Try re-writing the equation. The ~ equals the modulus in my equation

For 3x ~ 1 mod 6 to be true, 3x -1 = k6 must be true, where x and k are integers (with me so far?)

Then solve for x to see it can be an integer.

3x - 1 = k6

x - 1/3 = k2

x = k2 + 1/3

Therefore, for every case where k is an integer, x cannot be an integer so 3x ~ 1 mod 6 cannot be true.