Using inverses to solve congruences The example I am looking at is for the question "what are the solutions of the linear congruence 3x ≡ 4 (mod7) ?

Let's say we know -2 is an inverse of 3mod7.

So we multiply by it: -2 * 3x ≡ -2*4 (mod 7)

*The book then says the answer is x ≡ -8 ≡ 6 (mod 7)

Why? I don't understand why 6 is positive.
$-8$ is congruent to $-1$ modulo $7$, and $7-1 = 6$.

Is this what you don't understand?
I don't know if I have the right answer, but I know what you want to ask.

We know $3x\equiv4\pmod7$ from the question. We know $-6\equiv1\pmod7$ from the inverse, and we can get $-24\equiv4\pmod7$ by multiplying 4.
Then we can have $3x=-24$, namely $x=-8$ as a single constant solution.

However, -8 is not the only answer to solve this congruence.

we can take -8 as $x_0$ and say $3(x_0+7k)\equiv4\pmod7$ where k is any integer.
We want to know, -8 is congruent to what modulo m, then we find $6\pmod7$.

I don't know why people don't understand this question. It really troubles me.