The claim says, "half the class should be able to finish exam 3 for STA 3123 on-line, ... , in less than 145 minutes." Notice, the statement does not say the median is less than 145. It says that half the class will finish the exam in under 145 minutes. If the median is equal to 145, then half of the class will finish before 145 minutes (and half will finish after 145 minutes).
If you say the median is less than 145, then it would mean that more than half would finish in under 145 minutes. For example, if the median was 140 (which is less than 145), half of the class would finish in under 140 minutes. Some students would also finish somewhere between 140 and 145 minutes, which would mean more than half would finish before 145. The claim says half would finish before 145 minutes, not that more than half would finish. That is why you must say the median is equal to 145.
Notice the statement of the problem does not use the word "median." When a problem does not use the word "median," you need to be careful when writing the claim.