There are products out there esp. for ink like "ink out" for Pro Choice
and cyclone from Carpet Care Systems. Make sure you surrond the stain with water to help contain it. Put some of the product on a white cotton cloth and blot, don't rub, work from outside to inside. Keep blotting until no transfer takes place to the cotton. At this point most of the ink should be removed. You can put the product on the stain and GENTLY message with you fingers to work it in. Then reblot to remove more of it. If you have a spotter use it to remove any residue. At the end you may have to use Red relief to remove the left overs.
I love the tip from noweare about surrounding the spot with water! Depending on the type of fiber this may work well to keep it from being absorbed into the surrounding fibers. I also use this trick on concrete when working on oily parts in the garage: soak it and it won't absorb the oil near as much.
One thing I would add: I like to use my ink remover (p.i.g. or p.o.g.) and then extract with suction only, then repeat over and over. This keeps the ink remover from going to the backing and causing de-lamination. Then when you can get no more ink out rinse with a non-volatile solvent or a neutral spotter. You can also try a filtration soil remover such as "Filter out".
In this case Pros Choice Power Gel 2 parts works best. You have to take your time and work outside in. Using a truck mount or portable periodically to rinse and extract will really help the process. Careful not to over saturate as you will delaminate the backing of the carpet. These are the procedure we use at carpet cleaning mission viejo with great success throughout the many years we have been in business.
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