Finely slice the cabbage, green pepper, and onion into a large bowl - I use a mandoline for thin and even slices. If you do likewise, be very careful not to cut yourself.
Grate the carrot into the bowl with the cabbage mixture.
Sprinkle 3/4 cup of sugar on the cabbage mixture and toss.
In a non-reactive small saucepan over medium heat, combine remaining ingredients (1/4 cup sugar, vinegar, oil (original recipe called for 3/4 cup, which was overwhelmingly oily), salt, and dry mustard. Bring to a boil, stirring often.
Pour vinegar mixture over cabbage mixture. Toss well.
Cover and refrigerate for at least one hour before serving but this slaw will last for at least several days.