They do not always have to be exhaustive. Consider the following example. You want to know if your school have more than a half students interested in a football team. If you find siginicant evidence that there is, your school will try to fund a team.
Suppose your school is huge, so you take a random sample and survey them about their preference. Now, the null hypothesis is people are neutral, the alternative should be more than half of all students are interested. The reason we do not include less than half in the alternative is because we do not care if there is less than half. Note the objective of a hypothesis test is to see whether there is sufficient evidence for the alternative hypothesis. Now, if there is acutally less people interested in such a team, this would not be counted as evidence toward the alternative hypothesis. Remember, in this one-tail test, we only reject the null if there is sufficient eveidence that more than half people are interested, in other words. If less than half people in the sample favors such a team, then we would simply fail to reject the null hypothesis since we do not find evidence that the alternative is true. However, note that even if we fail to reject, we do not accept the null hypothesis, i.e. we do not say we find evidence that the null is true.