Jeg hader kortmisbruget.

Because Danish usually writes compound nouns as a single word.

So:

• kort = card
• misbrug = misuse / abuse
• kortmisbrug = card misuse / card abuse
• kortmisbruget = the card misuse / the abuse of cards

This is very normal in Danish. Where English often uses separate words, Danish often combines them.

It is a definite singular noun.

You can break it down like this:

• et kortmisbrug = a card misuse / a misuse of cards
• kortmisbruget = the card misuse

The ending -et is the definite article, similar to English the.

In Danish, the definite article is often added as a suffix to the noun.

So instead of saying a separate word for the, Danish often does this:

• et hus = a house
• huset = the house

The same pattern appears here:

• et kortmisbrug
• kortmisbruget

A separate word like det is used in some other situations, especially when there is an adjective:

• det alvorlige kortmisbrug = the serious card misuse

It is an et-word (a neuter noun).

That is why the indefinite form is:

• et kortmisbrug

and the definite form is:

• kortmisbruget

If it were an en-word, the definite ending would usually be -en instead.

Hader is the present tense of at hade = to hate.

So:

• at hade = to hate
• jeg hader = I hate

A useful thing for English speakers: Danish verbs do not change according to the person the way English verbs sometimes do. So you get:

• jeg hader = I hate
• du hader = you hate
• han/hun hader = he/she hates
• vi hader = we hate

The verb form stays the same.

Because this is the normal Danish main-clause word order:

subject + verb + object

So:

• Jeg = subject
• hader = verb
• kortmisbruget = object

This is very similar to English I hate card misuse / I hate the card abuse.

A useful Danish rule is that in main clauses, the finite verb usually comes in second position. In this sentence, the subject is first, so the verb naturally comes right after it.

Yes, that is grammatically possible, but it sounds more expanded and less compact.

• kortmisbruget is the neat compound form
• misbruget af kort means something like the misuse of cards

Danish often prefers compound nouns when they are established expressions, so kortmisbruget feels very natural.

In Danish, kort can have several meanings depending on context:

• kort = card
• kort = short
• et kort = a map

But in kortmisbrug, it almost certainly means card. The compound strongly points to things like bank cards, payment cards, or credit cards being misused.

So context tells you which meaning is intended.

Jeg is a famous tricky Danish word for learners.

In careful standard pronunciation, it is often close to yai or yigh, depending on accent and speech style. The important thing is:

• the j sounds like English y
• the g is not a hard English g

So do not pronounce it like jegg.

You will hear some variation in real speech, but the key point is that it sounds much softer than the spelling suggests.

Yes, hader is usually a strong word, just like English hate.

So Jeg hader kortmisbruget sounds forceful and emotional.

If you want something softer, you could say:

• Jeg kan ikke lide kortmisbrug. = I don’t like card misuse.
• Jeg synes, kortmisbrug er forfærdeligt. = I think card misuse is terrible.

So the sentence you were given has a strong tone.

Because Danish does not capitalize common nouns the way German does.

In Danish, you usually capitalize:

• the first word of the sentence
• proper names

So in this sentence:

• Jeg is capitalized because it starts the sentence
• kortmisbruget is not capitalized because it is an ordinary noun

That is completely normal in modern Danish.