Solve 2x-3/4-x<0

$$\frac { 2 x - 3 } { 4 - x } < 0$$

Solve for x

$$x\in \left(-\infty,\frac{3}{2}\right)\cup \left(4,\infty\right)$$

Solution Steps

For the quotient to be negative, 2x-3 and 4-x have to be of the opposite signs. Consider the case when 2x-3 is positive and 4-x is negative.

$$2x-3>0 4-x<0$$

The solution satisfying both inequalities is x>4.

$$x>4$$

Consider the case when 4-x is positive and 2x-3 is negative.

$$4-x>0 2x-3<0$$

The solution satisfying both inequalities is x<\frac{3}{2}.

$$x<\frac{3}{2}$$

The final solution is the union of the obtained solutions.

$$x>4\text{; }x<\frac{3}{2}$$