Hun er tålmodig nok til at lytte, når andre vil beklage sig over små problemer.

Because tålmodig is a predicate adjective after er.

In Danish, predicate adjectives usually have:

• the base form with singular common-gender nouns and with han/hun
• -t with singular neuter nouns
• -e with plural and with definite forms in many contexts

So:

• Hun er tålmodig = she is patient
• Barnet er tålmodigt = the child is patient
• De er tålmodige = they are patient

Since hun refers to one person, the base form tålmodig is correct.

Nok til at means enough to.

So:

• tålmodig nok til at lytte = patient enough to listen

This is a very common pattern in Danish:

• gammel nok til at køre bil = old enough to drive a car
• stærk nok til at løfte det = strong enough to lift it
• modig nok til at sige det = brave enough to say it

The structure is:

• adjective + nok + til at + infinitive

Because this is the fixed structure nok til at + infinitive.

Here:

• til connects the idea of being sufficient for something
• at lytte is the infinitive phrase, like to listen

So Danish says:

• Hun er stærk nok til at bære tasken
• Han er klog nok til at forstå det
• Hun er tålmodig nok til at lytte

You should learn nok til at as one chunk.

Because lytte means to listen, while høre means to hear.

That is a very important distinction:

• høre = receive sound, hear something
• lytte = actively pay attention, listen

In this sentence, she is making an effort to listen to other people, so lytte is the natural verb.

Compare:

• Jeg kan høre musik = I can hear music
• Jeg lytter til musik = I am listening to music

Good question. Danish very often uses lytte til when the thing listened to is stated directly:

• Hun lytter til musik
• Jeg lytter til radioen

But in your sentence, lytte is used more absolutely, without a direct object right after it:

• Hun er tålmodig nok til at lytte

Then the next clause explains the situation:

• når andre vil beklage sig over små problemer

So the sentence means she is patient enough to listen, specifically when others want to complain...

You could also sometimes see fuller expressions in other contexts, but here lytte without til is perfectly natural.

Here når means when in a general or repeated sense.

It suggests something that happens whenever this situation comes up:

• når andre vil beklage sig... = when others want to complain...

Why not the others?

• da usually refers to a specific past occasion: when in the sense of back when / at the time when
• hvis means if

So:

• Når andre vil beklage sig... = whenever others want to complain...
• Da andre ville beklage sig... = when others wanted to complain... on a specific past occasion
• Hvis andre vil beklage sig... = if others want to complain...

In this sentence, når is the right choice because it describes a general situation.

Because andre by itself can mean others.

So:

• andre = others
• de andre = the others

The difference is similar to English:

• others = people in general, unspecified
• the others = the remaining specific people in a known group

In your sentence:

• når andre vil beklage sig... = when others want to complain...

This is general, not a specific set of people, so andre fits well.

Here vil is best understood as want to or are inclined to.

Danish ville can mean:

• future: will
• desire/intention: want to
• tendency/willingness depending on context

In this sentence, andre vil beklage sig over små problemer is most naturally understood as:

• others want to complain about small problems
• or more loosely, others are going to complain about small problems

A learner should be careful not to always translate vil mechanically as English will. Context matters.

Compare:

• Jeg vil hjem = I want to go home
• Det vil regne i morgen = It will rain tomorrow

Beklage sig is a reflexive verb meaning to complain.

So:

• beklage by itself often means to regret or to express regret
• beklage sig means to complain

Examples:

• Jeg beklager fejlen = I regret / I apologize for the mistake
• Han beklager sig hele tiden = He complains all the time

The sig is the reflexive pronoun, like oneself in structure, though in English we usually do not translate it.

And over tells you what the complaint is about:

• beklage sig over små problemer = complain about small problems

Yes, often you could.

Both can mean complain about, but there can be a slight difference in tone:

• klage over is very common and direct
• beklage sig over can sound a bit more formal, literary, or expressive depending on context

Examples:

• De klager over vejret = They complain about the weather
• De beklager sig over vejret = They complain about the weather

In many everyday situations, klage over is probably the more common choice in spoken Danish, but beklage sig over is completely correct and natural.

Because problemer is plural, and adjectives in front of plural nouns usually take -e.

So:

• singular common gender: et lille problem? Actually en lille bil, en stor idé
• singular neuter: et lille problem
• plural: små problemer

That gives:

• et lille problem = a small problem
• små problemer = small problems

Why not de små problemer?

Because the sentence is talking about small problems in general, not specific ones already known to the listener.

So:

• små problemer = small problems
• de små problemer = the small problems

Because når andre vil beklage sig over små problemer is a subordinate clause.

In Danish, commas are commonly used to separate clauses, especially when a subordinate clause begins with a word like:

• at
• når
• hvis
• fordi
• som

So the comma helps show the structure:

• Hun er tålmodig nok til at lytte,
  når andre vil beklage sig over små problemer.

You will see commas used quite systematically in written Danish.

The sentence can be broken into three parts:

1. Hun er tålmodig nok

   • main clause
   • Hun = subject
   • er = verb
   • tålmodig nok = complement

2. til at lytte

   • infinitive phrase
   • explains what she is patient enough to do

3. når andre vil beklage sig over små problemer

   • subordinate clause
   • gives the situation in which she listens

So the overall pattern is:

• subject + verb + adjective + nok + til at + infinitive + subordinate clause

This is a very useful pattern to recognize in Danish.