Question 56·Medium·Linear Inequalities in One or Two Variables
A caterer has a budget of $1,200 for appetizers. Before ordering any food, the caterer must pay a $60 service fee. Mini sandwiches cost $3 each, and stuffed mushrooms cost $5 each.
Which inequality represents the possible numbers of mini sandwiches, , and stuffed mushrooms, , the caterer can order without exceeding the budget?
For budget and cost inequality problems, first subtract any fixed fees from the total budget to find how much is left for variable items. Then write an expression for the total cost by multiplying each item’s price by its quantity variable and adding these together. Finally, translate phrases like “does not exceed,” “at most,” or “within budget” into a inequality with this total cost on one side and the remaining budget on the other, and choose the answer that matches both the expression and the direction of the inequality.
Hints
Figure out how much money is left for food
Before any appetizers are bought, $60 is taken out of the $1,200 budget as a service fee. How much money remains for sandwiches and mushrooms?
Write the total food cost in terms of x and y
Mini sandwiches cost $3 each and stuffed mushrooms cost $5 each. How can you express the total food cost using and ?
Choose the correct inequality symbol
The caterer wants to stay within (not over) the food budget. Should the total food cost be greater than, less than, or equal to the remaining budget? Which symbol, or , matches that idea?
Check both the expression and the constant
Look for the answer choice that uses the correct total food cost expression from the previous hint and compares it to the correct amount of money left after the fee is paid.
Desmos Guide
Enter each answer choice as an inequality
In Desmos, type each option on its own line:
3x + 5y >= 11405x + 3y <= 12003x + 5y <= 11405x + 3y >= 1140Make sure Desmos shades a region for each inequality.
Test a combination that is clearly under budget
Pick a simple combination that is definitely within the budget, such as mini sandwiches and mushrooms (costing , well under the food budget). In Desmos, add the point (100, 0) and see which inequality regions include this point; those are the only candidates that could be correct.
Test a combination that is clearly over budget
Now choose a combination that is definitely too expensive, such as sandwiches and mushrooms (costing , above the food budget). Add the point (0, 300) in Desmos. The correct inequality will not shade this point; use this to eliminate any remaining incorrect inequalities.
Step-by-step Explanation
Account for the fixed service fee
The caterer starts with a budget of $1,200, but must first pay a $60 service fee. That money cannot be used for food.
So the amount of money left for appetizers is
This 1,140 is the maximum that can be spent on mini sandwiches and stuffed mushrooms together.
Write an expression for the food cost
Let be the number of mini sandwiches and be the number of stuffed mushrooms.
- Each mini sandwich costs $3, so sandwiches cost dollars total.
- Each stuffed mushroom costs $5, so mushrooms cost dollars total.
So the total food cost is .
Translate “without exceeding the budget” into an inequality
"Without exceeding the budget" means the food cost can be less than or equal to the amount of money left for food.
So we compare the total food cost to the maximum available using a ("less than or equal to") sign:
Match the inequality to the answer choices
From the answer choices, the inequality that shows the food cost is less than or equal to is
which corresponds to choice C.