Moja linijka jest dłuższa niż twoja, ale obie są już stare.

Because linijka is a feminine noun in Polish. A very common clue is the ending -a: many nouns ending in -a are feminine.

You can see that femininity affecting several words in the sentence:

• moja linijka — moja is the feminine form of my
• dłuższa — feminine singular form of longer
• twoja — feminine form of your
• obie — both for feminine things

So the grammar of the whole sentence is shaped by the gender of linijka.

Because adjectives and possessive words in Polish must agree with the noun in gender, number, and case.

Since linijka is:

• feminine
• singular
• nominative here

the possessive must also be feminine singular nominative:

• moja linijka = my ruler

Compare:

• mój długopis = my pen (masculine)
• moja linijka = my ruler (feminine)
• moje krzesło = my chair (neuter)

Dłuższa is the feminine singular form of the comparative adjective longer.

The basic adjective is:

• długi = long

Its comparative is:

• dłuższy = longer

Because it describes linijka, which is feminine singular, it changes to:

• dłuższa

So:

• linijka jest dłuższa = the ruler is longer

This is the normal Polish way to form this comparative, not a word-for-word equivalent of more long.

Niż means than in comparisons.

So:

• dłuższa niż twoja = longer than yours

This is one standard way to build comparisons in Polish.

There is also another common pattern with od:

• Moja linijka jest dłuższa od twojej.

That also means My ruler is longer than yours, but after od, Polish uses a different case, so twoja changes to twojej.

So both are correct:

• dłuższa niż twoja
• dłuższa od twojej

Because Polish often leaves out a repeated noun when it is obvious from context.

Here:

• moja linijka = my ruler
• twoja = your one / yours

So twoja really stands for twoja linijka, but repeating linijka would sound unnecessary.

English does the same kind of thing:

• My ruler is longer than yours

You do not need to repeat ruler.

Here twoja is in the nominative, because it stands for the omitted noun linijka in the comparison pattern with niż.

So the structure is essentially:

• Moja linijka jest dłuższa niż twoja linijka.

Then the repeated noun is omitted:

• Moja linijka jest dłuższa niż twoja.

If you used od instead of niż, the form would change:

• dłuższa od twojej

There, twojej is in the genitive.

Obie means both when talking about two feminine things.

Since the sentence is about two rulers, and linijka is feminine, Polish uses:

• obie = both

Compare some forms:

• obie linijki = both rulers (feminine)
• oba okna = both windows (neuter)
• obaj chłopcy = both boys (masculine personal)

You may also see obydwie, which means the same thing and is a slightly fuller form.

Because now the sentence is talking about both rulers, so the adjective must be plural.

Singular:

• linijka jest stara = the ruler is old

Plural:

• linijki są stare = the rulers are old

Since obie refers to two feminine, non-masculine-personal things, the adjective takes the plural non-masculine-personal form:

• stare

So:

• obie są już stare = both are already old

Because the subject changes from one thing to two things.

• moja linijka jest... — linijka is singular, so use jest
• obie są... — both means two things, so use są

This is just the present tense of być = to be:

• jest = is
• są = are

Już usually means already.

In this sentence it adds the idea that, by now, both rulers are old:

• obie są już stare = both are already old

It often gives a sense like:

• now
• by this point
• as it turns out

Without już, the sentence would still be correct:

• Moja linijka jest dłuższa niż twoja, ale obie są stare.

That simply sounds a bit less specific in nuance.

Not completely. Polish word order is more flexible than English word order, because endings carry a lot of the grammatical information.

The given sentence is natural and neutral:

• Moja linijka jest dłuższa niż twoja, ale obie są już stare.

But other orders are also possible for emphasis, for example:

• Moja linijka jest dłuższa niż twoja, ale już obie są stare.
• Obie są już stare, ale moja linijka jest dłuższa niż twoja.

The meaning stays similar, but the focus changes a little. For learners, the original order is a good standard pattern.

A few parts may look difficult to an English speaker:

• ł in dłuższa sounds like English w
• ż and rz sound like a voiced zh sound, similar to the s in measure
• ń in niż is a soft n
• j in Polish sounds like English y in yes

Very rough approximations:

• moja ≈ MO-ya
• linijka ≈ lee-NEEY-ka
• dłuższa ≈ DWOO-sha or DWOOZH-sha depending on how carefully you break it down
• niż ≈ neezh
• twoja ≈ TVO-ya
• obie ≈ O-byeh
• stare ≈ STA-reh

These are only approximations, but they help you get started.