What is a common value and how to find it? I have googled this term and did not find anything useful. This term is used multiple times in this problem explanation from the recent AMC. For example, I do not understand how the common value of $x+2$ and $3$ is $3$ and of $x+2$ and $y-4$ is $y-4$.

If two of the three terms are equal then the common value is just the value of the two terms that are equal.
1) If $3$ and $x + 2$ are the two terms that are equal, the common value is $k = 3 = x+1$.  (Which means $x = 1$ and $k =3 = x + 1 = 3$.)
2) If $3$ and $y - 4$ are the two terms that are equal, the common value is $k = 3 = y-4$.  (Which means $y = 7$ and $k = 3 = y-4 = 3$.
3) If $x+2$ and $y-4$ are the two terms that are equal, the common value is $k= x+ 2=y-4$.  (Which means the common value is 4 less than whatever $y$ is and 2 more than whatever $x$).
We are also told that the third value is at most this common value.
So
1) if $3 = x+2$ then $y-4 \le 3 = x+2$.
2) if $3 = y-4$ then $x+2 \le 3 = y-4$.
3) if $x+2 = y-4$ then $3 \le x+2 = y-4$.
In this case "common value" just means .... common value.  It is not a mathematical term.  It means exactly what it says.
Let Sam, Fred, and Jane be friends.  Two of them live at the same house.  The third lives down the street from the common address.  "common address" just means the address of the two people who live together. 
