Variable Elimination (VE) is a central component of Bayesian network (BN) inference. Two drawbacks of VE are that it has no graphical depiction and that currently its intermediate distributions have no clear semantics. To address these drawbacks, we suggest two approaches, called Graphical VE (GVE) and Semantic VE (SVE). GVE graphically depicts the BN conditional probability tables (CPTs) as Graphical CPTs (GCPTs). The salient characteristic of GVE is that it explains the intricate mathematical equations and notations using graphs without resorting the numerical values and evaluates the inference technique using Merge and Remove graphical operations. GVE may be useful to introduce BN inference to beginners. On the other hand, SVE is an extension of VE that ensures that its intermediate factors have clear semantics, namely, they are defined with respect to the joint probability distribution. SVE exhibits this favorable property by introducing the concept of a p-segment, which ensures well-defined semantics. When the p-segment condition is not satisfied, Bayes theorem can always be applied, iteratively if needed, until it is satisfied. A novel method is suggested for generating p-segments in a good manner. SVE improves clarity in BN inference by avoiding the term potential, where potential is the biggest hindrance to the comprehension of probabilistic inference in BNs.