Multiply the inequality by -1 to make the coefficient of the highest power in $6x-x^{2}$ positive. Since $-1$ is negative, the inequality direction is changed.
$$-6x+x^{2}>0$$
Factor out $x$.
$$x\left(x-6\right)>0$$
For the product to be positive, $x$ and $x-6$ have to be both negative or both positive. Consider the case when $x$ and $x-6$ are both negative.
$$x<0$$ $$x-6<0$$
The solution satisfying both inequalities is $x<0$.
$$x<0$$
Consider the case when $x$ and $x-6$ are both positive.
$$x-6>0$$ $$x>0$$
The solution satisfying both inequalities is $x>6$.
$$x>6$$
The final solution is the union of the obtained solutions.
