Question 104·Medium·Linear Inequalities in One or Two Variables
A local print shop charges a one-time setup fee of $28 and $1.35 per flyer. Maya has a budget of $140.
What is the greatest number of flyers she can have printed without exceeding her budget?
(Express the answer as an integer)
For inequality word problems with a fixed fee plus a per-item cost, always start by defining a variable for the number of items, then write a total cost expression (fixed fee + rate × number of items). Use the phrase “without exceeding” or “at most” to set up a ≤ inequality with the budget, solve it step by step, and finally interpret the result in context: if you are counting objects, take the greatest whole number that does not violate the inequality, not a rounded value that might go over the limit.
Hints
Write an expression for total cost
Combine the one-time setup fee and the per-flyer cost into a single expression in terms of the number of flyers .
Use the budget to form an inequality
Your total cost expression must be no more than $140. How do you write that as an inequality?
Solve and think about whole flyers
After solving the inequality for , remember that represents a count of flyers. How should you handle any decimal you get so that Maya does not go over her budget?
Desmos Guide
Compute the maximum flyers value
In Desmos, type the expression 112/1.35 (because ) and look at the value Desmos gives. This is the decimal upper bound for the number of flyers.
Interpret the Desmos result
From the decimal value you see, identify the greatest whole number less than or equal to that value; that whole number is the maximum number of flyers she can buy without exceeding her budget.
Step-by-step Explanation
Define the variable and set up an inequality
Let be the number of flyers Maya prints.
The total cost is: setup fee cost per flyer:
- Setup fee: $28
- Flyer cost: $1.35 per flyer, so for flyers
So the total cost is .
She does not want to exceed her budget of $140, so we write the inequality:
Isolate the flyer cost term
Subtract 28 from both sides of the inequality to get the term with alone on one side:
Solve for the number of flyers
Now divide both sides of the inequality by to solve for :
Use division (or a calculator) to find this value:
So must be less than or equal to about .
Interpret the answer in context
Maya cannot print a fraction of a flyer, and the cost must not exceed her budget. That means must be a whole number less than or equal to .
The greatest whole number less than or equal to is 82, so the maximum number of flyers she can have printed without going over budget is 82.