Solve 2x-1/1-x>1

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

Solve for x

$$x\in \left(\frac{2}{3},1\right)$$

Solution Steps

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

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

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

$$-x>-1$$

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

$$x<1$$

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

$$2x-1>1-x$$

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

$$2x+x>1+1$$

Combine like terms.

$$3x>2$$

Divide both sides by 3. Since 3 is positive, the inequality direction remains the same.

$$x>\frac{2}{3}$$

Consider condition x<1 specified above.

$$x\in \left(\frac{2}{3},1\right)$$

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

$$-x<-1$$

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

$$x>1$$

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

$$2x-1<1-x$$

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

$$2x+x<1+1$$

Combine like terms.

$$3x<2$$

Divide both sides by 3. Since 3 is positive, the inequality direction remains the same.

$$x<\frac{2}{3}$$

Consider condition x>1 specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \left(\frac{2}{3},1\right)$$