Solve 2x-5/x-1>0

$$\frac { 2 x - 5 } { x - 1 } > 0$$

Solve for x

$$x\in \left(-\infty,1\right)\cup \left(\frac{5}{2},\infty\right)$$

Solution Steps

For the quotient to be positive, 2x-5 and x-1 have to be both negative or both positive. Consider the case when 2x-5 and x-1 are both negative.

$$2x-5<0 x-1<0$$

The solution satisfying both inequalities is x<1.

$$x<1$$

Consider the case when 2x-5 and x-1 are both positive.

$$x-1>0 2x-5>0$$

The solution satisfying both inequalities is x>\frac{5}{2}.

$$x>\frac{5}{2}$$

The final solution is the union of the obtained solutions.

$$x<1\text{; }x>\frac{5}{2}$$