Remedy for Dry Itchy Scalp

I wonder if this would work for my shitzu. he has very dry skin and itches constantly no fleas or bugs of any kind.

I rinse my hair with Listerine after I shampoo it. I do not rinse it out. The Listerine gets rid of dandruff when I use it every time I shampoo my hair and it leaves my hair soft and shiny. And the odor evaporates too. I am a senior citizen and my hair is naturally dry.