Question 37·Easy·Linear Inequalities in One or Two Variables
A fitness studio charges an initial registration fee of $25 and $15 for each class attended. If Paul has at most $130 to spend on registration and classes, and classes must be purchased in whole numbers, what is the maximum number of classes Paul can attend?
For cost-and-budget word problems, first define a variable for the quantity being asked (here, the number of classes), then write the total cost as fixed fee + (rate × number of items). Translate phrases like at most into an inequality (≤ or ≥ as appropriate), solve the inequality step by step, and finally interpret the result in context—if the situation only allows whole numbers, take the greatest or least whole number that still fits the inequality and the question’s wording (maximum or minimum).
Hints
Turn words into an expression
Let be the number of classes. How can you write an expression for the total cost Paul pays, using the $25 registration fee and the $15 per class?
Use “at most” to write an inequality
Once you have the expression for the total cost, how do you show that this cost must be no more than $130 using an inequality symbol?
Solve the inequality step by step
After writing the inequality, first subtract the registration fee from both sides. What simpler inequality in terms of does that give you, and what do you get when you divide by 15?
Think about whole numbers and “maximum”
Your inequality will give a range of values for . Which largest whole number in that range satisfies Paul’s budget?
Desmos Guide
Enter the total cost expression
In Desmos, type the expression 25 + 15x. This represents the total cost in dollars when is the number of classes.
Use a table to test integer values
Click the small table icon next to the expression to create a table. In the -column, enter whole numbers starting from 0 and increasing. In the -column, Desmos will show the corresponding total cost.
Find the largest allowed integer
Look down the -values and find where they are less than or equal to 130. Identify the largest integer for which the total cost is still at most 130; that is the maximum number of classes Paul can afford.
Step-by-step Explanation
Define the variable and write the inequality
Let be the number of classes Paul attends.
The total cost is the registration fee plus the cost for each class:
- Registration fee: $25
- Class cost: $15 per class
So the total cost is .
Because Paul has at most $130 to spend, the inequality is:
Isolate the term with the variable
To solve the inequality, first get the term with alone on one side.
Subtract from both sides:
Solve for the number of classes
Now divide both sides of the inequality by to solve for :
This tells us that cannot be larger than 7.
Interpret the inequality and answer the question
The inequality means any number of classes up to and including 7 is allowed.
Because classes must be whole numbers, the maximum number of classes Paul can attend is 7.