Whether you need to "bound |x+3|" depends on what you are doing! If you are trying to prove that a certain limit is true at x= 3, then, yes, since you want |x-3|< constant, you are going to need a bound on |x+3|.
As for |2/(3x)||x- 1/2|< A, that is the same as |x- 1/2|< A/|2/(3x)|. Comparing with you inequality above, yes, the thing corresponding to |x+3| is |2/(3x)|. Why would you think it would be |3x/2|?