Solve 3+4x/x^2+x-1=3/x-x/x+1

$$\frac { 3 + 4 x } { x ^ { 2 } + x } - 1 = \frac { 3 } { x } - \frac { x } { x + 1 }$$

Solve for x (complex solution)

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

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^{2}+x,x,x+1.

$$3+4x+x\left(x+1\right)\left(-1\right)=\left(x+1\right)\times 3-xx$$

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

$$3+4x+x\left(x+1\right)\left(-1\right)=\left(x+1\right)\times 3-x^{2}$$

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

$$3+4x+\left(x^{2}+x\right)\left(-1\right)=\left(x+1\right)\times 3-x^{2}$$

Use the distributive property to multiply x^{2}+x by -1.

$$3+4x-x^{2}-x=\left(x+1\right)\times 3-x^{2}$$

Combine 4x and -x to get 3x.

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

Use the distributive property to multiply x+1 by 3.

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

Subtract 3x from both sides.

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

Combine 3x and -3x to get 0.

$$3-x^{2}=3-x^{2}$$

Subtract 3 from both sides.

$$3-x^{2}-3=-x^{2}$$

Subtract 3 from 3 to get 0.

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

Add x^{2} to both sides.

$$-x^{2}+x^{2}=0$$

Combine -x^{2} and x^{2} 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

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

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^{2}+x,x,x+1.

$$3+4x+x\left(x+1\right)\left(-1\right)=\left(x+1\right)\times 3-xx$$

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

$$3+4x+x\left(x+1\right)\left(-1\right)=\left(x+1\right)\times 3-x^{2}$$

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

$$3+4x+\left(x^{2}+x\right)\left(-1\right)=\left(x+1\right)\times 3-x^{2}$$

Use the distributive property to multiply x^{2}+x by -1.

$$3+4x-x^{2}-x=\left(x+1\right)\times 3-x^{2}$$

Combine 4x and -x to get 3x.

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

Use the distributive property to multiply x+1 by 3.

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

Subtract 3x from both sides.

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

Combine 3x and -3x to get 0.

$$3-x^{2}=3-x^{2}$$

Subtract 3 from both sides.

$$3-x^{2}-3=-x^{2}$$

Subtract 3 from 3 to get 0.

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

Add x^{2} to both sides.

$$-x^{2}+x^{2}=0$$

Combine -x^{2} and x^{2} 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.

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