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The Hacksaw Blade Method: You take a bare hacksaw blade and carefully cut a couple of notches on the inside of the broken pipe piece, being extremely careful not to cut into the threads of the fitting it’s stuck in. Then, you use a flathead screwdriver or chisel to tap the piece inward and peel it out. It’s surgery, not demolition.
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The Screwdriver/File Method: For smaller diameter pipes, you can sometimes heat up an old flathead screwdriver or a square file, press it into the plastic, let it cool, and then use it to back the broken piece out. This is a one-way trip for the tool, and it smells terrible.
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The Pliers Method: For larger pieces, you might be able to get a pair of needle-nose pliers inside, open them wide to press against the walls, and try to turn the piece out. Success rate: low. Frustration rate: high.
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Pros: You might be able to fix it without leaving the house. It's free if you have the tools.
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Cons: High risk of failure and, worse, high risk of damaging the outer fitting. One slip with the hacksaw and you’re no longer fixing a broken nipple; you’re replacing a valve inside a wall.