Antibiotics may be necessary for long-lasting sinus infections. Most sore throats are caused by viruses. Strep can only be diagnosed by a laboratory test. Colds, which sometimes last two weeks or more, are caused by viruses. It's normal for mucous to get thick and change color during a cold.
Antibiotics have no effect on colds.
When you or someone else in your family is prescribed an antibiotic, make sure the medication is taken exactly as directed, and completely finished. Otherwise only the weakest bacteria will be killed, leaving behind the stronger, more resistant bacteria. Also, never share antibiotics. Throw away any leftover medicine as soon as you are finished with it.