This is one thing that keeps bugging me, and maybe there's a direct answer.
Grammatically, which one is more correct of these two? Does it make a difference?
I tried not to do that.
I tried to not do that.
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As some others have said, both are correct, and it is not wrong to say
I tried to not do that.
However, that is not the full story. Searching the Corpus of Contemporary American for various phrases (not to hold vs to not hold; not to know vs to not know; not to go vs to not go) reveals that the not to <verb> form is far more common:
| Verb | Not to <verb> | To not <verb> | % | | Hold | 97 | 6 | 94 | | Know | 1130 | 69 | 94 | | Go | 452 | 57 | 88 |
(Note that I didn't search for "not to [any verb]", because that also picks up certain fixed expressions such as "not to mention ..." which might distort the picture.)
So it's clear that the not to <verb> form is far more common. Furthermore, looking at the context of a sample of the to not <verb> examples, most of them appeared to be in speech (either on the radio, or quoted in a magazine), or very informal writing.
Searching the British National Corpus gives an even clearer bias - there, not to <verb> dominates by about 99%.
So in general usage, it is clear that not to <verb> is preferred by most writers.
Why is this? And when should one choose one expression or the other? Naturally this is rather subjective, so take the following explanation as my personal view on the matter, but note that it is consistent with what a lot of other people think.
So when might one want to say to not <verb>?
The normal form of a negative infinitive is "not to X", in all contexts.
The form "to not X" is grammatical (notwithstanding the generations of people who have moaned about "splitting the infinitive"), but unusual, and would only be used in order to convey a special meaning.
So "I try not to care" would be normal, but "I try to not care" would be spoken with an emphasis on the "not", and would suggest that I am trying very hard to do something specific "not caring" instead of caring.
Some people will tell you that you should say "I tried not to do that."
However, this is what I think:
When you say, "My goal was to do X," it's clear what that you had a goal and that it was to do X. When you say, "My goal was not to do X," was you goal to ensure that you not do X ("I tried not to do X"), or was doing X just not a goal ("My goal was not to do X but to do Y")? In context it might be clear what you mean based on whether or not you have the "but to do Y" part.
In the example you gave, someone saying that doing "that" simply wasn't a goal of theirs might say "I did not try to do that."
Even if ambiguity does arise, my statement is that you can go right ahead and say "I tried to not do that" if you and your audience are fine with it (or maybe even if your audience isn't fine with it but you choose to ;). In fact, I might recommend it.
Older English grammars attempted to overlay Latin structure on modern English, which made no sense but turned split or cleft infinitives into a bugbear of a certain class of pedant. You can go with the first one in every case and, while it will sound unnatural or even give the wrong meaning in some contexts, it will never be marked as incorrect on a test. (Oxford)
In this particular case, both questions have a similar meaning and there's only a minor shift in emphasis: Can you afford not having-this-solution versus Can you afford not-having-this-solution.
In some other cases, the placement of the adverb actually affects the meaning.
Can you afford really to risk your children's future?
is, at a surface level, asking about risking your children's future at an extreme level instead of a moderate one. You could read it as asking about genuinely risking your children's future versus not doing so, but you'd need a dramatic and unnatural pause on both sides of the adverb to make it work.
Can you afford to really risk your children's future?
is only asking about genuinely risking your children's future and most native speakers will naturally opt for it when they speak.
The problem is that some sociolects (like @Ricky's) have so internalized the mistaken latinate rule that they really find split infinitives to be jarring to the eye and ear. Since English teachers and the upper class are disproportionately represented in those sociolects, many of the rest of us play along in formal situations like tests or theses while continuing to happily go our own way in day-to-day speech.
I think to properly vet this subject one should remember that there are many kinds of verbs (state, event, transitive, etc.). I've seen four possibilities.
It seems to me most people on this forum are discussing example number three because of the necessity of the word "to". But I think the bigger question is where one puts the blasted negation "not" when confronted with a complex sentence. If it really is a question of emphasis of meaning, it seems to be a very subtle affair, the likes of which make my brain want to turn into mush. It also makes me wonder if the rule of "no double negatives" is grammatically absolute. How is someone like myself supposed to teach this kind of thing to students whose native tongue (French for example) allows for double negatives, as well as only having one infinitive for the three that exist in English? You don't! That's what I say. Blast the complexities of grammar!
The first version is quite correct:
Can you afford not to take this approach?
The other one is deliberately contrived and would only be correct if the speaker accentuates the contrivance by inserting a pause between "to" and "not" and emphasizing "not":
Can you afford to ... not!!! ... take this approach?