Solve 3x+2/x-1>0

$$\frac { 3 x + 2 } { x - 1 } > 0$$

Solve for x

$$x\in \left(-\infty,-\frac{2}{3}\right)\cup \left(1,\infty\right)$$

Solution Steps

For the quotient to be positive, 3x+2 and x-1 have to be both negative or both positive. Consider the case when 3x+2 and x-1 are both negative.

$$3x+2<0 x-1<0$$

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

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

Consider the case when 3x+2 and x-1 are both positive.

$$x-1>0 3x+2>0$$

The solution satisfying both inequalities is x>1.

$$x>1$$

The final solution is the union of the obtained solutions.

$$x<-\frac{2}{3}\text{; }x>1$$