DS inequality clarification

[knight247]

Is x^2>x?
(1)x>0
(2)x<1

(1) If x=2 then x^2>x
If x=0.1 the x^2<x. INSUFFICIENT
(2) x<1

if x=0.1
If x=0.1 the x^2<x
If x=-5 then x^2>x. Getting conflicting answers on this one as well hence INSUFFICIENT

Combining both 0<x<1
x=0.9
x^2<x
x=0.1
x^2<x
SUFFICIENT. Hence C.

Just need an opinion on whether I've done all the calculations properly. Thanks

[force5]

Yes did it correct. C should be the answer.

[gmatboost]

I encourage everyone to memorize when
x < x^2
x > x^2
x = x^2

My preferred way to do this is to picture the graph of each one.
x is a line that moves up to the right.
x^2 is a classic parabola.

They meet at 0,0 and 1,1. Only between those points is the parabola below the line.

[GMATGuruNY]

Is x^2>x?
(1)x>0
(2)x<1

Another approach is to determine the critical points.

x²>x
x²-x > 0
x(x-1) > 0.

The critical points are 0 and 1.
These are the only values where x(x-1) = 0.
When x is any other value, x(x-1) < 0 or x(x-1) > 0.
To determine the range of x, test one value to the left and right of each critical point.

x<0.
Test x=-1 in x(x-1)>0.
(-1)(-1-1) > 0.
2 > 0.
This works.

0<x<1.
Test x = 1/2 in x(x-1)>0.
(1/2)(-1/2) > 0.
-1/4 > 0.
Doesn't work.

x>1.
Test x=2 in x(x-1)>0.
(2)(2-1) > 0.
2 > 0.
This works.

Question rephrased: Is x<0 or x>1?

Statement 1: x>0.
It's possible that x = 1/2 or that x = 2.
Insufficient.

Statement 2: x<1.
It's possible that x = 1/2 or that x = -2.
Insufficient.

Statements 1 and 2 combined:
0<x<1.
Thus, it's not possible that x<0 or that x>1.
Sufficient.

The correct answer is C.