Constraints, and Solutions the Situation Will Not Allow

In this lesson you will: Turn the limits described in a situation into one or more inequalities; Decide which values in a solution set are viable and which the situation rules out; Round in the direction the constraint requires, not the direction arithmetic habit suggests.

The algebra is more generous than the world

A middle school is sending 187 students on a trip. Each bus seats 44. Solving 44n >= 187 gives n >= 4.25.

Every number greater than 4.25 satisfies that inequality — 4.3, 5, 7.91, 1000. The inequality does not know that buses come one at a time. You do. The viable values are the whole numbers from 5 upward, and the answer to the question asked is 5.

Four questions to ask before you write the answer down

A.CED.3 asks you to represent constraints and then interpret solutions as viable or non-viable. In practice that is a short checklist, and it takes about ten seconds.

The fourth question is the one students skip. A situation often caps the variable from above as well as below, and a complete answer reports both ends.

Write every constraint down, including the quiet ones

Most constraints are stated: a budget, a capacity, a minimum order. Some are never stated because nobody would think to: the number of crates is a whole number, the number of hours is not negative.

Those unstated conditions are part of the model, not an afterthought. A delivery problem might carry three at once: 340+55c <= 1200, c >= 0, and c whole.

Common mistake — Rounding the way arithmetic habit says instead of the way the constraint says

A student gets n >= 4.25 buses and rounds to 4, because 4.25 rounds down to 4 in every rounding exercise they have ever done.

The habit is deeply trained and usually harmless. Nearest-value rounding is the right move when you are reporting a measurement. It is the wrong move when you are meeting a requirement.

Four buses seat 4 x 44=176 students. There are 187 students, so eleven of them are standing on the curb watching the buses leave.

The reverse error is just as common: getting k <= 7.43 notebooks on a $52 budget and rounding up to 8, which costs $56 and overdraws the account.

How to catch it: Put both neighboring whole numbers back into the situation and see which ones survive. For the buses, 4 leaves students behind and 5 does not, so the answer is 5. For the notebooks, 8 breaks the budget and 7 does not, so the answer is 7. You never have to remember a rounding rule if you test the two candidates.