For weeds you may want to try full strength vinegar. Just drip it on the undesirable weed, careful not to get the grass. Given time it will kill the weed.
I concur with frugalsunnie. Manually is a sure-fire way to get rid of them without any chemicals.
When we moved into our house there were massive shrubs lining the walk up to the house. It felt like walking down a dark tunnel. I didn't like the looks, plus it was a potential hiding spot for bad guys. So my mom and I dug them up. Those things must have been there for over 20 years (huge roots!). It took a lot of digging and prying with shovels and forks, but after much sweat we got it done.
The only way I know of is to take a skinny trowel or long toothed forked out there and dig them up as soon as you see them. It's a bit of work and is a lifetime commitment, but well worth the effort.
My new husband and I are reclaiming several garden beds and the lawn-one dandelion at a time, lol! We have agreed that every time we are outside we will dig at least one up apiece. When we leave the house we each dig one up and drop it in the bin. When we come back, we dig up another one each. Whenever we go out to hang washing out, pick some greens for the salad, visit over the fence with the neighbors, whatever, we dig up at least one dandelion.
It's working-slowly but surely, and one of these days we'll be on a maintenance plan instead of an eradication plan. :)
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