
jemandem ist übel
to feel nauseous
To have the feeling of nausea, as if you might have to vomit.
Nach der langen Autofahrt war mir furchtbar übel.
After the long car ride, I felt terribly nauseous.
nauseous / bad
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to feel nauseous
To have the feeling of nausea, as if you might have to vomit.
Nach der langen Autofahrt war mir furchtbar übel.
After the long car ride, I felt terribly nauseous.

bad / nasty
Morally bad, evil, or reprehensible.
Das war eine wirklich üble Nachrede, die ihm sehr geschadet hat.
That was a really nasty slander that harmed him greatly.

foul / unpleasant
Very unpleasant to the senses, especially in terms of smell or taste.
Aus dem Abfluss kam ein übler Geruch.
A foul smell was coming from the drain.

to hold something against someone
To be angry with someone because of an action or statement; to take something as a personal offense.
Ich hoffe, du nimmst es mir nicht übel, dass ich deine Einladung ablehnen muss.
I hope you don't hold it against me that I have to decline your invitation.
To express the feeling of nausea, German uses a fixed construction with the dative: 'jemandem ist übel' (someone feels nauseous). The person who feels sick is always in the dative.
Dem Kind wurde auf der langen Busfahrt übel.
The child felt nauseous on the long bus ride.
A typical mistake is using the nominative (*Ich bin übel*). That is wrong and would mean that you are a bad person. The correct form is always: 'Mir ist übel' (I feel nauseous).
Wenn ich zu viel Süßes esse, wird mir schnell übel.
If I eat too many sweets, I quickly start feeling nauseous.
The idiom 'jemandem etwas übel nehmen' (to hold something against someone) is a fixed structure. The person you resent something from is in the dative, and the action or statement that bothers you is in the accusative.
Er hat mir meine ehrliche Kritik sehr übel genommen.
He really held my honest criticism against me.
When 'übel' (bad/nasty) comes before a noun that describes an action or quality, it often has a moral meaning. Typical combinations are 'eine üble Tat' (a bad deed), 'üble Absichten' (evil intentions), or 'üble Gerüchte' (nasty rumors).
Über ihn werden in der Firma üble Gerüchte verbreitet.
Nasty rumors are being spread about him at the company.
bad
Schlecht and übel can morally condemn an action. Schlecht expresses a general negative evaluation, whereas übel usually conveys stronger condemnation. For particularly reprehensible behavior, übel is therefore more appropriate.

Es war übel von ihm, seinen Kollegen mit einer erfundenen Geschichte zu verleumden.
It was reprehensible of him to defame his colleague with a fabricated story.
Es war schlecht von ihm, seinen Kollegen vor allen zu kritisieren.
It was bad of him to criticize his colleague in front of everyone.