In an abstract algebra equation about groups, is "taking the inverse of both sides of an equation" an acceptable operation? I know you can right/left multiply equations by elements of the group, but was wondering if one can just take the inverse of both sides?

Recall these facts about the groups
Let A be a set, $\times a$ binary operation on A, and $a\in A$. Suppose that there is an identity element e for the operation. Then
- an element b is a left inverse for a if $b\times a=e$,
- an element c is a right inverse for a if $a\times c=e$,
- an element is an inverse(or two-sided inverse) for a if it is both a left and right inverse for a.
So in an abstract algebra equation about groups, taking the inverse of both sides of an equation is a valid statement. Since the inverse of an element exist iff its right inverse and left inverse exist. Consider an example.
Let G be the group with identity element e and binary operation $\times$.
Let a,b in G consider the equation $a\times x=b$. To solve the equation, take inverse of a both sides.
Case 1
${\displaystyle a^{-1}\cdot a\cdot x=a^{-1}\cdot b}$
${\displaystyle \Rightarrow e\cdot x=a^{-1}\cdot b}$
${\displaystyle \Rightarrow x=a^{-1}\cdot b}$
Case 2:
${\displaystyle a\cdot x\cdot a^{-1}=b\cdot a^{-1}}$
${\displaystyle \Rightarrow a\cdot a^{-1}\cdot x=b\cdot a^{-1}}$
${\displaystyle \Rightarrow e\cdot x=b\cdot a^{-1}}$
${\displaystyle \Rightarrow x=b\cdot a^{-1}}$