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Super market bags are made from a combination of LLDPE, LDPE, and HDPE. The same substances that typically go into Shrink Wrap, (primarily LDPE), Ziplock bags, and Garbage Bags (primarily LLDPE).

So you should absolutely recycle them in the same method you recycle grocery bags.

While all of those things CAN be recycled, most recyclers don't want them. These materials are hard to separate properly and result in a contaminated product that has no resale value.

Melting the bags releases toxic fumes. Use the bag whole.