Finding maximum and mimimum in inequality word problem

These word problems ask for the maximum or minimum of some quantity, subject to a constraint — a fixed total, a budget, or a limit.
The constraint ties the quantities together, so they cannot all grow at once.

The core move is a trade-off: to make one quantity as large as possible, make everything it competes with as small as possible.
If a fixed budget is split between two groups, spending the least on the first leaves the most for the second.

So push each variable to the extreme end of what the rules allow.
"As small as possible" means the smallest value still permitted by the constraints, and "as large as possible" means the largest permitted.
Read the conditions carefully to find those limits.

The same idea appears in disguise.
To be sure you are never late, plan for the slowest speed, since that gives the longest time.
To make one district's population as large as it can be relative to another, use the exact boundary the rule allows.

A budget of $\$500{,}000$ is split between hiring PhDs and non-PhDs. To make the non-PhD budget as large as possible, what should happen with PhD spending?
The total is fixed, so more for non-PhDs means less for PhDs.
So minimize PhD spending — hire the fewest PhDs at the lowest allowed cost — which leaves the most money for non-PhDs.

Tom must cover $20$ km to school, walking at a speed between $4$ and $6$ km/hr. To be sure he is never late, which speed should he plan for?
The latest he could arrive is at his slowest speed, which takes the longest.
So he plans for $4$ km/hr, giving a time of $\frac{20}{4} = 5$ hours.

Districts must be within $10\%$ of one another in population. If the least populated district has $x$ people, how large can the most populated district $y$ be?
The rule allows $y \le 1.1x$.
To make $y$ as big as possible, use the equality: $y = 1.1x$.