Solve x/x-1>1/4

$$\frac{ x }{ x-1 } > \frac{ 1 }{ 4 }$$

Solve for x

$$x\in \left(-\infty,-\frac{1}{3}\right)\cup \left(1,\infty\right)$$

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.

$$x>\frac{1}{4}\left(x-1\right)$$

Multiply out the right hand side.

$$x>\frac{1}{4}x-\frac{1}{4}$$

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

$$x-\frac{1}{4}x>-\frac{1}{4}$$

Combine like terms.

$$\frac{3}{4}x>-\frac{1}{4}$$

Divide both sides by \frac{3}{4}. Since \frac{3}{4} is positive, the inequality direction remains the same.

$$x>-\frac{1}{3}$$

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.

$$x<\frac{1}{4}\left(x-1\right)$$

Multiply out the right hand side.

$$x<\frac{1}{4}x-\frac{1}{4}$$

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

$$x-\frac{1}{4}x<-\frac{1}{4}$$

Combine like terms.

$$\frac{3}{4}x<-\frac{1}{4}$$

Divide both sides by \frac{3}{4}. Since \frac{3}{4} is positive, the inequality direction remains the same.

$$x<-\frac{1}{3}$$

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

$$x<-\frac{1}{3}$$

The final solution is the union of the obtained solutions.

$$x\in \left(-\infty,-\frac{1}{3}\right)\cup \left(1,\infty\right)$$