Ok, I'm dreading that this reply could potentially be quite lengthy, although perhaps it won't be as drastic as I fear. I then hope we can put this matter to rest, as I can't really keep devoting time to this.
Let's get some minutia out of the way. First, please don't cherry-pick an oversimplied example that was used merely to illustrate a point. Obviously in an actual calculation, I do and have factored in offense bonuses from mods; refer to the calculations in my original post in which I included not only a variable for flat bonuses, but also a variable for percent bonuses separately.
Second, I did notice a computational error in my calculation (though not an error of form). I redid my original calculations in the same way, except writing more legibly, and found I should have had a "-2z" term instead of the previous value. I have amended my original post to reflect this.
Now for the crux of the matter:
I think there are a couple of things going on here. One is that I think you get so set in your method or approach to doing things that you have difficulty seeing when an approach from a different angle is an equally valid setup--furthermore, the fact that you're evaluating the problem in chunks prevents you from seeing where our methods actually end up with the same procedure in the end--or at least, SHOULD...more on that below.
The second thing that I think is going on is that you left out or inadequately described details of your methodology (or perhaps they were buried in much earlier posts) and this lead to confusion on my part. Indeed, while your latest example helped with understanding the process you were undertaking, you actually still explain that process incorrectly with some of your statements.
Finally, I am sorry to point out that while working through your process, I did discover an error in your setup--likely because of your statement that you ignore the non-crit part of the equation when calculating the crit damage set. While you are correct that the set only affects the crit portion, the average damage under a crit set is very much a function of both non-crit and crit damage. To compare the average damage under each set, you need that portion. While you are correct that the offense set provides a 8.3% increase regardless of a crit (this is our earlier discussion that constants can be pulled out front), the increase in average damage is not 15.6%...which is the increase in just the crit portion. Just as adding offense bonuses from mods decreases the actual percentage increase of the offense set, the non-crit portion of the damage calculation decreases the average damage increase from the crit set in comparison to the increase it provides from the crit portion.
Interestingly enough, your final calculation does factor that non-crit portion back in (whether you realize it or not), and your method for finding the crit chance break point as the point where the two ratios become themselves a ratio equal to 1 would seem sound...
EXCEPT for the set up of those two ratios (probably as a result of initially trying to ignore that non-crit portion) results in a situation where you improperly split a fraction. You probably didn't even realize you were doing this, A) because your method of calculating out piecemeal values before setting up the next ration prevented you from realizing you were splitting fractions at all in the first place. B) Because who DOESN'T LOVE splitting fractions and isn't constantly thinking about them???? It even took a while for me working out your method on paper to realize that's why your step-by-step method was yielding different results than mine.
To be more clear, towards the end of your calculations, you have a situation resulting in ratio that, as a theoretical example, looks something like (a+b)/(c+d). Instead of taking a/(c+d) + b/(c+d), as would be correct, your final calculation essentially takes a/c + b/d, which is very incorrect. Again, you probably didn't realize you were doing this, as you were figuring quantities piecemeal and didn't see this as a problem involving partial fractions in the first place.
The good news is, you don't ACTUALLY have to split into partial fractions. Since you are comparing two values to one another and attempting to find the value of some variable which sets them equal to one another, you set the formula such that the quotient is equal to 1(the Identity of multiplication). You can then multiply both sides of the equation by the denominator. This does result in a situation, however, where the solution to the equation is to subtract like terms from each other--which is what my method was doing anyway when I subtracted one equation from the other and set the difference equal to 0 (the Identity of addition). I'm sorry you have such a grievance over linear equations. But this is simply the way you solve, algebraically, for unknown variables. I think you are focusing on this word, "inequality," and you are making it mean something in your head that it doesn't actually mean. When you compare two things...that's an inequality. When you want to find out when those things result in an EQUALITY, you set them equal to the appropriate Identity for the type of opration you are performing.
So enough with the theoretical, let's begin actually walking through your numbers.
Let's start with the endpoint of your calculations. You correctly determined that an offense set is worth 1.083 times the damage of what is provided without the set. This is a constant that is multiplied through both the crit and non-crit portions, so it therefore also holds true overall. You then determine that a crit damage set provides an increase of 2.22/1.92 = 1.15625 over not having the set. But this is ONLY true for the crit portion of the damage. So when you take 1.15625/1.083=1.0676, that represents the damage increase of the crit damage set over the offense set for the crit portion only . So in order to find the crit chance breaking point, you have to find the value of c (which is now invisible in the way you have this set up, making things difficult) for which this porportional increase would be equal to the proportional increase provided by the offense set over the critical damage set for the non-crit portion...or so you might think.
Here is where the fraction is inadvertently split incorrectly. We have already found the increase provided by the offense set for the non-crit portion: It is the same as that provided over the no-set bonus damage. So 1/1.083=0.92336 represents the loss of the crit damage set to the offense for the non-crit portion. So you intuit to find the value of c for which this ratio equals 1. Even keeping c invisible, you might figure that the bottom ratio only needs to travel 0.07664 to reach 1, and this distance is 53% of the way along the total distance, 1.0676-0.92336. Note that this is the same as 0.083/0.15625, which can be shown mathematically as the result of the ratio of the two ratios presented here. So this later result is how you arrived at the value of c.
But this is where you can see that a fraction was improperly split. And I apologize once again having to delve into the theoretical before getting back to more concrete numbers.
