If your mom is ill-tempered, she's cranky, or in a really bad mood. It's probably smarter to ask your cheerful grandma for a ride to school, instead of your ill-tempered mom.
Whenever you find yourself in an irritable, grouchy mood, you're ill-tempered. A really good kindergarten teacher will save his ill-tempered remarks for when he goes home after school, and an ill-tempered high school volleyball coach may end up with players quitting the team. Everyone's ill-tempered from time to time — the word combines ill, "bad," and tempered, from temper, or "mood," from the Latin root temperare, "be moderate."
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