My kitchen sink doesn't have water go down super fast anymore when I wash dishes. It eventually goes down but takes time now. Sometimes I turn on the sink incinerator so water goes down quicker. Also, the pipe is connected to my garage sink (for washing machine). Sometimes water might come up. Help

My kitchen sink used to be great and I would wash dishes and the water would go down the drain quickly. Never thought about it. Now as of late I notice when the water is on and there is soapsuds, it takes long for the water to drain. Is something wrong? I looked down the sink and I don't see anything caught. The sink incinerator seems to still function and I turn it on occasionally for the water to drain quicker.

I notice to that sometimes when the water in the kitchen is being used to wash dishes... sometimes the sink in the garage will have some water and soapsuds come up. The garage contains a sink that also is connected to the washing machine. When the washing machine is on it'll eventually drain water into the garage sink via a tube.

Is there a huge problem going on? What home remedies can I try to correct the slow water drain in the kitchen sink?