is positive, which of the following may be true?  [#permalink]

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New post 07 Aug 2019, 19:52
Hello can someone please explain for II and III please ?
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Re: If x is positive, which of the following may be true?  [#permalink]

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New post 07 Aug 2019, 19:58
ctmerrill50 wrote:
Hea234ven wrote:
How can option 2 may be correct? I have understood the point of 1 and 3


The answer to your question is that "X" can be a fraction.

For example with #2 when x=4/10, then 4/10<6/10<8/10

Same thing applies to number three but plug in 6/10 for X



I have understood it.
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Re: If x is positive, which of the following may be true?  [#permalink]

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New post 07 Aug 2019, 22:03
TheNightKing wrote:
If x is positive, which of the following may be true?

I. x < x^2 < 1 + x
II. x < 1 - x < 2x
III. 1 - x < x < 1/x

A. 1 Only
B. 1 and 2
C. 1 and 3
D. 2 and 3
E. 1 2 and 3


For what value of x can I and III be true? Can someone please explain. I can see that II can be true for fractions.
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Re: If x is positive, which of the following may be true?  [#permalink]

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New post 07 Aug 2019, 22:12
1
We need not have a single value of x for all 3 statements; as long as x is positive. Try the values: 1.1, 0.49 and 0.51 for the three statements respectively.

If the answer is not E, please ignore the above reasoning and help me with an explanation instead!!! xD :P
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Re: If x is positive, which of the following may be true?  [#permalink]

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New post 08 Aug 2019, 10:04
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If x is positive, which of the following may be true?

I. x < x^2 < 1 + x
II. x < 1 - x < 2x
III. 1 - x < x < 1/x


I. Substracting x
0<x^2-x<1
0<x(x-1)<1
For x=1.5 it is true

II. Adding x
2x<1<3x
x<1/2<3x/2
1/3<x<1/2
So for x=1/2.5=2/5 it is true

III. Adding x
1<2x<1/x + x
From the left: x>1/2
From the right, multipliyng by x (since we know that x is positive, there is no problem with the sign)
2x^2<1+x^2
x^2<1
-1<x<1
So, using both sides
1/2<x<1
Hence with x=3/4 it is true

Answer (E)
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Re: If x is positive, which of the following may be true?  [#permalink]

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New post 07 Sep 2019, 06:13
I. x < x^2 < 1 + x
Subtracting X across the inequation,
0 < X^2 - X < 1
= 0 < X (X-1) < 1...….X (X-1) will be less than 1 when X-1 is less than one. Hence this equation is true for all values of X from 1.1 to 1.9. Hence TRUE

II. x < 1 - x < 2x
We can split it in two parts. Firstly, X < 1-X and secondly 1-X < 2X
X < 1-X which gives X < 1/2
1-X < 2X, which gives X > 1/3.
This inequality hold true for all values 1/3 < X < 1/2. Hence TRUE

III. 1 - x < x < 1/x
Similar to equation II , 1 - X < X gives X > 1/2 and X < 1/X gives X^2 -1 < 0 (Cross multiplied since X is positive). X^2-1 < 0 can be written as (X-1)(X+1) < 0.. This gives us than -1 < X < 1.
Hence this inequality is true for all values of X between 1/2 to 1 exclusive. Hence TRUE

All three inequations hold, hence IMO its (E)
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Re: If x is positive, which of the following may be true?   [#permalink] 07 Sep 2019, 06:13

If x is positive, which of the following may be true?

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