The following examples illustrate a left-, right-, and two-tailed test. This makes the data analyst use judgment rather than mindlessly applying rules.
Similarly, for a large p-value such as 0.4, as opposed to a p-value of 0.056 (alpha = 0.05 is less than either number), a data analyst should have more confidence that she made the correct decision in not rejecting the null hypothesis.
It is the key to conducting the appropriate test.