Solve x/x-1>2

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

Solve for x

$$x\in \left(1,2\right)$$

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.

$$x>2\left(x-1\right)$$

Multiply out the right hand side.

$$x>2x-2$$

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

$$x-2x>-2$$

Combine like terms.

$$-x>-2$$

Divide both sides by -1. Since -1 is negative, the inequality direction is changed.

$$x<2$$

Consider condition x>1 specified above.

$$x\in \left(1,2\right)$$

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.

$$x<2\left(x-1\right)$$

Multiply out the right hand side.

$$x<2x-2$$

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

$$x-2x<-2$$

Combine like terms.

$$-x<-2$$

Divide both sides by -1. Since -1 is negative, the inequality direction is changed.

$$x>2$$

Consider condition x<1 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(1,2\right)$$