Show your attempted setup first. The useful feedback is often about the model, before any arithmetic begins.
Turn the wording into constraints
This original classroom exercise needs at least 23 blank cards. Cards are sold in packs of six for $5 and packs of ten for $9. Buy whole packs, with no delivery fees, taxes, discounts or stock limits in the exercise. Find the lowest stated pack cost.
Let x be the number of six-packs and y the number of ten-packs:
x and y are nonnegative integers
6x + 10y >= 23
Cost = 5x + 9y dollars; minimize this cost
The OpenStax guidance on inequality applications, checked October 24, 2026, explains why whole-number conditions affect solutions to real situations. The pack sizes and prices here are independently authored practice data.
"At least 23" permits 24 or 26 cards. Replacing the inequality with 6x + 10y = 23 makes this fixture impossible: the left side is even and 23 is odd. That would be an error in translation, not evidence that the original purchase cannot be made.
Find a feasible choice, then compare cost
Four six-packs provide 24 cards for $20, so a feasible solution exists. To prove it is the least expensive, exclude every choice that could cost less.
If y is at least 3, ten-packs alone cost at least $27, already more than $20. Therefore any cheaper competitor must have y equal to 0, 1 or 2. For each of those values, choose the smallest feasible whole x. Extra six-packs only increase cost under the stated positive prices.
| Ten-packs y | Minimum six-packs x | Cards | Cost |
|---|---|---|---|
| 0 | 4 | 24 | $20 |
| 1 | 3 | 28 | $24 |
| 2 | 1 | 26 | $23 |
The least cost is $20, achieved by four six-packs. The three-row comparison is complete because y at least 3 has already been excluded by its cost, and each remaining row uses its least-cost feasible x.
Buying more cards is allowed. Buying the smallest surplus is not the stated objective, although it happens to agree here. For a different set of prices, minimizing leftovers could choose a more expensive purchase.
Ask for a model audit, not just a number
Help review my math word-problem setup before solving it.
Problem: need at least23 cards. Packs:6 cards for5 dollars;
10 cards for9 dollars. Whole packs only, nonnegative quantities.
No additional costs or stock restrictions. Minimize stated pack cost.
My proposed setup and attempted answer: [PASTE]
Define each variable with its unit. Translate the quantity condition
and objective separately. Check whether I used equality incorrectly
or allowed a fractional pack. Find a feasible purchase and explain
why every cheaper possibility is excluded using a finite bound.
Show pack counts, card total and cost for the remaining candidates.
Substitute the chosen quantities into the original wording.
Do not invent prices, discount rules or a real supplier recommendation.
After feedback, give one changed practice problem and wait for my setup.
OpenAI's prompting documentation supports supplying the task and relevant constraints. Include the original wording so the model can compare your equation with what was actually asked.
Explain why the answer belongs to this problem
For the selected choice, x=4 and y=0 are permitted whole quantities. They provide 24 cards, meeting the 23-card minimum, and cost 4 times $5, or $20. The bounded comparison excludes every cheaper purchase under the exercise's assumptions.
Write that explanation in the assignment's required format. If the difficulty is a symbolic transformation after the model is correct, the algebra guide examines that separate step. If you want to practice rather than reveal the next solution, use one-step study prompts and attempt the new constraints yourself first.