>> How do I exclude the "Appointments" that are not adjacent to a "Discharge"? <<
Your problem is the true data model is wrong. An appointment is made up of two parts because of a temporal interval. It has a beginning and it has an end. The ending of such an event can be null if the event is still ongoing. Download a free copy of temporal queries in SQL by Richard Snodgrass. It's available as a PDF from the University of Arizona computer science department website.
The reason you made this common mistake is that you still think of a clipboard and a physical sign in/sign out sheet. You didn't even know that a table has to have a key so you declared it in such a way that it would be impossible for your DDL to ever have a key. You also use the term "adjacent" which is a spatial concept that doesn't apply to the table. A table is an abstract unordered set of rows. If this is a list of patient appointments, where is the patient identifier? Is that what "mtmrn" means? Why do you think that T# is a clear, precise and useful name for a table?
CREATE TABLE Appointments
(mtmrn VARCHAR (30) NOT NULL, --- no idea what this means
visit_date DATE NOT NULL,
PRIMARY KEY (mtmrn, visit_date), --- keys are not an option!
discharge_date DATE); -- NULLs mean not finished yet
Microsoft has had the ANSI/ISO standard row constructors for quite a few years now. There is no need to use the old Sybase punchcard style one at a time Insertions. Here is one possible way of arranging your data.
INSERT INTO Appointments
('x51xx31', '2020-03-24', '2020-04-19'),
('x51xx31', '2020-02-18', '2020-03-31'),
('x51xx31', '2020-03-10', '2020-04-16'),
('x51xx31', '2020-04-21', NULL),
('x51xx31', '2020-04-24', NULL),
('x51xx31', '2020-05-19', NULL),
('x51xx31', '2020-06-15', NULL),
('x51xx31', '2020-07-07', '2020-07-08'),
('x51xx31', '2020-08-17', NULL),
('x51xx31', '2020-09-02', NULL),
('x51xx31', '2020-09-29', NULL),
('x51xx31', '2020-09-30', NULL);
Please post DDL and follow ANSI/ISO standards when asking for help.