Let's call the the non-set bonus function for the basic damage calculation f (x) = x* (1-c)+x*c*1.92, where c is critical chance, and x is the total final offense number. Note that I am NOT leaving out the bonus offense from mods...I am simply incorporating them now into x so the EXAMPLE is easier to read...when we come back to actual numbers we will deal with those terms.
Why wait on adding them in? Because you have already calculated that the offense set only results in a 1.083 increase in the offense with the values you provided for this example. So we can use this ready value without having to muddle the example by recalculating.
So let us now call the offense set function, o (x) = 1.083x*(1-c)+1.083x*c*1.92.
The critical damage set function will be cd (x) = x*(1-c)+x*c*2.22.
In setting up the ratios the way you did, we are actually winding up with cd (x)/o (x), or,
(x *(1-c)+2.22*c*x)/(1.083x* (1-c)+1.92*1.08x*c).
And here is where the fraction is split. This SHOULD be evaluated as:
(x*(1-c))/(1.083x* (1-c)+1.92*1.08x*c) + (2.22*c*x)/(1.083x* (1-c)+1.92*1.08x*c),
thus keeping the whole denominator. Instead, the final step of your calculation essentially takes
(x *(1-c))/(1.083x*(1-c)) + (2.22*c*x)/(1.92*1.083x*c). This is, as you put it, improper math.
I demonstrate all this not to say your approach is somehow wrong. Your approach, conceptually, is a valid one. That is, you want to compare the proportional increase in damage for each of the different sets compared to not set, and then compare those values to one another and find the value of a variable, critical chance, for which this last ratio is equal to 1. I explained all this merely to point out that you made an error in setting up the problem--an error stemming from your assertion that the non-crit portion can be ignored with the crit damage set, even though the non-crit portion absolutely affects the average damage calculation.
Now I will endeavor to show how our two methodologies should actually arrive at the same result. You want to take the increase in damage from offense set over no set: o (x)/f (x). You want to do likewise with crit damage, so cd (x)/f (x). You then want to compare these values to one another, so (cd (x)/f (x))/(o (x)/f (x)). You can see that the two denominators in both the numerator and denominator of the resulting compound fraction actually cancel, by multiplying it by f (x)/f (x) (or 1). You wind up with cd (x)/o (x).
I've shown above that for this example, this is
(x *(1-c)+2.22*c*x)/(1.083x* (1-c)+1.92*1.08x*c).
The good news is, we don't ACTUALLY have to split this. Our goal is to determine the value of c for which this ratio equals 1 (meaning the two functions are equal in value). Or if you want to find all c for which one is greater than the other, you could turn the equals sign into one of your dreaded inequality signs (again, I'm not sure what you think is implied by this). So if cd (x)/o (x) = 1, we can simply multiply both sides of the equation by o (x).
So we have x*(1-c) + 2.22cx = 1.083x*(1-c) + 1.92*1.083cx.
With apologies for now having to do algebra like my approach did from the beginning, but this is simply the way you solve for unknowns.
x+1.22cx = 1.083x+0.99636cx
0.22634c = 0.083
c= 0.3667.
Let's now work with all the numbers, which I'm sure is what you are waiting for.
Let's here calculate x as the total culmination of all offense: x=2,888+3*0.085*2,888+3*0.045*2,888+990+185 = 5,189.32.
With an offense set, xo = 2,888*1.15+3*0.85*2,888+3*0.045*2,888+990+185 = 5,622.52.
In now comparing cd (x)/o (x), we have
(5,189.32*(1-c) + 5,189.32*2.22c)/(5,622.52*(1-c)+5,622.52*1.92c).
Setting this equal to 1, then multiplying both sides by the denominator and distributing, we have
5,189.32-5,189.32c+11,520.29c = 5,622.52 - 5,622.52c + 10,795.2384c
=
6,330.97c-5,172.718c = 5,622.52-5,189.32
=
1,158.252c = 433.2
c=0.374.
Note that the slight descrepancy from above is due to rounding error; when I redo the previous example with 1.083688 instead I got about 37.5, and I'm sure carrying to more decimal places in both examples would yield convergent results.
Anyway, you can see how we end up with subtractingthe terms of one side from the other, which is what I did in the beginning after setting the two equations equal to one another.
Now, let's plug these values into my formula from way back:
c= 1/ (1.5+2*(0.85*3+0.45*3)+(2*(990+185)/2888)-0.42) = 1/2.6737 = 37.4.
Although I think we could honestly dump some of the terms in that expression, if, when computing offense bonuses from mods, we take a hand calculation on the non offense set first and then just include it as one term.
But, all of these theoretical discussions are just that, because they assume those other mod bonuses as equal. In practice, as another user pointed out, these will be different based on the mods you have available. So really, you just need to put the sets on, look at the different offense values, and set up cd (x) and o (x) directly, keeping in mind that characters may get crit damage up for a portion of the time. Including the extra mod bonuses is useful for deciding if you want to farm new mods and want to guage how much your offense secondaries would need to be, which is also a variable you could solve for.
In short, are approaches (linear algebra or piecemeal comparison of ratios) should yield the same result, and are both valid conceptualizations of the same problem. However, I felt it dutiful to point out that in setting up your comparisons, your decisiom to ignore non-crit damage as an equalizer resulted in a mathematically unsound comparison, thus throwing off your calculations.
