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Salt in that example would be a hypertonic solution.  A hypertonic solution is when the solution has a higher salt concentration compared to the concentration of the salts within the cells.  Water moves through permeable cell membranes through a process called osmosis.  Osmosis is when water will move from areas of lower concentration to areas of higher concentration.  So in this case, if you add a hypertonic solution (the salt solution) to your mouth, water will move from your gums (lower concentration) out to the salt solution (higher concentration).  This will reduce the swelling of your gums.

Swelling of the gums is known as gingivitis. It can be caused by numerous factors including infections, poor hygiene, trauma (e.g. such as aggressive brushing), etc. The gums are a soft tissue similar to skin but without the keratin covering the top layer. Keratin on the skin provides protection from abrasion as well as protection from dehydration, which the gums don't have–and normally do not require as they 'live' in a humid environment provided by saliva. The extra-cellular fluid between cells in the gums is maintained by certain molecules which through osmotic pressure retain water molecules in this space. Since saliva and this fluid have the same osmotic pressure there is no net movement of water. During inflamation, certain molecules with osmotic potential increase in this space, as does irrigation of blood through vasodilation as well as a decreased capillary permeability (leakage of blood fluid into the extravascular space). This inflamation can be reduced by a) removing the cause e.g. infection, b) reducing the inflamation with, e.g. an anti-inflammatory drug, or c) by actively pulling the accumulated water in the extra-cellular space out. This can be done with salt-water rinses of the mouth that are hypertonic (i.e. higher osmotic pressure than the extracelular space which would require more than 10 g of NaCl per litre of water). Since there is no keratin to protect from dehydration, the salt in the rinse would 'pull' water from the extra-cellular (as well as cellular) space in the gums and thus reduce inflammation, albeit only temporarily.

The salt water is hypertonic to the cells, becuase of osmosis the water in the gums is leaving (being pulled away) therefore the swelling in your gums will reduce 

Salt water is not only a natural disinfectant helping to keep everything clean in your mouth it also draws out the fluid (swelling) in your gums. Salt water has a lower water potential than the fluid in your gums. So the fluid moves out of the gums and into the salt water from high water potential to low water potential down a water potential gradient. 

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