Answer: There are several approaches to getting rid of weeds around existing plantings. Usually a combination of methods will work over time. First of all, don't let them go to seed if at all possible; clip off the seedheads if you can't otherwise remove the weed plant. This will reduce the number of weeds in the next generation.
Next, get rid of the existing weeds by cultivating, smothering, hand pulling or chemical controls. Cultivating (scratching the soil with a clawed tool or hoe) works best on barely germinating and small seedlings, but it can also be used to "exhaust" larger weeds. If you are persistent about breaking off the tops, eventually the roots will run out of energy and die.
Catching weeds when they are small is probably the most effective control method. You can also try smothering tiny seedlings with an extra layer of mulch applied right over top of them. In fact, an annual mulching can help a great deal towards weed control, and those that do struggle up through it are usually far easier to pull.
Larger weeds usually need to be pulled out or cut off by hand. Hand pulling works best when the soil is moist because you should get the entire root out along with the top. (Roots left in the ground will resprout.) If you have accidentally broken off a top or are faced with deeply rooted weeds such as dandelions, you may be able to prevent them from resprouting by covering the remaining root with a few layers of newspaper topped by a thick layer of mulch.
In some cases the situation may warrant use of an herbicide such as glyphosate. This is difficult to use around established plants because it is non-selective, which means it kills all plants, not just weeds. If you use this, apply it carefully being sure to follow the label instructions. Also be sure to allow it ample time to work; it may take ten days or so for the plant to translocate it to the roots effectively.
Good luck with your weeds -- just remember that the problem is always worst the first year or two after starting a garden; eventually it gets easier, especially as your established plants fill in and leave less room for the weeds to take hold.
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