Solve x+6/x^2-4+1/x+2=2/x-2

$$\frac { x + 6 } { x ^ { 2 } - 4 } + \frac { 1 } { x + 2 } = \frac { 2 } { x - 2 }$$

Solve for x (complex solution)

$$x\in \mathrm{C}\setminus -2,2$$

Solution Steps

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x^{2}-4,x+2,x-2.

$$x+6+x-2=\left(x+2\right)\times 2$$

Combine x and x to get 2x.

$$2x+6-2=\left(x+2\right)\times 2$$

Subtract 2 from 6 to get 4.

$$2x+4=\left(x+2\right)\times 2$$

Use the distributive property to multiply x+2 by 2.

$$2x+4=2x+4$$

Subtract 2x from both sides.

$$2x+4-2x=4$$

Combine 2x and -2x to get 0.

$$4=4$$

Compare 4 and 4.

$$\text{true}$$

This is true for any x.

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

Variable x cannot be equal to any of the values -2,2.

$$x\in \mathrm{C}\setminus -2,2$$

Solve for x

$$x\in \mathrm{R}\setminus 2,-2$$

Solution Steps

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right), the least common multiple of x^{2}-4,x+2,x-2.

$$x+6+x-2=\left(x+2\right)\times 2$$

Combine x and x to get 2x.

$$2x+6-2=\left(x+2\right)\times 2$$

Subtract 2 from 6 to get 4.

$$2x+4=\left(x+2\right)\times 2$$

Use the distributive property to multiply x+2 by 2.

$$2x+4=2x+4$$

Subtract 2x from both sides.

$$2x+4-2x=4$$

Combine 2x and -2x to get 0.

$$4=4$$

Compare 4 and 4.

$$\text{true}$$

This is true for any x.

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

Variable x cannot be equal to any of the values -2,2.

$$x\in \mathrm{R}\setminus -2,2$$