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I would use an integrated circuit called a voltage comparator to solve this problem. It compares one voltage to another to set the output on or off. You can set the fixed voltage with a voltage divider circuit which can be variable resistor connected between battery voltage and ground.
I once solved a computer problem with this circuit. The computer had a very weak output from the "Integrated Woz Machine" chip and the disk drive couldn't detect it. I used the voltage comparator to strengthen the output to the voltage expected by the disk drive. Replacing the bad computer chip would have been very difficult.
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