Question 124·Easy·Linear Inequalities in One or Two Variables
A taxi service charges a flat fee of $3 plus $2 per mile traveled. If Raj can spend at most $25 on his ride, which inequality models the possible number of miles, , he can travel?
For word problems about cost with a flat fee plus a per-unit charge, first write the total cost as "flat fee + (rate × quantity)," using the given variable for the quantity. Then translate key phrases: "at most," "no more than," and "up to" mean you should use ; "at least" and "no less than" mean . Finally, write one inequality comparing your total-cost expression to the given limit and choose the option that matches it exactly.
Hints
Identify the total cost
First, combine the flat fee and the per-mile charge into a single expression that gives the total cost in terms of .
Understand the words "at most"
Ask yourself: does "at most $25" allow amounts bigger than 25, smaller than 25, or both? Which inequality symbol matches that idea?
Put it together
Once you have the expression for total cost and know which inequality symbol to use, write one inequality that compares the total cost to 25.
Desmos Guide
Graph the cost as a function of miles
In Desmos, type y = 3 + 2x. This graph shows the total taxi cost (y) for each number of miles (x).
Graph Raj's maximum budget
Type y = 25 to graph a horizontal line representing Raj's maximum allowed spending.
Interpret the inequality visually
Look at where the line y = 3 + 2x is on or below the line y = 25. The x-values in that region are the miles Raj can travel without exceeding $25; match this situation to the algebraic inequality that says the total cost is less than or equal to 25.
Step-by-step Explanation
Write an expression for the total cost
The taxi charges a flat fee of $3 and $2 per mile.
- Let be the number of miles.
- The cost for the miles is .
- Adding the flat fee, the total cost is .
Translate "at most $25" into math
The phrase "at most $25" means the amount can be less than or equal to $25.
So the inequality relating Raj's total cost and $25 is:
Combine the expression and the inequality
Substitute the expression for total cost into the inequality from Step 2.
We have total cost , so:
This inequality models all possible numbers of miles Raj can travel without spending more than $25.