Solve for x
x\in \left(-\frac{1}{3},\frac{1}{2}\right)
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3x+1<0 5-10x<0
For the product to be positive, 3x+1 and 5-10x have to be both negative or both positive. Consider the case when 3x+1 and 5-10x are both negative.
x\in \emptyset
This is false for any x.
5-10x>0 3x+1>0
Consider the case when 3x+1 and 5-10x are both positive.
x\in \left(-\frac{1}{3},\frac{1}{2}\right)
The solution satisfying both inequalities is x\in \left(-\frac{1}{3},\frac{1}{2}\right).
x\in \left(-\frac{1}{3},\frac{1}{2}\right)
The final solution is the union of the obtained solutions.
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