A system of inequalities models several limits that must hold at once. Phrases like no more than and at least tell you which direction each inequality points. Translate the weight limit and the comparison between counts separately, then put them together. The main trap in a “times as many” statement is multiplying the wrong count.
Hints
- Hint 1
For the weight limit, multiply each item's weight by its count, then add. The total weight can equal the van's capacity but cannot exceed it. Which inequality symbol allows equality?
- Hint 2
In three times as many pouches as boxes, the boxes are the starting count. Multiply the number of boxes by to find the minimum number of pouches. Which variable counts pouches?
Step-by-step
Translate each limit
Step 1Model the van's weight limit
No Desmos needed. You're translating the limits, not solving for the counts. Each box weighs ounces, so boxes weigh ounces. The pouches weigh ounces altogether. No more than ounces includes a full van, so the total must satisfy
- Step 2
Model the minimum number of pouches
The boxes number , so three times that count is . At least three times as many pouches means can equal or be greater: . In “times as many pouches as boxes,” multiply the box count, not the pouch count. Together, these inequalities form the system for the given nonnegative integer counts. Choice D.