Is x between 0 and 1?

Question

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.

bianalyst · Answer

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?

BrentGMATPrepNow · Answer

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:  A

RELATED VIDEO
 https://www.youtube.com/watch?v=Od98BOicIq4

BrentGMATPrepNow · Answer

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

bianalyst · Answer

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?

BrentGMATPrepNow · Answer

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:  A

RELATED VIDEO
 https://www.youtube.com/watch?v=Od98BOicIq4

BrentGMATPrepNow · Answer

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
 https://www.youtube.com/watch?v=wyR1Oj7AvzM

bianalyst · Answer

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!

bianalyst · Answer

Found Bunuel's graphical explanation here for anyone interested: https://gmatclub.com/forum/if-x-is-an-i ... ml#p731476

BrentGMATPrepNow · Answer

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

KanishkM · Answer

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

A

bumpbot · Answer

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Is x between 0 and 1?

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