Solve for x_0
x_{0}\in \left(0,1\right)
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x_{0}\left(x_{0}-1\right)<0
Factor out x_{0}.
x_{0}>0 x_{0}-1<0
For the product to be negative, x_{0} and x_{0}-1 have to be of the opposite signs. Consider the case when x_{0} is positive and x_{0}-1 is negative.
x_{0}\in \left(0,1\right)
The solution satisfying both inequalities is x_{0}\in \left(0,1\right).
x_{0}-1>0 x_{0}<0
Consider the case when x_{0}-1 is positive and x_{0} is negative.
x_{0}\in \emptyset
This is false for any x_{0}.
x_{0}\in \left(0,1\right)
The final solution is the union of the obtained solutions.
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