Solve 1/x+frac{x^{2}-x-1}{x^{3}+x^2-x-1}=x-1/x^2+x^2-x+1/x^3-x

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

Solve for x

$$x=-\frac{1}{2}=-0.5$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -1,0,1 since division by zero is not defined. Multiply both sides of the equation by \left(x-1\right)x^{2}\left(x+1\right)^{2}, the least common multiple of x,x^{3}+x^{2}-x-1,x^{2},x^{3}-x.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

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

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

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

Use the distributive property to multiply x^{3}+2x^{2}+x by x-1 and combine like terms.

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

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

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

Combine x^{4} and x^{4} to get 2x^{4}.

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

Combine x^{3} and -x^{3} to get 0.

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

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

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

Multiply x-1 and x-1 to get \left(x-1\right)^{2}.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-1\right)^{2}.

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

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

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

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

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

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

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

Combine x^{4} and x^{4} to get 2x^{4}.

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

Subtract 2x^{4} from both sides.

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

Combine 2x^{4} and -2x^{4} to get 0.

$$-2x^{2}-x=-2x^{2}+1+x$$

Add 2x^{2} to both sides.

$$-2x^{2}-x+2x^{2}=1+x$$

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

$$-x=1+x$$

Subtract x from both sides.

$$-x-x=1$$

Combine -x and -x to get -2x.

$$-2x=1$$

Divide both sides by -2.

$$x=\frac{1}{-2}$$

Fraction \frac{1}{-2} can be rewritten as -\frac{1}{2} by extracting the negative sign.

$$x=-\frac{1}{2}$$