Solve for x
x\in (-\frac{5}{2},4]
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x-4\geq 0 2x+5<0
For the quotient to be ≤0, one of the values x-4 and 2x+5 has to be ≥0, the other has to be ≤0, and 2x+5 cannot be zero. Consider the case when x-4\geq 0 and 2x+5 is negative.
x\in \emptyset
This is false for any x.
x-4\leq 0 2x+5>0
Consider the case when x-4\leq 0 and 2x+5 is positive.
x\in (-\frac{5}{2},4]
The solution satisfying both inequalities is x\in \left(-\frac{5}{2},4\right].
x\in (-\frac{5}{2},4]
The final solution is the union of the obtained solutions.
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