0.9999.... = 1.0

[ebw]

dsgrue3 said:

ebw said:
I'm saying that "x = ±2" does not mean "both -2 and 2 are solutions". 

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. 

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.

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).