My kitchen drain will go down if I run the water slow but when the dish washer pumps the water out it backs up in the sink. The vent for the drain is under the sink with a cap on it.
How do I fix it?

Do you have a clean out under the sink? If you do it might be possible to run a small drain cleaning cable down the drain to where it is plugged up. Most of the drain cables are at most 25 feet so if it is within 25 feet you should be able to unstop the sink. Running the cable through the clean out enables you to run some water in the sink after you have made it drain, doing this will wash out the junk that you have knocked loose and if you keep turning the cable while running water will clean the drain even more.
If you do not have a clean out you can try to run the cable down through the basket strainer in the sink. If it will not go that way then disconnect the trap and run the cable through the arm coming out of the wall, the only problem with this is that you can no9t run water as you turn the cable to wash out the junk.

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Originally Posted by ronfjack

My kitchen drain will go down if I run the water slow but when the dish washer pumps the water out it backs up in the sink. the vent for the drain is under the sink with a cap on it.
How do I fix it?

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Sounds like a partial blockage in the horizontal drain inside the wall.
You have a AAV (Air Admittance Valve) under your sink. Remove it and send a snake into the wall. The good news is that you'll only have to put out about 6 feet more before you hit the vent stack. Flush with a tub of hot water. Good luck, Tom

I use hot boiling water and bleach in my drains to keep them fresh and soap & grease build up free, also I think it takes care of hairs, I can say I have not called a plumber in 9 years knock on wood and I have 3 teenagers, 4 cats 2 dogs, lots of hair and cleaning :) also I love this plumbing section. You guys are so smart, professional and nice even when you disagree.