Solve 1/alpha+1+1/beta+1=beta+1+alpha+1/(alpha+1)(beta+1)

$$\frac { 1 } { \alpha + 1 } + \frac { 1 } { \beta + 1 } = \frac { \beta + 1 + \alpha + 1 } { ( \alpha + 1 ) ( \beta + 1 ) }$$

Solve for α

$$\alpha \neq -1$$

$$\beta \neq -1$$

Solution Steps

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

$$\beta +1+\alpha +1=\beta +1+\alpha +1$$

Add 1 and 1 to get 2.

$$\beta +2+\alpha =\beta +1+\alpha +1$$

Add 1 and 1 to get 2.

$$\beta +2+\alpha =\beta +2+\alpha$$

Subtract \alpha from both sides.

$$\beta +2+\alpha -\alpha =\beta +2$$

Combine \alpha and -\alpha to get 0.

$$\beta +2=\beta +2$$

Reorder the terms.

$$\text{true}$$

This is true for any \alpha .

$$\alpha \in \mathrm{R}$$

Variable \alpha cannot be equal to -1.

$$\alpha \in \mathrm{R}\setminus -1$$

Solve for β

$$\beta \neq -1$$

$$\alpha \neq -1$$

Solution Steps

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

$$\beta +1+\alpha +1=\beta +1+\alpha +1$$

Add 1 and 1 to get 2.

$$\beta +2+\alpha =\beta +1+\alpha +1$$

Add 1 and 1 to get 2.

$$\beta +2+\alpha =\beta +2+\alpha$$

Subtract \beta from both sides.

$$\beta +2+\alpha -\beta =2+\alpha$$

Combine \beta and -\beta to get 0.

$$2+\alpha =2+\alpha$$

Reorder the terms.

$$\text{true}$$

This is true for any \beta .

$$\beta \in \mathrm{R}$$

Variable \beta cannot be equal to -1.

$$\beta \in \mathrm{R}\setminus -1$$