Question 64·Easy·Linear Inequalities in One or Two Variables
A gym charges a one-time sign-up fee of $40 and $15 per month for membership. Victoria can spend at most $115 on her membership.
If represents the number of months of membership Victoria can afford, which inequality represents this situation?
For inequality word problems, first isolate the quantity being limited (here, total cost) and write it as an expression in terms of the given variable (fixed fee plus rate times number of units). Then translate key phrases like "at most," "no more than," or "at least" into the correct inequality symbols ("at most" and "no more than" mean ; "at least" and "no less than" mean ). Finally, combine the expression and the inequality symbol, and double-check that the coefficient on the variable matches the per-unit rate and the constant term matches the one-time amount before matching your result to the answer choices.
Hints
Separate the fixed and monthly charges
Identify which part of the cost is paid once and which part is paid every month. How will the monthly part use the variable ?
Build the cost expression
Write an expression that adds the one-time fee to 15 dollars times the number of months . That expression represents the total cost.
Interpret "at most" correctly
Think about the phrase "at most $115". Does it mean the total cost is less than, greater than, or less than or equal to $115? Which inequality symbol fits that idea?
Match your inequality to a choice
Once you have your inequality for the situation, compare it carefully with each answer choice. Pay attention to which number is multiplied by and which number is the fixed fee.
Desmos Guide
Enter expressions for each option
In Desmos, on four separate lines, type the left-hand side of each choice: 40m + 15, 15 + 40m, 40 + 15m, and 15m + 115. When Desmos offers to create a slider for , accept it so you can change .
Check the one-time sign-up fee using
Use the slider to set . For each expression, look at the value it produces when . The correct model should give the one-time sign-up fee (since with 0 months, you only pay the sign-up fee). Identify which expression does that.
Use the budget limit
Once you know which expression correctly models the total cost, compare it to the answer choices and pair it with the statement that says this cost is no more than $115 (uses ). Desmos helps you confirm the correct structure for the left side; the inequality symbol and 115 come from the wording of the problem.
Step-by-step Explanation
Identify the parts of the cost
The gym charges two things:
- A one-time sign-up fee of $40.
- A monthly fee of $15 for each month.
If is the number of months, then the total cost is:
- the fixed $40 sign-up fee plus
- $15 times the number of months .
So the total cost in dollars can be written as an expression involving .
Write an expression for the total cost
The monthly fee applies every month, so it should be multiplied by :
- Monthly part: (15 dollars per month, for months)
- One-time part: 40 (paid once)
So the total cost is the sum of these:
Translate "at most $115" into an inequality
"At most $115" means the total cost cannot be more than $115.
Mathematically, "at most $115" means the total cost is less than or equal to 115, which we write using the symbol :
Combine the cost expression with the inequality
Now substitute the expression for total cost into the inequality from the previous step:
This matches answer choice C, so the correct inequality is .