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These are commonly sold at stores that sell moving supplies, including U-Haul locations. You'll also be able to find mattress covers at a local hardware store. They commonly sell for about $5$10 USD. During the process of being moved, the sides and top of your mattress may come in contact with floors, stairs, the ground outside, and the inside of a moving van (or the top of your car). You can use a plastic mattress cover to prevent your mattress from collecting dirt and permanent stains. If you're simply moving the mattress from one room to another within the same house or apartment, you don't need to purchase or use a mattress cover. Before you begin to package and move the mattress, it needs to be lifted off of the bed frame or box spring it's been resting on. Also remove any blankets or quilts. You can probably move the mattress with the fitted sheet and mattress pad still on, but remove all other sheets. Start by standing the mattress on its side and slipping the end farthest from the zipper over the base of your mattress. Work the cover up over the mattress by tugging on the top of the cover, and then on the bottom. Keep sliding the cover up until the entire mattress is inside. If you have someone with you, ask them to hold the head of the mattress a few inches up in the air. Once your mattress is completely in the moving cover, you can adjust the corners of the cover as needed. Make sure that there are no spots where the plastic is stretched thin around the mattress inside and might tear. Then, zip the bag closed. If your mattress cover doesn't have a zipper, you'll need to use packaging tape to make sure that the cover is completely sealed. Fold over the top flap of the plastic cover, and tape down all of the loose edges so no dust can make its way inside. Before you begin to move the mattress, make sure that you have a clear path to the vehicle or rental truck you'll be placing the mattress in. Clear any boxes or furniture out of the path you'll be walking through, and prop doors open so you won't have to open them while holding the mattress. Even if you're only moving the mattress from one room to another within a house or apartment, still clear a path. You could easily trip over a stray side table or chair. If there is more than one route outside, figure out which route to take before you begin walking. Communicate this to the person helping you move the mattress.
Purchase a plastic mattress cover. Take the mattress off of the bed. Slip the mattress cover over the mattress. Zip the bag shut, or tape it closed with packaging tape. Prepare the way to the vehicle.