Hvis jeg arbejder sjusket, laver jeg let fejl, men min søster er dygtig til at opdage dem.

Because Danish main clauses follow the V2 rule: the finite verb usually comes in the second position.

In this sentence, the whole if-clause Hvis jeg arbejder sjusket takes the first position. So the next element, the finite verb of the main clause, must come next:

• Hvis jeg arbejder sjusket, laver jeg let fejl.

So the order is:

1. Hvis jeg arbejder sjusket
2. laver
3. jeg
4. the rest of the sentence

This is very common in Danish when a sentence begins with something other than the subject.

After hvis meaning if, Danish uses normal subordinate-clause word order:

• Hvis jeg arbejder sjusket

That is:

• hvis = if
• jeg = subject
• arbejder = verb
• sjusket = adverb

So unlike the main clause, there is no inversion here. You do not say Hvis arbejder jeg...

A useful contrast:

• Hvis jeg arbejder sjusket, laver jeg let fejl.
• I dag laver jeg let fejl.

In the first part, after hvis, the subject comes before the verb. In the main clause, the verb follows the V2 rule.

Here sjusket functions like an adverb, meaning sloppily, carelessly, or in a sloppy way.

It describes how I work:

• arbejder sjusket = work sloppily

This is very common in Danish: an adjective form is often used adverbially.

Compare:

• Han skriver pænt. = He writes neatly.
• Hun arbejder hurtigt. = She works quickly.
• Jeg arbejder sjusket. = I work sloppily.

With sjusket, the form already ends in -t, so it looks the same as the adverbial form.

Here let means easily, not light.

So:

• jeg laver let fejl = I easily make mistakes

This can confuse learners because let can have different meanings depending on context:

• en let taske = a light bag
• en let opgave = an easy task
• jeg bliver let træt = I get tired easily
• jeg laver let fejl = I easily make mistakes

So in this sentence, let is an adverb.

Because lave fejl is the normal Danish expression for make mistakes.

So:

• lave fejl = make mistakes
• begå fejl = commit mistakes/errors, a bit more formal

But gøre fejl is generally not the standard expression.

This is just one of those verb-noun combinations you need to learn as a set, much like collocations in English.

Because fejl is being used as an indefinite plural in a general sense: mistakes in general, not specific mistakes.

So:

• laver jeg let fejl = I easily make mistakes

Compare:

• Jeg laver fejl. = I make mistakes.
• Jeg laver nogle fejl. = I make some mistakes.
• Jeg laver fejlene. = I make the mistakes.
  This last one would sound strange in most contexts unless you mean specific known mistakes.

So the bare plural fejl is the natural choice here.

Because the sentence expresses a general truth, habit, or usual result, not a one-time event.

• Hvis jeg arbejder sjusket, laver jeg let fejl
  = If I work sloppily, I easily make mistakes.

Danish often uses the present tense for this kind of general statement, just like English does.

It does not mean the speaker is necessarily doing it right now. It means this is what generally happens.

Dygtig means skilled, good, or capable.
The pattern dygtig til at + infinitive means good at doing something.

So:

• min søster er dygtig til at opdage dem
• my sister is good at noticing them

Useful patterns:

• Hun er dygtig til matematik. = She is good at math.
• Han er dygtig til sprog. = He is good at languages.
• Hun er dygtig til at tegne. = She is good at drawing.

So with a noun, you often get dygtig til + noun.
With a verb, you usually get dygtig til at + infinitive.

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

• opdage = discover / notice / detect
• at opdage = to discover / to notice / to detect

After dygtig til, when the next thing is a verb, Danish normally uses:

• dygtig til at + infinitive

So:

• dygtig til at opdage
• good at noticing

Even though English says good at noticing with an -ing form, Danish uses til at + infinitive here.

Because dem is the object form of they.

Here dem refers back to fejl:

• fejl = mistakes
• dem = them

Since they are the thing being noticed, Danish uses the object pronoun:

• min søster er dygtig til at opdage dem

Compare:

• De er store. = They are big.
  Here de is the subject.
• Jeg ser dem. = I see them.
  Here dem is the object.

So in this sentence, dem is correct because the sister is noticing them.

Because men is a coordinating conjunction meaning but. It joins two main clauses, and it does not itself force inversion.

So the second clause keeps normal main-clause order:

• men min søster er dygtig til at opdage dem

That is simply:

• min søster = subject
• er = verb

Compare:

• Jeg er træt, men min søster er frisk.

Inversion happens when something else takes the first position in a main clause, but men does not count in the same way as a fronted adverbial or subordinate clause.

Arbejder is the present tense of arbejde:

• at arbejde = to work
• jeg arbejder = I work / I am working

This is a very common Danish pattern:

• infinitive often ends in -e
• present tense often ends in -r

Examples:

• at bo → bor
• at lære → lærer
• at arbejde → arbejder

So Hvis jeg arbejder sjusket literally starts with If I work sloppily.

In this sentence, opdage means something like notice, spot, or detect.

So:

• min søster er dygtig til at opdage dem means she is good at noticing/spotting the mistakes.

Opdage can also mean discover, depending on context, but here it clearly means noticing the mistakes.

So a learner should understand it broadly as:

• opdage fejl = spot mistakes / notice mistakes

That is a very natural combination in Danish.