A simple fraction with a complex answer: possibly right under my nose

Question

I've been doing some calculations in fractions, and found this equation pop up to calculate my answer:

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

the initial equation is

$$\frac{2(x-1)}{\frac{4(x+1)}{2}}+x=4x+9(-4x-2)-2(-17x+34)+61+6$$

(I used a random number generator)I started tackling it by solving the right side

$$ \begin{align} \cdots&=4x+9(-4x+2)-2(-17x+34)+61+6\\ &=4x+(-36x)+18-(-34x)-68+61+6\\ &=4x+(-36x)+18-(-34x)-68+61+6\\ &=4x-36x+1+34x-68+61+6\\ &=2x\\ \end{align} $$

then i simplified it even further using the other side as well, getting:

$$ \begin{align} \frac{2(x-1)}{\frac{4(x+1)}{2}}+x&=2x\\ \frac{2(x-1)}{2(x+1)}+x&=2x\\ \frac{x-1}{x+1}+x&=2x\\ \frac{x-1}{x+1}&=x\\ \end{align} $$

This is my problem. so what is $x???$ also, did I do this correctly? if not, could you solve the equation for me, and then still solve this annoying equation?

Answer 1

$$\frac{x-1}{x+1} = x$$ Multiply both sides by $x+1:$ $$ x-1 = x(x+1) $$ $$ x-1 = x^2 + x $$ $$ -1 = x^2 $$ $$\text{etc.} $$

Answer 2

The right hand side of the equation is $$x-19$$ not $2x $ as you wrote.

the equation is

$$\frac {x-1}{x+1}=x-19$$