Identifying constraints of a variable

Some questions pin down a variable only indirectly, through constraints such as "odd", "distinct", "positive integers", an ordering like $a < b$, or a stated mean or median.
The skill is to turn each constraint into a bound and then choose values at the extremes.

Take a simple one: $2a + 1 < 10$, where $a$ must be an odd number. What is the largest possible value of $a$?
Solving the inequality, $2a < 9$, so $a < 4.5$.
The odd numbers below $4.5$ are $1$ and $3$, so the largest possible value is $a = 3$.

Now a mean-and-median example: three distinct positive integers have a mean of $10$, so they add up to $30$. What is the largest possible value of the smallest of them?
To keep the smallest as large as possible, bunch the three as close together as "distinct" allows: $s$, $s + 1$, $s + 2$.
Then $s + (s + 1) + (s + 2) = 30$ gives $3s + 3 = 30$, so $s = 9$, and the integers are $9, 10, 11$.

So the recipe is always the same: translate each constraint into a limit, then to make one quantity extreme, push the others as far as the rules allow.
To maximize one value, make the others as small as possible; to minimize it, make the others as large as possible.
The "distinct", "integer", and ordering conditions decide exactly how far you can push.

In $3b - 2 < 20$, where $b$ is an even number, what is the largest possible value of $b$?
Solving, $3b < 22$, so $b < \frac{22}{3} \approx 7.33$.
The largest even number below $7.33$ is $b = 6$.

Three distinct positive integers have a sum of $24$. What is the largest possible value of the smallest of them?
To keep the smallest as large as possible, make the three as close as "distinct" allows: $s$, $s + 1$, $s + 2$, summing to $24$.
Then $3s + 3 = 24$, so $s = 7$, and the integers are $7, 8, 9$.

Two positive integers $a$ and $b$ satisfy $a < b$ and $a + b = 9$. What is the smallest possible value of $b$?
To make $b$ small, make $a$ as large as possible while keeping $a < b$.
Since $a$ and $b$ are integers with $a < b$ and $a + b = 9$, the largest $a$ can be is $4$ (giving $b = 5$); any larger $a$ would break $a < b$, so the smallest possible $b$ is $5$.