Solve x/2x-6-x/2x+2=2x/(x+1)(x-3)

$$\frac { x } { 2 x - 6 } - \frac { x } { 2 x + 2 } = \frac { 2 x } { ( x + 1 ) ( x - 3 ) }$$

Solve for x (complex solution)

$$x\in \mathrm{C}\setminus 3,-1$$

Solution Steps

Variable x cannot be equal to any of the values -1,3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-3\right)\left(x+1\right), the least common multiple of 2x-6,2x+2,\left(x+1\right)\left(x-3\right).

$$\left(x+1\right)x-\left(x-3\right)x=2\times 2x$$

Use the distributive property to multiply x+1 by x.

$$x^{2}+x-\left(x-3\right)x=2\times 2x$$

Use the distributive property to multiply x-3 by x.

$$x^{2}+x-\left(x^{2}-3x\right)=2\times 2x$$

To find the opposite of x^{2}-3x, find the opposite of each term.

$$x^{2}+x-x^{2}+3x=2\times 2x$$

Combine x^{2} and -x^{2} to get 0.

$$x+3x=2\times 2x$$

Combine x and 3x to get 4x.

$$4x=2\times 2x$$

Multiply 2 and 2 to get 4.

$$4x=4x$$

Subtract 4x from both sides.

$$4x-4x=0$$

Combine 4x and -4x 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 -1,3.

$$x\in \mathrm{C}\setminus -1,3$$

Solve for x

$$x\in \mathrm{R}\setminus 3,-1$$

Solution Steps

Variable x cannot be equal to any of the values -1,3 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-3\right)\left(x+1\right), the least common multiple of 2x-6,2x+2,\left(x+1\right)\left(x-3\right).

$$\left(x+1\right)x-\left(x-3\right)x=2\times 2x$$

Use the distributive property to multiply x+1 by x.

$$x^{2}+x-\left(x-3\right)x=2\times 2x$$

Use the distributive property to multiply x-3 by x.

$$x^{2}+x-\left(x^{2}-3x\right)=2\times 2x$$

To find the opposite of x^{2}-3x, find the opposite of each term.

$$x^{2}+x-x^{2}+3x=2\times 2x$$

Combine x^{2} and -x^{2} to get 0.

$$x+3x=2\times 2x$$

Combine x and 3x to get 4x.

$$4x=2\times 2x$$

Multiply 2 and 2 to get 4.

$$4x=4x$$

Subtract 4x from both sides.

$$4x-4x=0$$

Combine 4x and -4x 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 -1,3.

$$x\in \mathrm{R}\setminus -1,3$$