Solve x+1/x-1>-1

$$\frac { x + 1 } { x - 1 } > - 1$$

Solve for x

$$x\in \left(-\infty,0\right)\cup \left(1,\infty\right)$$

Solution Steps

Denominator x-1 cannot be zero since division by zero is not defined. There are two cases.

$$x-1>0 x-1<0$$

Consider the case when x-1 is positive. Move -1 to the right hand side.

$$x>1$$

The initial inequality does not change the direction when multiplied by x-1 for x-1>0.

$$x+1>-\left(x-1\right)$$

Multiply out the right hand side.

$$x+1>-x+1$$

Move the terms containing x to the left hand side and all other terms to the right hand side.

$$x+x>-1+1$$

Combine like terms.

$$2x>0$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x>0$$

Consider condition x>1 specified above.

$$x>1$$

Now consider the case when x-1 is negative. Move -1 to the right hand side.

$$x<1$$

The initial inequality changes the direction when multiplied by x-1 for x-1<0.

$$x+1<-\left(x-1\right)$$

Multiply out the right hand side.

$$x+1<-x+1$$

Move the terms containing x to the left hand side and all other terms to the right hand side.

$$x+x<-1+1$$

Combine like terms.

$$2x<0$$

Divide both sides by 2. Since 2 is positive, the inequality direction remains the same.

$$x<0$$

Consider condition x<1 specified above. The result remains the same.

$$x<0$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\infty,0\right)\cup \left(1,\infty\right)$$