Solve 10x-x^2<0

$$10 x - x ^ { 2 } < 0$$

Solve for x

$$x\in \left(-\infty,0\right)\cup \left(10,\infty\right)$$

Solution Steps

Multiply the inequality by -1 to make the coefficient of the highest power in 10x-x^{2} positive. Since -1 is negative, the inequality direction is changed.

$$-10x+x^{2}>0$$

Factor out x.

$$x\left(x-10\right)>0$$

For the product to be positive, x and x-10 have to be both negative or both positive. Consider the case when x and x-10 are both negative.

$$x<0 x-10<0$$

The solution satisfying both inequalities is x<0.

$$x<0$$

Consider the case when x and x-10 are both positive.

$$x-10>0 x>0$$

The solution satisfying both inequalities is x>10.

$$x>10$$

The final solution is the union of the obtained solutions.

$$x<0\text{; }x>10$$