There are several ways to handle a personal matter over work, depending on the nature of the circumstance and the work environment. If it's the death of a family member, the matter can be prioritized, and time off from work may be allowed for bereavement. However, there are instances when personal matters have to be put on the back burner because work has to take precedence.
An example of prioritizing work over a personal matter is when one is working and gets a break-up email from one's significant other. It's normally uneasy, but that situation has to be handled after the work shift has ended because employers are not amenable to the idea of having an employee neglect work because of a relationship break-up. Whether one should prioritize work over life circumstances also depends on certain work environment factors, such as if one works in an office or freelance. If a person works freelance, he can manage his time accordingly and set aside work to attend to whatever personal matters come up. How well a person gets along with his boss is a significant factor, too, especially if he feels confident that he can talk to his boss about the problem. On the other hand, a person who has a strained work relationship with his boss might have difficulty getting away from the office even if the circumstances warrant it.