У этой старой книги потрёпанная обложка, но страницы чистые.

Russian very often expresses possession with у + genitive.

So instead of saying the book has a cover, Russian commonly says something closer to at the book there is a cover.

• у этой старой книги = this old book has / belonging to this old book
• This pattern is used not only with people, but also with things.

It is one of the most common ways to say have in Russian.

Here книги is genitive singular, because it follows у.

The noun книга has these relevant forms:

• nominative singular: книга
• genitive singular: книги
• nominative plural: книги

So yes, книги can mean either:

• of the book (genitive singular), or
• books (nominative plural)

In this sentence, it must be genitive singular because:

1. it comes after у, which requires the genitive
2. этой старой also agrees with a singular feminine noun here

Because they both describe книги, and they must agree with it in:

• gender: feminine
• number: singular
• case: genitive

So:

• эта → этой
• старая → старой
• книга → книги

That gives:

• у этой старой книги

A learner-friendly way to think about it is: after у, the whole noun phrase changes into the genitive.

Because обложка is the thing being talked about as what the book has.

In this possession pattern, the possessor goes after у in the genitive, but the possessed thing is usually in the nominative:

• у книги обложка
• у него брат
• у дома крыша

So here:

• у этой старой книги = possessor
• потрёпанная обложка = the thing possessed

That is why обложка stays nominative singular.

Because in the present tense, Russian usually omits is / are.

So:

• страницы чистые literally looks like pages clean
• but it means the pages are clean

The same idea applies to the first clause. Russian often leaves out the present-tense verb when it is obvious from the context.

You could compare:

• У книги есть обложка. = The book has a cover.
• У книги потрёпанная обложка. = The book has a worn cover.

With есть, the focus is more on existence.
Without есть, the sentence often feels more descriptive and natural in context.

Because of agreement.

старой is feminine singular genitive, which matches книги.

But обложка is feminine singular nominative, so if old described cover, it would have to be:

• старая обложка

So:

• этой старой книги = of this old book
• потрёпанная обложка = a worn cover

The endings show which words belong together.

Потрёпанная matches обложка, so it is:

• feminine
• singular
• nominative

because обложка is feminine singular nominative.

In everyday learning terms, you can treat потрёпанный / потрёпанная / потрёпанное like an adjective meaning something like:

• worn
• shabby
• frayed
• battered from use

It comes historically from a participle, but for most learners it is easiest to think of it as an adjective here.

Because страницы is the subject of the second clause, and чистые agrees with it.

• страницы = plural
• чистые = plural

So this is a normal agreement pattern:

• страница чистая = the page is clean
• страницы чистые = the pages are clean

In the plural, adjectives do not show grammatical gender, so you only really notice number and case here.

Both can be possible in Russian, but they are not exactly the same in style.

• страницы чистые = very normal, neutral, common
• страницы чисты = shorter form, often more literary, formal, or stylistically marked

Modern everyday Russian often prefers the long form in sentences like this.

So страницы чистые sounds natural and straightforward.

Yes. Russian word order is flexible, and changing it usually changes emphasis, not the core meaning.

The given sentence is natural and neutral:

• У этой старой книги потрёпанная обложка, но страницы чистые.

You might also hear things like:

• У этой старой книги обложка потрёпанная, но страницы чистые.
• Страницы у этой старой книги чистые, но обложка потрёпанная.

These versions shift the focus a bit. The original puts the idea together neatly as a worn cover versus clean pages.

So the meaning stays similar, but the information spotlight moves.

Grammatically, yes. But it sounds less natural in this context.

Есть is most useful when you want to stress that something exists or is present:

• У книги есть обложка. = the book does have a cover.

But in your sentence, the important point is not that the book possesses a cover at all. The important point is what kind of cover it has.

So Russian usually prefers:

• У этой старой книги потрёпанная обложка...

That sounds more descriptive and idiomatic.

Because но means but, and here it joins two clauses:

1. У этой старой книги потрёпанная обложка
2. страницы чистые

In Russian, a comma is normally used before но when it connects two clauses.

So the punctuation is completely standard.