A two part question.
1
True or False: when working with an equation or inequality, everything that you do is either:
- a substitution, or
- an operation performed on each side
Note that algebraic or numerical simplifications are substitutions – $2+2=4$, so we're free to substitute $4$ where $2+2$ is present in an equation.
2
Is there a generic name for 'performing an operation on both sides' of an equation/inequality?
Transformation? Equivalence derivation? I think that I've heard 'transpose' for the case of adding/subtracting something to both sides in order to eliminate it from one side, but I don't believe that this is used for, say, doubling or squaring both sides.
Best Answer
Is there a generic name for 'performing an operation on both sides' of an equation/inequality?
Yes. The name is algebra (maybe "doing algebra" is more appropriate and/or sounds better).
Here there are three sources:
EDIT
@BillDubuque pointed out that a correct answer for the question should give a name for the following rule: equalities are preserved by "performing an operation on both sides" (see the comments in this post).
In the Patrick Suppes terminology, the name of this rule can be Consequence of the Rule Governing Identities.
Remark 1: Given an operation $f$, let $S$ be the formula $f(z)=f(z)$. Then, by the RGI, $$x=y\quad\Longrightarrow\quad f(x)=f(y).$$ In words, "equalities are preserved by performing an operation on both sides".
Remark 2: The RGI also justifies general substitutions in equalities. For example, $x+y=2$ and $x=y+3$ implies $(y+3)+y=2$ (here, $S$ is the formula $x+y=2$). To understand the word "general" see the comments in the ASKASK's answer.
Remark 3: Other names (probably, the usual ones) are Replacement, Substitution Property and Substitution by equality (see other answers and comments).