Given a set A of nonnegative integers, define the sum set A + A = {a_i + a_j | a_i, a_j ∈ A} and the difference set A − A = {a_i − a_j | a_i, a_j ∈ A}. The set A is said to be sum-dominant if |A + A| > |A − A|. In answering a question by Nathanson, Hegarty used a clever algorithm to find that the smallest cardinality of a sum-dominant set is 8. Since then, Nathanson has been asking for a human-understandable proof of the result. We offer a computer-free proof that a set of cardinality less than 6 is not sum-dominant. Furthermore, we prove that the introduction of at most two numbers into a set of numbers in an arithmetic progression does not give a sum-dominant set. This theorem eases several of our proofs and may shed light on future work exploring why a set of cardinality 6 is not sum-dominant. Finally, we prove that if a set contains a certain number of integers from a specific sequence, then adding a few arbitrary numbers into the set does not give a sum-dominant set.