<s>[INST] Is this proof for 1/4 mod 9 = x, correct? Find an integer x so (1/4) mod 9 = x
Proof:
> 1/4 mod 9 = x 
> 1 mod 9 = 4 * x
> - using x = 7 - 
> 1 mod 9 = 28 
> 28 mod 9 = 1 (to validate)
> - using Euler division theorem m = nq + r - 
> 28 = 9(3) + 1
> r = 1, so 1 = 1

I appreciate the feedback.
 [/INST] Here is a systematic approach to finding the inverse.
Since $\gcd(4, 9) = 1$, $4$ has a multiplicative inverse modulo $9$.  To find it, we must solve the equivalence 
$$4x \equiv 1 \pmod{9}$$
To find the solution, we can use the Euclidean Algorithm.
$$9 = 2 \cdot 4 + 1$$
Solving for $1$ yields
$$9 - 2 \cdot 4 = 1$$
Hence, $1 \equiv -2 \cdot 4 \pmod{9}$, so
$$x \equiv -2 \equiv 7 \pmod{9}$$
Therefore, 
$$4^{-1} \equiv \frac{1}{4} \equiv 7 \pmod{9}$$
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