When negating a "complex" mathematical statement, which part of the statement should be negated? This is a general question, so I will provide an easy example, which everyone can reference for clarity. However, I am really looking for a general rule.
Consider the following implication that I am pulling from Daniel Solow's book How to Read and Do Proofs:
If $r$ is a real number such that $r^2 =2$, then $r$ is not rational.
Dr. Solow asks the reader to write the inverse of this implication.
I wrote it as follows:
If $r$ is not a real number that satisfies $r^2=2$, then $r$ is rational.
However, in the solutions manual, the answer is:
If $r$ is a real number such that $r^2 \neq 2$, then $r$ is rational.
In thinking about this a little bit, I am fairly certain these are not equivalent statements (though perhaps they are). In my version, I basically negated the "set from which we are drawing our number $r$ from"...i.e. I've made it so that it can be any number besides a real number. In Solow's correct answer, he negated the "property that the number exhibits"...i.e. being equal to $2$. 
There are many other instances where I observe this sort of discrepancy of "what to negate within a complex sentence". Is there a general rule of thumb that someone could share with me so that I do not make this mistake again?

I agree with Reese that natural language is a weak model for logic, so let's translate the sentence into the propositional calculus.  If we take $R=(r\in\mathbb R)$, $S=(r^2=2)$, and $Q=(r\in\mathbb Q)$, then the original statement is
$$R\to(S\to \neg Q)$$
("such that" can also be read as $(R \wedge S)\to\neg Q$, but those are logically equivalent, so we're fine.)   Since an implication is false if and only if it's antecedent is true and its consequent is false, we can simplify the negation of this statement as follows:
$$\neg (R\to(S\to\neg Q))$$
$$R\wedge\neg (S\to\neg Q)$$
$$R\wedge S\wedge Q$$
So given that, the negation of the statement is "$r$ is a real number such that $r^2=2$ and $r$ is rational."
