Solve for x
x\in \left(-\infty,-\frac{2}{5}\right)\cup \left(\frac{3}{4},\infty\right)
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5x+2<0 4x-3<0
For the product to be positive, 5x+2 and 4x-3 have to be both negative or both positive. Consider the case when 5x+2 and 4x-3 are both negative.
x<-\frac{2}{5}
The solution satisfying both inequalities is x<-\frac{2}{5}.
4x-3>0 5x+2>0
Consider the case when 5x+2 and 4x-3 are both positive.
x>\frac{3}{4}
The solution satisfying both inequalities is x>\frac{3}{4}.
x<-\frac{2}{5}\text{; }x>\frac{3}{4}
The final solution is the union of the obtained solutions.
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