Take the corned beef brisket out of the package and save the spice packet. Rinse the corned beef to get out some of the salt from the brining process. (Optional: If you are really watching your salt intake, soak the corned beef in water for 30 minutes before cooking then rinse.)
Place the corned beef brisket into the slow cooker pot. Sprinkle spice packet over the top of brisket.
Add enough room temperature water to cover brisket (about 6 cups, no more than 8 or you won’t be able to fit the remaining ingredients). Secure lid on slow cooker or multipurpose cooker and cook on low heat for 4 ½ hours.
While the corned beef is cooking, wash and cut the potatoes into halves then cut each half into eight equal pieces. If you have some smaller potatoes, cut them so they are about the same size and cut pieces from the larger potatoes.
Wash, peel, and cut the carrots into about 1 ½-inch pieces. Cut the thicker top end of the carrots into half lengthwise. Cut the cabbage into wedges.
After the corned beef has cooked 4 ½ hours, add the potatoes and carrots and cook for another 2 hours on low heat.
After the 2 hours of cooking with the potatoes and carrots have ended (6 ½ hours total), add the cabbage wedges to the slow cooker and cook for 30 minutes on low for firm cabbage, 1 hour for soft cabbage.
Once the vegetables are done, remove the corned beef brisket from the slow cooker and transfer to a cutting board. Let this cool for at least 10 minutes before cutting.
Pour the vegetables and cooking liquid into a colander. (Alternatively, you can use a slotted spoon or strainer to remove the vegetables if you plan on cooking more cabbage, see tips below.)
Slice the corned beef against the grain into ¼-inch to ½-inch thick slices. Serve with the vegetables and a mustard or creamy horseradish (optional).