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x\left(2x-5\right)>0
Factor out x.
x<0 x-\frac{5}{2}<0
For the product to be positive, x and x-\frac{5}{2} have to be both negative or both positive. Consider the case when x and x-\frac{5}{2} are both negative.
x<0
The solution satisfying both inequalities is x<0.
x-\frac{5}{2}>0 x>0
Consider the case when x and x-\frac{5}{2} are both positive.
x>\frac{5}{2}
The solution satisfying both inequalities is x>\frac{5}{2}.
x<0\text{; }x>\frac{5}{2}
The final solution is the union of the obtained solutions.