Solve 2x-1/2x+1<1

$$\frac { 2 x - 1 } { 2 x + 1 } < 1$$

Solve for x

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

Solution Steps

Denominator 2x+1 cannot be zero since division by zero is not defined. There are two cases.

$$2x+1>0 2x+1<0$$

Consider the case when 2x+1 is positive. Move 1 to the right hand side.

$$2x>-1$$

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

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

The initial inequality does not change the direction when multiplied by 2x+1 for 2x+1>0.

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

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

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

Combine like terms.

$$0<2$$

Consider condition x>-\frac{1}{2} specified above.

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

Now consider the case when 2x+1 is negative. Move 1 to the right hand side.

$$2x<-1$$

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

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

The initial inequality changes the direction when multiplied by 2x+1 for 2x+1<0.

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

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

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

Combine like terms.

$$0>2$$

Consider condition x<-\frac{1}{2} specified above.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

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