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Is x between 0 and 1?

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Is x between 0 and 1?  [#permalink]

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New post 05 Jun 2017, 14:17
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Is \(x\) between 0 and 1?

1) \(x^{2}<x\)

2) \(x\) is positive

Can someone please explain the official answer? This is from the Veritas Prep Data Sufficiency book.
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Re: Is x between 0 and 1?  [#permalink]

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New post 05 Jun 2017, 14:39
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What I don't get is why intuitively we know that x cannot be negative, since x^2 is always positive, but algebraically,

x^2<x
x^2-x<0
x(x-1)<0
x<0 or x<1

^here, x<0 or x<1. This shows x can be negative. Where am I going wrong?
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Re: Is x between 0 and 1?  [#permalink]

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New post 05 Jun 2017, 14:41
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bianalyst wrote:
Is \(x\) between 0 and 1?

1) \(x^{2}<x\)

2) \(x\) is positive



Target question: Is x between 0 and 1?

Statement 1: x² < x
Since we know that x² is POSITIVE, we can safely divide both sides of the inequality by x²
When we do this, we get: 1 < 1/x
If 1/x is greater than 1, we know that x must be POSITIVE (i.e., x > 0).
Since x is POSITIVE, we can safely multiply both sides of the inequality by x to get: x < 1
Combine the two inequalities to get: 0 < x < 1
In other words, x IS between 0 and 1
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x is positive
This doesn't tell us much.
It could be the case that x = 1/2, in which case, x IS between 0 and 1
Or it could be the case that x = 2, in which case, x is NOT between 0 and 1
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer:

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Re: Is x between 0 and 1?  [#permalink]

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New post 05 Jun 2017, 14:46
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bianalyst wrote:
What I don't get is why intuitively we know that x cannot be negative, since x^2 is always positive, but algebraically,

1. x^2<x
2. x^2-x<0
3. x(x-1)<0
4. x<0 or x<1

^here, x<0 or x<1. This shows x can be negative. Where am I going wrong?


Everything is correct up to step #3.
Step #4 is incorrect.

You are taking a rule about EQUATIONS and applying it to INEQUALITIES (which we can't do)
IF it we had the EQUATION x(x-1) = 0, then we could conclude that x = 0 or x = 1

However, we can't make the same kind of conclusion when it comes to inequalities.
In fact, if we test a value like x = -1, we see that the step #3 inequality, x(x-1)<0, does not hold true. So, we can't conclude that x<0

Cheers,
Brent
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Re: Is x between 0 and 1?  [#permalink]

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New post 05 Jun 2017, 14:54
Thanks - had no idea it only applied to equations. I'm trying to find a sure-fire algebraic approach to finding the range of values we can get from these types of inequalities questions; what approach would you recommend? I can see that you're determining the sign of x first and then dividing/multiplying to "solve" the inequality. Is this the general approach I should go for with these types of questions? Are there other ways to solve inequalities?
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Re: Is x between 0 and 1?  [#permalink]

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New post 05 Jun 2017, 14:57
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bianalyst wrote:
Is \(x\) between 0 and 1?

1) \(x^{2}<x\)

2) \(x\) is positive



Here's a even faster solution.

Target question: Is x between 0 and 1?

Statement 1: x² < x
For all values of x, x² will equal either zero or some positive number.
Since x is greater than x², we can be certain that x is positive.
Since x is POSITIVE, we can safely divide both sides of the inequality by x to get: x < 1
If x is positive AND x < 1, then we can be certain that x IS between 0 and 1
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x is positive
This doesn't tell us much.
It could be the case that x = 1/2, in which case, x IS between 0 and 1
Or it could be the case that x = 2, in which case, x is NOT between 0 and 1
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer:

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Re: Is x between 0 and 1?  [#permalink]

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New post 05 Jun 2017, 15:00
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bianalyst wrote:
Thanks - had no idea it only applied to equations. I'm trying to find a sure-fire algebraic approach to finding the range of values we can get from these types of inequalities questions; what approach would you recommend? I can see that you're determining the sign of x first and then dividing/multiplying to "solve" the inequality. Is this the general approach I should go for with these types of questions? Are there other ways to solve inequalities?


This question can be treated as a quadratic inequality question, or it can be treated as a general inequality question.
This one is pretty straightforward, so I used the latter approach.
If you're given a more complex quadratic inequality, (e.g., x² - 5x - 6 > 0), then I suggest the quadratic inequality approach (see my video below).

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Re: Is x between 0 and 1?  [#permalink]

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New post 05 Jun 2017, 15:22
GMATPrepNow wrote:
bianalyst wrote:
Thanks - had no idea it only applied to equations. I'm trying to find a sure-fire algebraic approach to finding the range of values we can get from these types of inequalities questions; what approach would you recommend? I can see that you're determining the sign of x first and then dividing/multiplying to "solve" the inequality. Is this the general approach I should go for with these types of questions? Are there other ways to solve inequalities?


This question can be treated as a quadratic inequality question, or it can be treated as a general inequality question.
This one is pretty straightforward, so I used the latter approach.
If you're given a more complex quadratic inequality, (e.g., x² - 5x - 6 > 0), then I suggest the quadratic inequality approach (see my video below).

RELATED VIDEO


Thanks, I actually found an easier solution that works for me - from my high school days I know what quadratic graphs look like, so I know in this case that this is a positive quadratic (x^2), and drawing out the parabola instantly tells me the range of x!
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Re: Is x between 0 and 1?  [#permalink]

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New post 05 Jun 2017, 15:28
Found Bunuel's graphical explanation here for anyone interested: https://gmatclub.com/forum/if-x-is-an-i ... ml#p731476
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Re: Is x between 0 and 1?  [#permalink]

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New post 05 Jun 2017, 16:20
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bianalyst wrote:
Thanks, I actually found an easier solution that works for me - from my high school days I know what quadratic graphs look like, so I know in this case that this is a positive quadratic (x^2), and drawing out the parabola instantly tells me the range of x!


I'm impressed that you remember how to graph parabolas from high school!! I'm sure most people can't make the same claim! :-D

Cheers,
Brent
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Re: Is x between 0 and 1?  [#permalink]

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New post 26 Jan 2019, 04:26
bianalyst wrote:
Is \(x\) between 0 and 1?

1) \(x^{2}<x\)

2) \(x\) is positive



Now x^2 will always be +ive, and the only region where

Statement 1 is true, is in between 0 and 1

Because higher the value of exponent lower is the value in the region of 0<x<1

Example 0.10 > 0.01,when x =0.1

Statement B doesn't tell us anything

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Re: Is x between 0 and 1?   [#permalink] 26 Jan 2019, 04:26
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