Question 23·Medium·Linear Inequalities in One or Two Variables
A school pays a flat fee of $120 to rent a bus plus $8 for each student who goes on a field trip. If the school can spend no more than $400 on transportation, which inequality represents the possible numbers of students, , who can attend the trip?
For word problems that ask for an inequality, first define the variable and write an algebraic expression for the quantity described (here, total cost = per-student cost times number of students plus flat fee). Next, translate key phrases like “no more than,” “at least,” and “less than” into inequality symbols (, , , ). Finally, combine the expression with the correct inequality symbol and match it exactly to one of the answer choices.
Hints
Build the cost expression
What expression represents the total cost if each student costs $8 and there is also a flat fee of $120? Think “per student” plus “flat fee.”
Focus on the phrase “no more than $400”
Does “no more than $400” mean the cost must be greater than, less than, or equal to 400? Should the symbol be , , , or ?
Combine expression and inequality symbol
Once you have the cost expression and know which inequality symbol to use for “no more than $400”, write them together in one inequality, then look for the option that matches it exactly.
Desmos Guide
Graph the total cost as a function of students
In Desmos, enter y = 8x + 120 to represent the total cost for students.
Graph the spending limit
Add the horizontal line y = 400 to represent the maximum allowed cost of $400.
Use the graph to understand the inequality
Look at where the line is below or on the line . These points show the values of where the total cost does not go over $400. Translate this relationship into an inequality, then choose the option that matches it.
Step-by-step Explanation
Translate the situation into an expression
The school pays a flat fee of $120 plus $8 for each student.
If is the number of students, then the per-student cost is , and adding the flat fee gives the total cost expression .
Interpret the phrase “no more than $400”
“No more than $400” means the cost can be less than or equal to 400.
So if represents the total cost, then .
Substitute the cost expression into the inequality
Use the expression from Step 1 for the total cost and place it into the inequality from Step 2.
This creates an inequality comparing the total cost to 400 using the symbol .
Match the result to the answer choices
Substituting for in gives:
This matches the answer choice .