Solve x-4/2x-5<0

$$\frac { x - 4 } { 2 x - 5 } < 0$$

Solve for x

$$x\in \left(\frac{5}{2},4\right)$$

Solution Steps

For the quotient to be negative, x-4 and 2x-5 have to be of the opposite signs. Consider the case when x-4 is positive and 2x-5 is negative.

$$x-4>0 2x-5<0$$

This is false for any x.

$$x\in \emptyset$$

Consider the case when 2x-5 is positive and x-4 is negative.

$$2x-5>0 x-4<0$$

The solution satisfying both inequalities is x\in \left(\frac{5}{2},4\right).

$$x\in \left(\frac{5}{2},4\right)$$

The final solution is the union of the obtained solutions.

$$x\in \left(\frac{5}{2},4\right)$$