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Well, you may have got the splinter out but I am afraid you most likely have developed an infection of the tissue surrounding the area where the splinter was. This type of infection is also known as "cellulitis".
Cellulitis is a term used to describe an infection of the skin. Cellulitis may be present in the skin on any part of the body. Certain conditions/situations predispose one to developing cellulitis. These include diabetes, eczema, burns, swelling of the extremities (edema or lymphedema), poor circulation, burns, surgery, IV drug use or occupational exposure to skin irritants. With cellulitis, there is bacteria present which is causing the underlying symptoms. The type of bacteria present differs depending on the part of the body/skin affected. Symptoms present with cellulitis include a generalized, confluent red rash which blanches with pressure, tenderness, swelling and occasionally drainage from the area involved. The area is warm and fever or chills are commonly present. The doctor may due a culture of the affected area to help determine the type of bacteria causing the cellulitis. Antibiotics are prescribed for a period of 10-14 days and in severe cases, hospitalization for IV antibiotics and surgical consultation may be necessary.
I worry a bit more about infections of the hand, in that, if tendons are involved, the damage to the tissue can be even more severe. You need to have this looked at TODAY. If you can't get into your doctor's office, go to the ER. This is most definitely something you are going to need treatment for ASAP.