Solve x+4/x-5<0

$$\frac { x + 4 } { x - 5 } < 0$$

Solve for x

$$x\in \left(-4,5\right)$$

Solution Steps

For the quotient to be negative, x+4 and x-5 have to be of the opposite signs. Consider the case when x+4 is positive and x-5 is negative.

$$x+4>0 x-5<0$$

The solution satisfying both inequalities is x\in \left(-4,5\right).

$$x\in \left(-4,5\right)$$

Consider the case when x-5 is positive and x+4 is negative.

$$x-5>0 x+4<0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(-4,5\right)$$