A system of inequalities describes counts that must meet two limits at once. With tiered pricing, count the first group separately from the boxes beyond it, then add any one-time charge. More than excludes its cutoff, while at most includes it. Pricing every box at the additional-box rate undercounts the first group's cost.
Hints
- Hint 1
The lower rate applies only after the first boxes. If is the total number of boxes, subtract to find how many boxes get that rate.
- Hint 2
A one-time fee is added once, not multiplied by . Add the cost of the first boxes, the cost of the remaining boxes, and the shipping fee.
- Hint 3
A strict inequality leaves out its boundary: more than uses , not . For the box-count condition, does the team get to buy exactly boxes?
Step-by-step
Count each price tier, then apply the limits
Step 1Count the boxes at the lower price
The first boxes cost each, so only the boxes beyond the first get the price. Since counts all the boxes, the number at the lower price is , not .
- Step 2
Write the total cost
Multiply each price by the number of boxes at that price, then add the one-time shipping fee once:
- Step 3
Distribute the additional-box price
Distribute across to separate the part that changes with :
- Step 4
Group the fixed amounts
Move the box-count term to the front. Group the constant terms, the amounts that don't change with :
- Step 5
Find the constant in Desmos
Type in Desmos. It displays , so the total cost is . The isn't only shipping; it also accounts for the higher price of the first boxes.
- Step 6
Apply the budget
The budget is at most $250, so the total may equal $250 but can't exceed it: . Using would allow purchases over budget.
- Step 7
Include the box-count condition
The team purchases more than boxes, so , not . Both conditions must hold: . This system represents the possible numbers of boxes the team can buy. Choice D.