I would think that if I was a deaf mute I would carry a card that said that along with a note book and pencil for communication with those who don’t understand sign language but that is Monday morning quarterbacking.
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Most deaf people today socialize almost exclusively with other deaf people or friends who sign.
Years ago, deaf people learned to read lips and even speak, once signing became available deaf people didn’t see the necessity of reading lips or even speaking, signing is dramatically easier and is many cases faster.
About 50 years ago while in college I befriended a deaf/dumb kid. He could speak a little and could read lips pretty good because his parents put him in a deaf school. He didn’t know know about signing until one day he met someone in school who signed. That was it, he no longer wanted anything to do with the school and was very angry with his parents for hiding this means of communicating from him. He refused to communicate with his parents unless it was signing. He and I communicated mostly by writing notes. Watching him with his deaf friends was amazing, they communicate very easily and fast with each other. They have many universal slang signs that basically substitute for a whole sentence. The problem is that when signers only are away from signers they are like a fish out of water, nearly helpless.
Most of the deaf I have associated with have a horrible handwriting (scribble).
The ability to read lips depends on how well the individual can be assisted with hearing aids (there is a large and varied spectrum of decibel loss in deafness, as well as the type of deafness. Some levels of deafness cannot be helped with even the most powerful hearing aids).
Lip reading often provides less than 40% of what is communicated; throw in a mustache or someone who speaks fast and it gets even worse! Deaf individuals supplant the ability to read lips with their hearing aid(s).
My wife has enough hearing in one ear to be able, with her digital hearing aid, to discern just enough to carry on a conversation with most hearing people without signing. Again, it depends: mustache, crowded room, fast talker, and things go downhill fast.