Solve for x
x\in \left(-\frac{1}{2},4\right)
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2x+1>0 x-4<0
For the quotient to be negative, 2x+1 and x-4 have to be of the opposite signs. Consider the case when 2x+1 is positive and x-4 is negative.
x\in \left(-\frac{1}{2},4\right)
The solution satisfying both inequalities is x\in \left(-\frac{1}{2},4\right).
x-4>0 2x+1<0
Consider the case when x-4 is positive and 2x+1 is negative.
x\in \emptyset
This is false for any x.
x\in \left(-\frac{1}{2},4\right)
The final solution is the union of the obtained solutions.
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