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

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

Solve for x (complex solution)

$$x\in \mathrm{C}\setminus -2,0$$

Solution Steps

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+2\right), the least common multiple of 2x+4,x^{2}+2x,2x.

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

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

$$x+x^{2}-2+\left(x+2\right)\left(x+1\right)=2x\left(x+2\right)$$

Use the distributive property to multiply x+2 by x+1 and combine like terms.

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

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

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

Combine x and 3x to get 4x.

$$4x+2x^{2}-2+2=2x\left(x+2\right)$$

Add -2 and 2 to get 0.

$$4x+2x^{2}=2x\left(x+2\right)$$

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

$$4x+2x^{2}=2x^{2}+4x$$

Subtract 2x^{2} from both sides.

$$4x+2x^{2}-2x^{2}=4x$$

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

$$4x=4x$$

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 -2,0.

$$x\in \mathrm{C}\setminus -2,0$$

Solve for x

$$x\in \mathrm{R}\setminus -2,0$$

Solution Steps

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 2x\left(x+2\right), the least common multiple of 2x+4,x^{2}+2x,2x.

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

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

$$x+x^{2}-2+\left(x+2\right)\left(x+1\right)=2x\left(x+2\right)$$

Use the distributive property to multiply x+2 by x+1 and combine like terms.

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

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

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

Combine x and 3x to get 4x.

$$4x+2x^{2}-2+2=2x\left(x+2\right)$$

Add -2 and 2 to get 0.

$$4x+2x^{2}=2x\left(x+2\right)$$

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

$$4x+2x^{2}=2x^{2}+4x$$

Subtract 2x^{2} from both sides.

$$4x+2x^{2}-2x^{2}=4x$$

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

$$4x=4x$$

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 -2,0.

$$x\in \mathrm{R}\setminus -2,0$$