Lets say
10x = 9.99999999 ... , then x= 0.999999999 ....
Thus, 9x = 10x - x = 9.9999999... - 0.9999999.... = 9
thus, since 9x = 9 , then x=1
Wait a second, I thought x=0.99999999 ....
0.999999... does not equal 1
Decimal representations are not unique. Every number that terminates with an infinite string of zeros can be represented by an expression that terminates with an infinite string of 9's. To understand this further, you would need to learn some topology and/or analysis. The equality comes from something called the $\displaystyle \varepsilon$-principle. Given two numbers $\displaystyle x,y$, if for all $\displaystyle \varepsilon>0$, $\displaystyle |x-y|<\varepsilon$, then $\displaystyle x=y$. Here, $\displaystyle |x-y|$ is the absolute value function, so I am referring to real numbers. If you are using a different metric $\displaystyle d$, then if $\displaystyle d(x,y)<\varepsilon$ for all $\displaystyle \varepsilon>0$, you have $\displaystyle x=y$.
You have to understand the basic rule of mathematics that if a = b and b = c, then a = c. In other words, if two things are equal to the same thing, then they are equal to each other.
Since you have shown that x is equal to both 0.9999... and 1, then that means 0.9999... = 1.
Think of it this way. If two number are different, then there are an infinite amount of numbers between them. The problem with .99999... and 1 is that there are NO numbers in between. This means that they are the same number.