This has been bothering me since I read the OP.

Seems like there should be an obvious flaw and something about going from 10x = 9.999... to 9x = 9 rubs me the wrong way. But when we simply do as he says and subtract x, it yields 9x = 9.999... - x, then substituting the x yields the 9. So everything seems okay, despite substituting on only one side of the equation, which seems incredibly odd.

I think the issue is with that step though, because:

9(0.999...) = 9.999... - x does not seem valid. If we substitute here we have 9.999... = 9.

I've had 3 calculuses in years past. 

You're making a bit of a false dichotomy.  I'm saying that "x = ±2" does not mean "both -2 and 2 are solutions".  If I say that "J'ai faim" does not "I'm thirsty" in French, it doesn't mean I disagree with the result, only that it is not the correct interpretation.

The "both -2 and 2 are solutions" interpretation is *inclusive*, in that it actively claims that -2 works.  It is a common misconception that when you apply algebraic manipulations to solve an equation, you are creating solutions from the equations at hand.  In fact it is better to think of it as a winnowing process: initially x could be any real number, and you cannot hope to check all of them one at a time.  The equations are clues which one actively uses to narrow down the possibilities to a manageable few.

[Many students "learn" to just throw out solutions "that don't make sense" without any understanding of the process.  Those "extra" solutions are not really created from thin air: they were already among the infinite number of possibilities at the start, and the method simply failed to rule them out at an earlier stage.  Sometimes the manipulations are fully reversible and there is a perfect correspondence between the solutions you find and the actual solutions.  But this tends to happen only in trivially simple cases, and since these tend to be the first cases a learner sees, this misconception is easy to pick up.]

What I am saying is that the correct interpretation is *exclusive*, in other words "no number other than 2 or -2 is a solution".  This makes no claim that -2 is a solution, only that -1 and 3 and uncountably many more numbers cannot be equal to x.  But neither does it claim that -2 is NOT a solution.  Starting from the assumption "x=2" and deducing that "x=±2" is not a contradiction (starting from "x=±2" and deducing that "x=2" would be flawed logic, but that is another discussion entirely).

x = +/-2 is shorthand for +2 or -2, meaning they are both solutions. It's just the absolute value function f(x) = |x| where f(x) = 2, meaning -2 is indeed a solution for x. f(-2) = |-2| = 2.

I think you're more referring to what Jaydi said where x = 2 is not actually equivalent to x^2 = 4. The problem is, as I've repeated again and again that x = 2 is not a function. It's simply a declaration for the variable x as the value 2.

Good.

Just think of x^2 as a function of x, which I'm sure you already do. If you do that then yes, two different inputs (2 and -2) will have the same output (4), but that does not imply that they are equivalent. They just happen to have the same output when inputted in the quadratic function.

Just like the sin function. The input 0 will have the same output as pi, which has the same output as 2pi, which has the same output as 3pi, 4pi, 5pi, etc. All of these have the same output (0). But no one would ever say 0 = pi = 2pi. No, but they would say sin(0) = sin(pi), etc

Similarly, people just because X^2 = (-X)^2, that doesn't mean x = -x. It just mean the ^2 function is symmetrical across the Y axis and thus the inputs x & -x will have the same outputs, without being the same thing on their own.

But you're tying to go the other way. When you go from 4 to x=2 or x=-2, you're going from the output to the input (because you're going from the squared value to the square root, or from y to x). If you do this, then you're finding the inverse of the quadratic function. And when you take the inverse of a function, you have to limit the domain to where it is one-to-one before doing so. So you have to limit the quadratic function to [0, infinty) or (-infinity, 0]. That way, the function you use will give you one result.

Just like the sin function. If you want to take the inverse of a value, then you need to limit the domain to make the function one-to-one. If you didn't then it would be possible for you to input 1 into the inverse function, and get back pi/2, 3pi/2, 5pi/2, etc...implying that these values are all equal. But that's because you didn't restrict the sin function to where it was one-to-one. If you did then you would get one value.

That's why you never see the square root function graph including both y and -y values. The square root function is an inverse of the quadratic function. And to get the inverse of the quadratic function, you must restrict the domain to either all positive or all negative values, so it's one-to-one. THEN, you can draw the inverse graph, which would only include one square root for every square.

I fail to see how you get 9.999... = 9 if you substitute there. Substitution brings the equation to this form:

    9(0.999...) = 9.999... - 0.999... = 9,

from which we get (divide by 9) 0.999... = 1. Right?

I agree that in order to evaluate the function 1/(x^2) you need to limit it to a non-zero evaluation for x^2, however you are making a major error.

What was given was this: x^2 = 4. The function is then f(x) = (x^2) - 4  not f(x) = (x^-2) - 4 (inverse function) so no limits need be imposed. 

Resolving f(x)=(x^2) - 4 is simply a quadratic at y intercept -4, which has x intercepts at 2 and -2 confirming them as solutions.

I suggest you read what Jaydi said about "or", especially the point about the sentence "you're alive or you're dead".   It is not meant to imply they are both solutions, unless it is in the highly specific context that you are giving an answer to the specific question "what are the solutions to...?"  The mathematical sentence "x = ±2" does not by itself carry the connotation you ascribe to it that both options are equally possible (in light of other pieces of information).  There is nothing to criticise about the deduction "x=2, therefore x^2 = 4, therefore x=±2"; aside from not being very informative it's perfectly correct.  The inequivalence of "x^2=4" and "x=2" is a red herring: all logical deductions only lay claim to the forward direction (some just happen to be reversible).

I see my error. I made 9 * 0.999... to 9.999... for some reason.