If Ho is greater than or equal to (or less than or equal to), it is a one-sided test. It is also possible to reject or accept the original hypothesis, and there is no such thing as "false". If the original hypothesis is rejected, then the alternative hypothesis indicates the direction, that is, one of A >: B or a < B has been determined during the investigation.
Bilateral tests are more likely to reject Ho and get significant results. In fact, the comparison between A and B can be judged by their mean values, which is one of the reasons why bilateral tests are more commonly used.