Wash a few times with no detergent or bleach, or anything. Use warm water. Hang in sun. I do this automatically all the time and my whites are quite acceptable.
Oops! I do put detergent in sometimes, but the dingy look, I believe, comes from a build up of too much soap and synthetics. Hope this works for you.
If you are using a dryer all the time for drying your clothes, you might try finding a way to use the sun to help whiten some of the cotton fabrics. I'm not sure that the sun will help too much when it comes to the polyesters and other man-made fibers, but I know for sure that the sun alone will help whiten cotton towels, "t" shirts, socks, bed and table linens, as well as dish or tea towels.
Boiling cotton on the stove will help too, but not everyone has the tubs or buckets to do that with today. You might also try using a little of the electric dishwashing powder, Cascade in some loads to see if it helps.
Water is so often the culprit when it comes to whitening clothes today. Water which contains a lot of iron is just always going to dull the whites and even the pastel colored clothing. You might try adding Oxyclean to those washer loads.
Good luck. I've been there and done that, and it can be a troublesome problem. I understand. I grew up during the time of when a woman was judged partially by the quality of her laundry abilities, and the way the clothes looked hanging out on clotheslines.
Bluing is another product that can be used with your laundry to whiten whites. It is in the laundry section of most stores where I live. It is an old fashioned product; comes in a bottle. The brand that is sold here is Mrs. Stewart's Bluing.
There is a product at Walmart and other places called "Iron Out" It takes about i/2 cup per white load and your whites will glisten. I am a nurse and I used it in my uniforms. You don't have to use it each time, just occasionally.
Here is a link that discusses ways to make clothes whiter...
It discusses boiling water and soaking clothes in the pot of water with lemon slices, and also using a small amount of dishwasher detergent in the laundry. Also vinegar is discussed. My favorite is lemon juice soak overnight as discussed in the above link.
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