Solve for x
x\in \left(\frac{3}{2},4\right)
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4-x<0 2x-3<0
For the quotient to be positive, 4-x and 2x-3 have to be both negative or both positive. Consider the case when 4-x and 2x-3 are both negative.
x\in \emptyset
This is false for any x.
2x-3>0 4-x>0
Consider the case when 4-x and 2x-3 are both positive.
x\in \left(\frac{3}{2},4\right)
The solution satisfying both inequalities is x\in \left(\frac{3}{2},4\right).
x\in \left(\frac{3}{2},4\right)
The final solution is the union of the obtained solutions.
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