I hate black plastic, absolutely hate what it does to garden soil. I would never, ever subject my living soils to that torture. To say that the only choices are chemicals or black plastic is absolutely ridiculous.
SeniorCitizen suggested a third option which I completely endorse and have used successfully for many years. Cover crops are absolutely remarkable at reducing/eliminating weeds. Planted thickly cover crops prevent weed seed germination and growth. Not only that but they add vital nutrients to the soils and drive away pests like nematodes. So many benefits for cover crops, so easy to use.
I use them year around. In the fall/winter, I like the clovers, the small grains, turnips, the elbon rye which form carpets of cover that few weeds can penetrate. Nematodes flee the soils that have these covers on them.
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In summer, rotations of legumes like cow peas and beans are just absolutely fabulous for soil building and weed prevention. I have had up to four cycles of plant, grow, shred, grow, shred, grow and shred in one growing season only planting once the entire season and never weeding one time. Plus I harvest some peas and beans as you go but leave enough for regeneration. It absolutely works. It is fantastic in the results it produces. Its the easiest labor saving way to build garden soils while reducing/eliminating weeds and many pests.
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My soils are alive, thriving and virtually weed free. I started gardening in an old hay field about 40 years ago...and without cover crops would have long ago abandoned that site because of the weeds. Now it is virtually weed free. It never needs artificial fertilizers...and is never sprayed with chemicals. There is no need to. I can't imagine gardening without cover crops. Way too much work, too many weeds, too many pests.
To suggest that this is not a viable alternative is utterly incorrect.