Solve for x
x\in \left(1,2\right)
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x-1<0 2-x<0
For the product to be positive, x-1 and 2-x have to be both negative or both positive. Consider the case when x-1 and 2-x are both negative.
x\in \emptyset
This is false for any x.
2-x>0 x-1>0
Consider the case when x-1 and 2-x are both positive.
x\in \left(1,2\right)
The solution satisfying both inequalities is x\in \left(1,2\right).
x\in \left(1,2\right)
The final solution is the union of the obtained solutions.
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