My question is: Is there any disadvantages to thinking about Algebra
like this? Is there anything later in my math education that will
require me to know that I am subtracting or adding 2x to get rid of it
on this side?
As a college algebra instructor, I'm involved with remediation efforts for hundreds of students each year who have graduated high school but can't get started with college math, mostly due to incorrect concepts picked up in their prior schooling. So I would say "yes". There are some shortcuts that teachers can take to get students to pass some specific tests or programs that they are involved in; but the incorrect concepts definitely make things more difficult for students, sometimes overwhelmingly so, later on. (A majority of students that land in college remediation programs never get college degrees.)
The first thing that I would point out is that the "apply inverse operations to both sides" idea is generalizable to any mathematical operation; this allows you to cancel additions, subtractions, multiplications, divisions, exponents, radicals... even exponential, logarithmic, and trigonometric functions. (With appropriate fine print: no division by zero, square roots to both sides creates two plus-or-minus solutions, trigonometric inverses creates infinite cyclic solutions, etc.)
In contrast, the "move over and change the sign" method is not generalizable, as it only works for addend terms. This sets students on a course that requires memorizing many apparently different rules, one for each operation, which is much harder. When solving $2x = 10$, how is the multiplier of 2 canceled out? Must we remember to move it and turn it into the reciprocal 1/2? Will the students mistakenly change the sign and multiply by -1/2? Or add or multiply by -2 (I see this a lot)? How do we remove the division in $\frac{x}{2} = 5$ (probably some other rule)? How will we remember the seemingly totally different rule to solve $x^2 = 25$?
By way of analogy, I have college students who never memorized the times tables; they did manage to get through high school by repeatedly adding on their fingers, and can get through perhaps the first part of an algebra course that way. But then we start factoring and reducing radicals: "What times what gives you 54?" I might ask; "I have no idea!" will be the answer (this happened this past week; and here's a student who has effectively no chance of passing the rest of the course).
In summary: There are shortcuts or "tricks" that can get a student through a particular exam or test, which prove to be detrimental later on, as the "trick" fails in a broader context (like in this case, with any operations other than addition or subtraction). This then sets a student on a road to memorizing hundreds of little abstract rules, instead of a few simple big ideas, and at some point that complicated ad-hoc structure comes crashing down. Be polite and don't fight with your teacher to change things; but make sure to pick up a broader perspective for yourself, and share it with other students if they're willing, because you will need it later on. Take the opportunity to think about how you could improve on teaching the material, and then you may be on the path to being a master teacher yourself someday, and helping lots of people who need it.
The answer from u/lurking_quietly:
What makes Euler's totient function important is that for all positive $n\geq 2$, $\varphi(n)$ counts the number of elements of $\mathbb Z/n\mathbb Z$ which admit multiplicative inverses (i.e., $\varphi(n)$ counts the number of distinct units in this ring.)
Rather than have your students compute $\varphi(n)$ with no motivation, you might first ask them to compute the size of the unit group, $|U(\mathbb Z/n\mathbb Z)|$ for various values of $n\geq2$. When you later define the totient function via $\varphi(1) := 1$ and
$$\varphi(n) := |\{ k : 1≤k≤n \text{ and } \gcd(k,n) = 1 \}|,$$
then your students might better appreciate that you're computing something with relevance by introducing this definition of the totient function. (Note: $n=1$ merits emphasis as a special case. The definition above does indeed recover $\varphi(1)=1$, as desired. But in general, the "correct" definition of the totient function would be $\varphi(n) = |U(\mathbb Z/n\mathbb Z)|$. For the case $n=1$, this might be potentially ambiguous if we require $0\neq 1$ in the quotient ring, something very common in the definition of fields, for example.)
$$\rule{100pt}{1pt}$$
You can also introduce the totient function in the context of something like the finite sequence
$$\frac 1n, \frac 2n, ..., \frac{n-1}{n}, \frac nn.$$
For a fixed positive integer $n\geq 2$, how many elements for this sequence of order n have denominator $k$ when the fraction $j/n$ expressed in lowest terms? Answer: if $k|n, \varphi(k)$; otherwise, zero. This is related to the divisor sum identity given on the Wikipedia page for the totient function. There are multiple ways to verify this, ranging from the direct to Möbius inversion.
There are other directions you can go, too. For example, if your students are familiar with a little bit of ring theory, then you can explore how not only is $\varphi$ a multiplicative function, but its multiplicativity is closely related to the Chinese Remainder Theorem over the integers. If $m, n$ are positive integers greater than $1$ and $\gcd(m,n)=1$, then not only do we have
$$\varphi(mn) = \varphi(m) \varphi(n),$$
but we have the stronger result that
$$\mathbb Z/mn\mathbb Z \cong \mathbb Z/m\mathbb Z × \mathbb Z/n\mathbb Z$$
which restricts to an isomorphism of the unit groups of the respective rings:
$$U(\mathbb Z/mn\mathbb Z) \cong U(\mathbb Z/m\mathbb Z) × U(\mathbb Z/n\mathbb Z).$$
Roughly speaking, this means that when $m, n$ are coprime, not only is the number of units modulo $mn$ the same as (the number of units modulo $m$)$\times$(the number of units modulo $n$), but we also have that a unit modulo $mn$ is expressible in a unique way as the product of a unit modulo $m$ and a unit modulo $n$.
If you're even more ambitious (and have enough time), you might even consider possible generalizations to the totient function, too. For example, what might something like "$\varphi(1+4i)$" mean? One natural idea would be to try to count the number of units $|U(\mathbb Z[i]/(1+4i)\mathbb Z[i]|$, or the number of units in the Gaussian integers modulo $1+4i$. Or, alternatively, say that $p$ is a positive integral prime, and consider the polynomial ring $\mathbb Z/p\mathbb Z[x]$. What might "$\varphi(p(x))$" mean in context? Suggestion: set $(p(x)) := |U(\mathbb Z/p\mathbb Z[x]/(p(x))|$, the size of the group of units for polynomials in one variable with coefficients in $\mathbb Z/p\mathbb Z$, all modulo the polynomial $p(x)$.
I hope something in the above proves useful. Good luck!
Best Answer
It sounds like you want the sign function. Considering the restriction that $x \neq 0$, you could just write
$$\frac{x}{|x|}$$