Solve -x^2+2x>0

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

Solve for x

$$x\in \left(0,2\right)$$

Solution Steps

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

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

Factor out x.

$$x\left(x-2\right)<0$$

For the product to be negative, x and x-2 have to be of the opposite signs. Consider the case when x is positive and x-2 is negative.

$$x>0 x-2<0$$

The solution satisfying both inequalities is x\in \left(0,2\right).

$$x\in \left(0,2\right)$$

Consider the case when x-2 is positive and x is negative.

$$x-2>0 x<0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(0,2\right)$$