My earlier answer was deleted even though it is expressly a YEC answer. Bolded for clarity because a moderator overlooked this fact and I can't undelete.
How do Young Earth Creationists go about showing that this attempt isn't successful?
Because the doctrine of ex nihilo is false.
Interestingly both the current mainstream Christian argument and the secular pseudo-science that evolved from it claim that matter poofed out of nothing, the only difference being that mainstream Christians say it was God who spoke the universe into being, while secularists say it was some physically impossible "singularity" or something like that. Both are incorrect.
The materials of which our bodies are made are in fact very old. It is literally true that we are made from stardust. This does not make atheists like Carl Sagan right.
The age of the universe is in fact infinite. But no one can falsify that this Earth was formed from those materials around 6000 years ago. The Earth is quite young.
This is simply using the correct definition of the word "create".
This makes the theology of the Church of Jesus Christ of Latter-day Saints the only one that is compatible with known conservation laws of physics. It is the only belief system that affirms conservation laws even when we include what people call "science" and "physics" (because many "scientists" will tell you to your face that "the laws of physics cease to apply" in the purported big bang, which is a disingenuous argument). Ex nihilo theories presuppose conservation (and the laws of physics generally) to be false. But there is no evidence for such an opinion anywhere. It is just a made-up idea. Language itself does not admit any such thing as "ex nihilo creation".
My take on the subject is my own, but the very meaning of the Hebrew word bara' used in Genesis 1:1 already proves that ex nihilo always was a false doctrine. Every argument to the contrary requires special pleading.
Therefore in answer to the question, How old is the universe, or the matter in the universe?:
The answer is 'infinite'.
The Bible gives us an idea of when this Earth was formed, but it does not say when the stars and other worlds were made. It only says "God made the stars also". Of course He did!
Once we take the Scriptures and the known meanings of words to be true, we can dispense with the debate and strife on the subject.
We can then ask a more intelligent question: When was the Earth organized into a globe orbiting the Sun?
Since modern theories don't even allow the Earth to exist, the question of when the Earth was organized is up for grabs. Two hundred years from now someone discovering the ruins of our civilization might find some few articles in an excellent state of preservation. The question of when they were made in part begs the question of how they were made. One must actually have a model for how a thing is made, what it is made of and how it ages in a natural lifecycle in order to begin to estimate its age from its present appearance and condition. Which of us were there when this Earth was formed? Have we witnessed even one complete lifecycle of a planet like our own to use as a model for the age of our own? Until secular "scientists" have an actual, plausible model for how the Earth was formed, they will have to punt on this question. In fact the video documentary link above highlights dozens of evidences that the Solar System is much, much, much younger than modern scholars say it is.
Even if one has a model for how a thing is made and how it normally ages, there is still no such thing (short of a manufacturer's label) as an infallible way to tell how old a manufactured article is in general. If you have a hubcap made twenty years ago, can you reliably differentiate it from one made last Thursday?
This Earth bears its manufacturer's label in Genesis chapter 1.
The Creation is ongoing. My physical body has the appearance of being young because it is young. The materials it is made of are ancient; the matter it is composed of has always existed. Fossils having some extraordinary age is an illusion; the only reason why many think they must be terribly old is because Darwinian textbooks say they are, while ignoring the vastly more credible hypothesis that there was in fact a global Flood.
So everything materially is old eternally, and yet part of it may be newly organized (literally, "created" into something, of things which are not seen as Paul states, speaking of the Creation) as of last Thursday or 6000 years ago and this violates nothing but some people's preconceptions.
if the world was created 6,000 years ago
I have every reason to believe that it was.
I believe this Earth was formed about 6000 years ago as the Bible states, not millions or billions of years ago.