Prove 5x-4 is even iff 3x+1 is odd.
I have two questions: First, for example, could we assume that 5x-4 is even such that
5x-4 = 2k, for some integer k
and then manipulate the above equation to produce 5x-4, which would be conducted as follows
5x-4 = (3x + 2x) + (-5 + 1) = 2k
3x + 1 = 2k - 2x + 5 = 2(k-x+2)+1,
which shows that 3x+1 is odd, since (k-x+2) is an integer.
Would the above sequence of steps be acceptable?
My second question is, How might this proof be demonstrated by cases? I understand that there are easier methods of proof, but we were asked in class to do a proof by cases.