Solve 1/x(x+1)=1/x-1/x+1

$$\frac { 1 } { x ( x + 1 ) } = \frac { 1 } { x } - \frac { 1 } { x + 1 }$$

Solve for x (complex solution)

$$x\in \mathrm{C}\setminus 0,-1$$

Solution Steps

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+1\right), the least common multiple of x\left(x+1\right),x,x+1.

$$1=x+1-x$$

Swap sides so that all variable terms are on the left hand side.

$$x+1-x=1$$

Subtract 1 from both sides.

$$x-x=1-1$$

Subtract 1 from 1 to get 0.

$$x-x=0$$

Combine x and -x to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Variable x cannot be equal to any of the values -1,0.

$$x\in \mathrm{C}\setminus -1,0$$

Solve for x

$$x\in \mathrm{R}\setminus 0,-1$$

Solution Steps

Variable x cannot be equal to any of the values -1,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+1\right), the least common multiple of x\left(x+1\right),x,x+1.

$$1=x+1-x$$

Swap sides so that all variable terms are on the left hand side.

$$x+1-x=1$$

Subtract 1 from both sides.

$$x-x=1-1$$

Subtract 1 from 1 to get 0.

$$x-x=0$$

Combine x and -x to get 0.

$$0=0$$

Compare 0 and 0.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Variable x cannot be equal to any of the values -1,0.

$$x\in \mathrm{R}\setminus -1,0$$