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

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?

• The lines after "I started tackling it by solving the right side" are wrong as well — 61(x+1) should expand to 61x+61 Apr 27, 2017 at 23:56
• @TobyMak i made an error, and changed it back to normal. thanks for pointing it out Apr 27, 2017 at 23:57
• $4x+9(-4x-2)-2(-17x+34)+61+6=2x-19$ Apr 28, 2017 at 6:30

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

• great, great!! my soultion is not even real!! i now gotta change my 'real analysis' tag Apr 27, 2017 at 23:44
• If you replace "etc." with "$x=\pm i$", you have a complete answer lol Apr 27, 2017 at 23:46
• also, good call on the \cdots idea. i didn't know it existed! Apr 27, 2017 at 23:50
• @AlexanderDay : Actually this is neither real analysis nor complex analysis; rather it is algebra. Real analysis deals with limits and continuity and integration and differentiation, and complex analysis deals with consequences of the holomorphic nature of complex-valued functions of a complex variable. $$\S$$ As for typography, I use \ldots between commas, as in $1,\ldots, n,$ and \cdots between binary relation or binary operation signs as in $A+\cdots+Z$ or $A < B < \cdots < Z.$ \vdots and \ddots also exist and are sometimes used in matrices. Apr 28, 2017 at 0:35

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

the equation is

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

• sorry, and thanks. but this is a comment, not an answer Apr 27, 2017 at 23:42
• you right member seems not correct. Apr 28, 2017 at 0:04
• $1-68+61+6=67-67=0$, which gets $2x\pm 0$ Apr 28, 2017 at 0:31