Question 86·Medium·Linear Inequalities in One or Two Variables
A ride-share app charges a base fee of $1.80 plus $0.60 per mile. For a certain trip, the total charge was at most $12. Let m represent the number of miles for that trip. Which inequality represents all possible values of m?
For word problems that ask for an inequality, first build a clear algebraic expression for the total quantity using the given rates and base amounts (fixed part plus variable part times the variable). Then translate phrases like “at most,” “no more than,” “at least,” or “no less than” into the correct inequality symbol (≤ or ≥). Finally, match that full inequality—both the expression and the direction of the inequality sign—to the answer choices, watching for common traps like reversing the inequality or misplacing the fixed fee.
Hints
Identify the fixed and variable parts of the cost
One part of the cost is the same no matter how many miles you travel, and the other part changes with the number of miles. Which numbers in the problem represent these two parts?
Write an expression for the total cost
If is the number of miles, how do you express the per-mile cost in terms of , and then how do you include the base fee to get the total cost?
Match the phrase “at most $12” to a symbol
Does “at most $12” mean the total cost is less than , greater than , or less than or equal to ? Which inequality symbol matches that idea?
Desmos Guide
Model the total cost as a function of miles
In Desmos, type y = 0.60x + 1.80 to represent the total ride cost (y) as a function of miles (x), based on the description: base fee plus $0.60 per mile.
Represent the maximum allowed cost
Type a second line y = 12 to represent the maximum total charge. Look at where the line for the total cost is below or on the horizontal line to understand which -values (miles) are allowed.
Step-by-step Explanation
Translate the wording into a cost expression
The app charges two parts:
- A fixed base fee of $1.80 (this does not depend on miles).
- A per-mile fee of $0.60 for each mile.
If is the number of miles, the cost for the miles is . So the total cost for the trip is:
Interpret the phrase “at most $12”
“At most $12” means the total cost can be less than or equal to $12, but not more than $12.
In symbols, that is written with a sign:
Write the inequality using your expression
Now replace “total cost” with the expression from Step 1.
We get:
So the inequality that represents all possible values of for this trip is . This corresponds to choice D.