Question 108·Hard·Linear Inequalities in One or Two Variables
In a fundraiser, a team plans to purchase boxes of candy to resell. The wholesaler charges $4 per box for the first 20 boxes and $2.50 per box for each additional box beyond the first 20. There is also a one-time shipping fee of $35. The team’s budget for buying the candy is at most $250. If the team purchases more than 20 boxes, which system of inequalities represents the possible numbers of boxes, , the team can buy?
For word problems with inequalities, first translate phrases about amounts directly into inequality symbols: "more than" usually means a strict , "at least" means , and "at most" or "no more than" means . Next, carefully build the algebraic expression for total cost by separating fixed costs (like shipping or the first block of items) from variable costs (the per-item price times the number of items). Once you have the expression, attach the correct inequality sign based on the budget or limit, and then combine it with any other conditions (such as minimum or maximum quantities) to form a system. Finally, match your system to the answer choices without wasting time solving for the variable unless the question specifically asks for that.
Hints
Start with the number of boxes
Focus on the phrase "more than 20 boxes." Should this be written with or ? Think about whether 20 is allowed or not.
Build the cost expression step by step
Find the total fixed cost first: how much do the first 20 boxes cost at 4 dollars each, and what happens when you add the 35-dollar shipping fee? Then, how do you represent the cost of each additional box beyond 20 at 2.50 dollars each?
Turn the budget into an inequality
The team’s budget is "at most 250 dollars." Which inequality symbol (, , , or ) matches "at most" when comparing the total cost to 250?
Combine both conditions
Once you have an inequality for the number of boxes and another for the total cost, think about how to write them together as a system that must be true at the same time.
Desmos Guide
Graph the total cost as a function of boxes
In Desmos, enter the expression C(b) = 2.5b + 65 (or use x instead of b). This represents the total cost (in dollars) of buying boxes when is greater than 20.
Graph the budget limit
On a new line, enter y = 250. This is a horizontal line showing the maximum budget of 250 dollars.
Interpret the region that satisfies the inequalities
Look at where the graph of is on or below the line (this shows "at most 250 dollars"), and focus only on values of that are greater than 20. Then choose the answer option whose inequalities describe exactly that situation.
Step-by-step Explanation
Translate the phrase about the number of boxes
The problem says the team purchases more than 20 boxes.
- "More than 20" means the number of boxes must be strictly greater than 20, not equal to 20.
- So this condition is written as the inequality .
Write an expression for the total cost when b>20
There are three parts to the cost:
- First 20 boxes at 4 dollars each: the cost is dollars.
- Additional boxes beyond 20 at 2.50 dollars each: there are such boxes, so their cost is dollars.
- Shipping fee: a flat 35 dollars.
So the total cost is:
- .
Now simplify this expression:
- .
So the total cost in terms of (when ) is dollars.
Translate the budget condition into an inequality
The budget is at most 250 dollars.
- "At most 250" means the total cost cannot be more than 250; it can be less than or equal to 250.
- Using the cost expression from the previous step, this becomes the inequality
.
This inequality represents the budget limit.
Combine both conditions into a system and match the choice
We now put together both conditions:
- Number of boxes: (more than 20 boxes)
- Budget limit: (total cost at most 250 dollars)
So the system of inequalities that represents the situation is
- and ,
which corresponds to choice A.