For the quotient to be $≤0$, one of the values $9x$ and $5x-12$ has to be $≥0$, the other has to be $≤0$, and $5x-12$ cannot be zero. Consider the case when $9x\geq 0$ and $5x-12$ is negative.
$$9x\geq 0$$ $$5x-12<0$$
The solution satisfying both inequalities is $x\in \left[0,\frac{12}{5}\right)$.
$$x\in [0,\frac{12}{5})$$
Consider the case when $9x\leq 0$ and $5x-12$ is positive.
$$9x\leq 0$$ $$5x-12>0$$
This is false for any $x$.
$$x\in \emptyset$$
The final solution is the union of the obtained solutions.
