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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 $y-4$.

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    $\begingroup$ Here, the "common value" means something akin to "the number they are both equal to". When x+2 and 3 are paired up, x+2=3, so 3 is the common value because they are both equal to 3. When we talk about a common value of x+2 and y-4, we have x+2=y-4 so either would be the common value. It's useful language for the second part of the description of points in S. $\endgroup$ – emma Feb 11 '17 at 23:50
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    $\begingroup$ In that article they are simply saying two of a, b and c are equal. The "common value" is just whatever the two terms are equal to. So if two of 3, x + 2, and y-4 or equal then either i) 3 = x+2 or ii) 3 = y-4 or iii) y-4 = x+2. If i) then the common value is whetever 3 and x + 2 equal, if iii) then it is whatever y-4 and x+2 equal. It isn't one or the other of the two. It is both of the two as the two are equal. That's all. $\endgroup$ – fleablood Feb 11 '17 at 23:55
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This is not about mathematical terms, but just English expression:

"We have three quantities A, B, and C, two of which are equal, and the third of which is no greater than this common value"

means

  • $A = B \geq C$

or

  • $A = C \geq B$

or

  • $A \leq B = C$
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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.

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