Solve for x
x\in \left(-\frac{1}{5},0\right)
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5x^{2}+x<0
Multiply the inequality by -1 to make the coefficient of the highest power in -5x^{2}-x positive. Since -1 is negative, the inequality direction is changed.
x\left(5x+1\right)<0
Factor out x.
x+\frac{1}{5}>0 x<0
For the product to be negative, x+\frac{1}{5} and x have to be of the opposite signs. Consider the case when x+\frac{1}{5} is positive and x is negative.
x\in \left(-\frac{1}{5},0\right)
The solution satisfying both inequalities is x\in \left(-\frac{1}{5},0\right).
x>0 x+\frac{1}{5}<0
Consider the case when x is positive and x+\frac{1}{5} is negative.
x\in \emptyset
This is false for any x.
x\in \left(-\frac{1}{5},0\right)
The final solution is the union of the obtained solutions.
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