Ten kubek jest brudny, weź inny.

Because kubek is a masculine singular noun, and the demonstrative has to agree with it.

• ten = masculine singular
• ta = feminine singular
• to = neuter singular

So:

• ten kubek = this mug/cup
• ta filiżanka = this cup
• to krzesło = this chair

Also, to is often used by itself in sentences like To jest kubek = This is a mug, but before kubek you need ten.

Here kubek is the subject of the sentence: This mug is dirty. Subjects are normally in the nominative case, so the basic dictionary form kubek is used.

In other words:

• Ten kubek = the subject
• jest brudny = tells us something about that subject

Because brudny has to agree with kubek.

Since kubek is:

• masculine
• singular
• nominative

the adjective also appears in the masculine singular nominative form:

• brudny

Compare:

• ten kubek jest brudny
• ta filiżanka jest brudna
• to naczynie jest brudne

The ending changes to match the noun’s gender and number.

In standard Polish, yes, jest is normally necessary.

• Ten kubek jest brudny = natural, standard Polish
• Ten kubek brudny = nonstandard, very colloquial, or something you might hear in very reduced speech

So unlike Russian, for example, Polish usually keeps the present-tense form of to be in normal sentences.

Weź is the 2nd person singular imperative of wziąć, which means to take.

So weź means:

• take
• pick up
• grab

This verb is a bit irregular, so learners often notice it:

• infinitive: wziąć
• imperative: weź

It is the form you use when telling one person to take something.

This is mainly about aspect.

• wziąć is perfective
• brać is imperfective

The imperative weź sounds natural when you mean take another one now, as a single completed action.

• Weź inny. = Take another one.

Bierz inny is possible in some contexts, but it has a slightly different feel and is less neutral here. It can sound more like go ahead and take another one or focus on the action in progress rather than the completed choice.

In this sentence, weź is the most natural choice.

Yes, the full version could absolutely be weź inny kubek.

But Polish often leaves out a noun when it is already obvious from context. So:

• weź inny
  literally = take another/different one

English often uses one in this situation. Polish usually does not need a separate word like one here. The adjective can stand by itself if the noun is understood.

Here inny is in the accusative singular masculine inanimate, because it refers to the object you are being told to take: another mug.

The full version is:

• weź inny kubek

Now the important part: for masculine inanimate nouns, the accusative looks the same as the nominative.

So:

• nominative: inny kubek
• accusative: inny kubek

That is why you do not see innego here.

You would get innego with many masculine animate nouns, for example:

• weź innego psa = take another dog

It can mean both, depending on context.

In this sentence, weź inny most naturally means:

• take another one
• take a different one

Polish inny covers both ideas here, so English can translate it in either way depending on what sounds best.

Because the sentence has two linked parts:

• Ten kubek jest brudny = a statement
• weź inny = a command

Polish commonly joins short clauses like this with a comma, especially when the second part follows naturally from the first.

You could also make the connection more explicit:

• Ten kubek jest brudny, więc weź inny.
  = This mug is dirty, so take another one.

But the version without więc is very natural.

By itself, weź inny is a direct command to one person. It is natural with friends, family, children, or in casual situations.

To make it softer, you could say:

• Weź może inny. = Maybe take another one.
• Weź inny, proszę. = Take another one, please.
• Proszę wziąć inny kubek. = a more polite/formal version

So the original sentence is not rude by itself, but its tone depends on context and relationship.