Solve a+1/a-1>0

$$\frac { a + 1 } { a - 1 } > 0$$

Solve for a

$$a\in \left(-\infty,-1\right)\cup \left(1,\infty\right)$$

Solution Steps

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

$$a+1<0 a-1<0$$

The solution satisfying both inequalities is a<-1.

$$a<-1$$

Consider the case when a+1 and a-1 are both positive.

$$a-1>0 a+1>0$$

The solution satisfying both inequalities is a>1.

$$a>1$$

The final solution is the union of the obtained solutions.

$$a<-1\text{; }a>1$$