Question 12·Hard·Linear Inequalities in One or Two Variables
A community center is purchasing folding chairs and folding tables for a fundraiser. Because of limited storage space, the total volume of the shipment must be no more than 120 cubic feet. Each chair occupies cubic feet of space, and each table occupies cubic feet of space.
For every table purchased, the center wants at least 3 chairs so that there will be enough seating.
Let be the number of chairs and be the number of tables, where and are non-negative integers.
Which of the following systems of inequalities best represents this situation?
For word problems that ask for a system of inequalities, first clearly define each variable, then translate each sentence into math one at a time. Use “number of items × amount per item” to get totals, and pay close attention to phrases like “no more than” (≤), “at least” (≥), and “at most” (≤). Once you have each inequality, match them carefully to the answer choices, checking that coefficients (like 4.5 and 12 here) and inequality directions (≥ vs ≤) exactly reflect the wording; you usually don’t need to solve the system, just model the situation correctly.
Hints
Hint 1: Focus on the volume limit
Translate the storage condition first: how do you express the total volume from chairs and tables, and what inequality symbol matches “no more than 120 cubic feet”?
Hint 2: Express “at least 3 chairs for every table”
Think about how many chairs are needed for 1 table, 2 tables, 3 tables, etc. What general expression relates (chairs) and (tables) so that there are at least 3 chairs per table?
Hint 3: Check both parts of each answer choice
For each choice, look separately at the volume inequality and the chairs/tables inequality. Does the first one correctly use for chairs, for tables, and “no more than 120”? Does the second one really mean “at least 3 chairs for each table”?
Desmos Guide
Represent the variables in Desmos
Use for chairs and for tables in Desmos (so corresponds to and to ).
Test the volume inequality for a choice
For any choice you want to test, type its volume inequality into Desmos, replacing with and with , for example: 4.5x + 12y <= 120. Check that the shaded region makes sense for the total volume (for instance, more chairs and tables eventually move outside the region).
Test the chairs-per-table inequality for a choice
Type the second inequality from that same choice, again replacing with and with , such as x >= 3y or x <= 3y. Look at the shaded region and think: does every point in this region have at least 3 chairs for every table, or does it allow fewer chairs than that?
Compare choices
Repeat the previous two steps for each answer choice. The correct system will be the one whose combined shaded region includes only the points that satisfy both conditions: the total volume is no more than 120 cubic feet and the number of chairs is at least 3 times the number of tables.
Step-by-step Explanation
Define the variables and identify the two constraints
We are told:
- = number of chairs
- = number of tables
There are two separate ideas in the situation:
- A volume (storage) limit: the shipment’s total volume must be no more than 120 cubic feet.
- A seating requirement: there must be at least 3 chairs for every table.
We will write one inequality for each of these.
Write the inequality for the volume limit
Each chair uses cubic feet, so chairs use cubic feet.
Each table uses cubic feet, so tables use cubic feet.
Total volume is therefore
The phrase “no more than 120 cubic feet” means the total volume cannot exceed 120, so we use a ≤ inequality:
Write the inequality for the seating requirement
The phrase “for every table … at least 3 chairs” means:
- If there is 1 table, there must be at least 3 chairs.
- If there are 2 tables, there must be at least 6 chairs.
- In general, if there are tables, there must be at least chairs.
That means the number of chairs must be greater than or equal to :
Match the system to the answer choices
From the two constraints, we have the system:
- Volume:
- Seating:
Looking at the options, the system that has both of these inequalities together is:
which is choice A.