In Chinese, as in other languages, adjectives can come before or after a noun:
That woman (is) beautiful.
That beautiful woman.
Have a look here for some tips.
English has all kinds of suffixes to change words from noun to verb to adjective to adverb and back again. Chinese does not do that. This begs the question, 'How do you identify which word is what in Chinese sentences?'
Do you live in luxury, not caring about the needy?
You can translate 生活奢侈 as 'live luxuriously' or 'live in luxury'
他的生活很奢侈。 His life(style) is luxurious.
Whereas 'His life is luxury.' means 'the goal and sole aim of his life is to attain and enjoy luxury'
奢侈品 is a compound noun 'luxury goods'. Chinese makes no distinction between luxury and (luxury + -ous) = luxurious and in 'luxury goods', neither does English! While the phrase 奢侈的品 is unlikely or at least unusual, 奢侈的 is not unusual.
Who is paying for her extravagant holiday?
However, in 2。, 奢侈的 is probably not an adjective.
Can you tell the difference between generosity and luxury?
Not: *Can you tell the difference between generosity and luxurious difference.
So, how do you know what kind of word you are dealing with? Could you write a computer program to do that?