What values of x satisfy x^2<x ?

value of x that satisfy x^2 < x.
x (x - 1) < 0 or x = 0 or x = 1
on the graph we plot a parabola (for quadratic equation x^2 - x < 0) as shown with roots at 1 and 0. the graph will open up since x^2 is positive.
from the graph we see that area less than 0 lies between 0 and 1
hence for the above equation to have "< 0" or NEGATIVE values, the range of x = 0 < x < 1

had it been x^2 - X > 0 the graph would have remained the same - opening up since x^2 is positive and roots = 0 and 1. but the region selected would have been the area of the curve over x-axis i.e. x > 1 or x < 0