Algebra -assumptions

[chamb878]

I would really appreciate help with this problem! :)

Q Is x between 0 and 1?
1) x^2 is less than x
2) x^3 is positive
76. A
How can we assume that x is a positive number?
I know we are looking to see whether x is a fraction and a positive, just don't understand why we can assume x is positive.

[Binit]

Hi,
Here is how I approach this:
1) x^2 < x.
this implies 0<x<1. (For any other range of x, x^2>x. I can elaborate this: suppose, x=2 =>x^2=4; suppose x=-1/2,=>x^2=1/4>-1/2; suppose x=-2,=>x^2=4>-2.. I can't think of any other significant range of x).
So, (1) is sufficient.
2) x^3 is positive.
this only implies x to be positive. x could be anything: 1/2, 1, 45, 500,000 etc. So, (2) isn't sufficient.
Ans. (A)

[RonPurewal]

they don't have to specify that x is positive, because each statement implies that x is positive.

for statement 2 this is probably pretty clear-- if the cube of a number is positive, then the number must be positive.
(by contrast, if they told you "x^2 > 0", then the only thing you would know is that x ≠ 0.)

but ... look at statement 1 again.
if x is any negative number, then
... x^2 is positive
... x is negative
under these circumstances it's impossible to have x^2 < x, since you can't find a positive number that's smaller than a negative number.