Solve -1+2k/k<0

$$- \frac { 1 + 2 k } { k } < 0$$

Solve for k

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

Solution Steps

Divide both sides by -1. Since -1 is negative, the inequality direction is changed. Zero divided by any non-zero number gives zero.

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

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

$$2k+1<0 k<0$$

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

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

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

$$k>0 2k+1>0$$

The solution satisfying both inequalities is k>0.

$$k>0$$

The final solution is the union of the obtained solutions.

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