Solve x/2x+1>0

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

Solve for x

$$x\in \left(-\infty,-\frac{1}{2}\right)\cup \left(0,\infty\right)$$

Solution Steps

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

$$x<0 2x+1<0$$

The solution satisfying both inequalities is x<-\frac{1}{2}.

$$x<-\frac{1}{2}$$

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

$$2x+1>0 x>0$$

The solution satisfying both inequalities is x>0.

$$x>0$$

The final solution is the union of the obtained solutions.

$$x<-\frac{1}{2}\text{; }x>0$$