Purity: Purification of Clothing or Carpets from Impure Dust by Shaking
Practical Laws of Islam as per the teachings of Ayatullah Sistani
Ruling 218
If impure dust settles on dry clothing, a carpet, or similar thing, in the event that the object is shaken in a way that the amount of dust that is certain to have been impure falls off, the clothing, carpet, or similar thing is pure and it is not necessary to wash it.
-Ayatullah Sistani, Practical Laws of Islam, Purification (Taharah)