Question 53·Medium·Linear Inequalities in One or Two Variables
A moving company charges a flat fee of $120 plus $25 per hour of work. If a customer can spend no more than $350 on the service, which inequality models the possible number of hours, , that the company can work for the customer?
For word problems that ask you to write an inequality, first define the variable clearly (here, the number of hours). Next, build an expression for the total quantity in question by combining the fixed part (flat fee) and the variable part (hourly rate times hours). Finally, translate key phrases like "no more than," "at least," and "at most" into the correct inequality symbols and place the total expression on one side and the limit on the other, then match that inequality to the choices.
Hints
Build the cost expression
Focus first on how to express the total cost in terms of . There is a flat fee and a per-hour fee—how do you combine these?
Handle the "per hour" part
'$25 per hour' means you multiply 25 by the number of hours . So the hourly part of the cost is , which you then add to the $120 flat fee.
Interpret 'no more than $350'
Ask yourself: if the customer can spend no more than $350, is the total cost allowed to be greater than $350? Decide whether the inequality symbol should show the cost being greater than $350 or being $350 or less.
Desmos Guide
Graph the cost as a function of hours
In Desmos, let represent hours (instead of ) and enter the equation
y = 25x + 120
This line shows the total cost for each number of hours .
Graph the spending limit
On a new line, enter the horizontal line
y = 350
This represents the maximum amount the customer can spend.
Use the graph to understand the inequality
Look at where the cost line is on or below the line . Those -values are the hours that keep the cost within the budget. In words: 'total cost is at most $350.' Then match that description to the answer choice that uses times the number of hours plus on one side and 350 on the other side.
Step-by-step Explanation
Write an expression for the total cost
The moving company charges a flat fee of $120 plus $25 per hour.
- Let be the number of hours.
- The hourly part of the cost is (because it is $25 per hour times hours).
- Add the flat fee: total cost .
Translate the spending limit into an inequality
The phrase 'can spend no more than $350' means the total cost can be $350 or less, but not above $350.
- In inequality language, that means the total cost is less than or equal to $350.
Combine the expression and the inequality
Put the total cost expression on one side and 350 on the other, with the correct inequality symbol:
- Total cost:
- Spending limit: 350
- Relationship:
This matches answer choice D) .