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Water the lawn thoroughly, apply fertilizer, and overseed if necessary to promote healthy growth.

Weeds can grow through rubber mulch, but it depends on if you removed all the weeds in your garden beforehand. Weeds already in the soil can grow even when using rubber mulch. However, that doesn’t mean that your garden will be littered with rubber and weeds, but it does mean that the occasional weed might still pop up. Another way you can increase your weed suppression is by using landscape fabric as a second layer. 

No, it's not really possible to completely prevent weed growth. Weeds are incredibly resilient—all they usually need is a tiny bit of dirt to sprout, so their eventual appearance is pretty much inevitable. However, you can definitely keep them at bay by getting rid of weeds as soon as you notice them. The longer you let them sit and grow roots, the more they'll spread.

There are organic herbicide options that will kill grass but not flowers, including white vinegar or a dish soap solution. Commercial selective herbicides can also target grass over other plants, but read the instructions carefully to avoid damaging your garden. Herbicides that kill grass but not flowers usually include ingredients like clethodim and sethoxydim. Otherwise, your best bet will be to weed by hand.