Solve -5x/x-5<0

$$\frac { - 5 x } { x - 5 } < 0$$

Solve for x

$$x\in \left(-\infty,0\right)\cup \left(5,\infty\right)$$

Solution Steps

For the quotient to be negative, -5x and x-5 have to be of the opposite signs. Consider the case when -5x is positive and x-5 is negative.

$$-5x>0 x-5<0$$

The solution satisfying both inequalities is x<0.

$$x<0$$

Consider the case when x-5 is positive and -5x is negative.

$$x-5>0 -5x<0$$

The solution satisfying both inequalities is x>5.

$$x>5$$

The final solution is the union of the obtained solutions.

$$x<0\text{; }x>5$$