Hvis du er ærlig og høflig, selv når du er uenig, er det lettere at løse problemet.

Hvis means if. It introduces a condition:

Hvis du er ærlig og høflig ... = If you are honest and polite ...

So the sentence sets up a condition first, and then gives the result in the main clause.

This is because Danish main clauses follow the verb-second (V2) rule.

The whole opening part — Hvis du er ærlig og høflig, selv når du er uenig — comes before the main clause. Since that first part takes the first position, the finite verb of the main clause must come next:

..., er det lettere at løse problemet.

If you started with the main clause instead, you would say:

Det er lettere at løse problemet, hvis du er ærlig og høflig ...

Selv når means even when.

• når = when
• selv adds the sense of even

So:

selv når du er uenig = even when you disagree

It adds emphasis: not only in easy situations, but also in difficult ones.

Because the sentence contains more than one clause, and each clause needs its own finite verb.

Here you have:

• Hvis du er ærlig og høflig
• selv når du er uenig
• er det lettere at løse problemet

So er appears in each clause. This is completely normal in Danish, just as English can repeat are/is in a long sentence.

Because they are predicative adjectives: they come after er and describe the subject du.

With du, Danish normally uses the basic adjective form:

• du er ærlig
• du er høflig
• du er uenig

Compare:

• det er ærligt
• det er høfligt
• de er ærlige
• de er høflige

So the form in the sentence is the normal one after du er.

Uenig means in disagreement or not in agreement — in natural English, usually just disagreeing.

It is the opposite of enig (in agreement).

Examples:

• Jeg er enig. = I agree.
• Jeg er uenig. = I disagree.

You can also add what or whom you disagree with:

• Jeg er uenig med ham. = I disagree with him.
• Jeg er uenig i det. = I disagree about that / I disagree with that.

In your sentence, that extra information is left out because it is not important.

Lettere is the comparative form of let here, meaning easier.

So:

• let = easy
• lettere = easier

In this sentence:

det er lettere at løse problemet = it is easier to solve the problem

Danish often forms the comparative with -ere, especially with short adjectives.

Here det works like the English it in sentences such as:

It is easier to solve the problem.

This is sometimes called a formal subject or dummy subject. The real idea comes in the infinitive phrase:

at løse problemet = to solve the problem

So literally, the structure is very close to English: it is easier to solve the problem.

Because at is the Danish infinitive marker, like English to.

• løse = solve
• at løse = to solve

After lettere, Danish uses an infinitive phrase:

lettere at løse problemet = easier to solve the problem

So at is required here.

Because Danish usually marks the definite form by adding the article to the end of the noun.

• et problem = a problem
• problemet = the problem

So problemet is the normal way to say the problem.

A separate word like det is used in other structures, for example with an adjective:

• det store problem = the big problem

But with a plain noun by itself, Danish normally uses the ending: problemet.

It can mean either one, depending on context.

In a sentence like this, du often has a general meaning, similar to English you in advice or general statements:

Hvis du er ærlig og høflig ...
= If you are honest and polite ...

This can mean:

• the person being spoken to directly, or
• people in general

So the sentence can sound like advice, not just a statement about one specific person.

Because Danish does not need så there, even though English sometimes uses then after if.

So this is perfectly normal:

Hvis du er ærlig og høflig ..., er det lettere at løse problemet.

You can also say:

Hvis du er ærlig og høflig ..., så er det lettere at løse problemet.

Here så means something like then, but it is optional. Leaving it out is very common and natural.