If m is an even integer, v is an odd integer, and m > v> 0, which of

Greyfield , it looks as if you are interpreting one little part incorrectly. The wording is terse. And to use \(x\) can be confusing here.

We are looking for a specially defined kind of number, defined by the prompt. Not a value.

The answers are supposed to tell us how many
of those specially defined numbers there are -- not the actual values of those specially defined numbers.

We need the number of numbers between yet two other numbers on a number line.

We choose a value for \(m\). We choose a value for \(v\). We count how many even integers exist between our \(m\) and our \(v\).
Then we plug our \(m\) value and \(v\) values into the answer choices.
We are looking for the answer choice that "spits out" the number that matches the quantity of even integers that we counted.

Try rewriting this:
...which of the following represents the number of even integers less than m and greater than v

to THIS:
...which of the following represents how many even integers exist that are less than m and greater than v

You wrote
Mmm... no. \(x\) is how many even integers there are.

If we pick values, in one case, the number of integers will be even.
If we increase m, the number of integers will be odd.
Count how many integers there are between your chosen values.
When you plug in your \(m\) and \(v\)
Option B will always give you that quantity, that number of integers.

• Let's use the numbers in your quote, m = 8, v = 3

<----(v = 3)-----4-----5-----6-----7-----(m=8)----->

Between the integer 3 and the integer 8, there are two even integers: 4 and 6.
How many even integers?
Two: (1) the number four; and (2) the number six.

The number of even numbers is 2.

Answer B) \(\frac{(m-v-1)}{2}\)

Plug in your numbers:
\(\frac{(8-3-1)}{2}=\frac{4}{2}=2\)
Perfect.

That is, on the number line, between 8 and 3, there are two even numbers: 4 and 6.
Both are less than m=8 and greater than v=3.

Answer B does not tell you "4 and 6." You have to see that fact yourself.

• Case #2: Your numbers: m = 10, v = 3

<----(v = 3)----4----5----6----7----8-----9-----(m=10>

Between 10 and 3, there are, just as you calculated from B, three even integers: 8, 6, and 4.
All are less than 10 and greater than 3.

Once you have figured out how many of those special integers there are, you cannot "go again"
and plug your "new" choices into (B) to find out which even integers they are.
If you choose different values for the variables, although B's answer or output will change,
it will still match your answer -- your new answer, which has also changed.

v is any odd number. m is any even number. m > v > 0

I could use m = 100 and v = 7. The answer will be B.

The number of even numbers will change depending on what you plug in for m and v,
but option B will "spit out" the number of numbers that matches the values you choose.

Hope that helps.