A gym charges a $25 signup fee each month plus $10 for every visit. If Paula can spend at most $100 on the gym this month, which inequality models the possible number of visits, , she can make?
For word problems that ask you to write an inequality, first define the variable clearly (here, is the number of visits). Next, build an expression for the total quantity in question by adding the fixed amount and the per-unit amount (fixed fee plus cost per visit times number of visits). Finally, translate key phrases: "at most" or "no more than" mean , "at least" or "no less than" mean , and then relate your expression to the given limit with the correct inequality symbol before matching it to the choice with the same structure.
Hints
Identify the two parts of the cost
Separate the gym charges into the fixed amount paid once in the month and the amount that depends on how many times Paula visits.
Write the total cost in terms of v
Use for the number of visits. How can you combine the fixed signup fee and the per-visit charge into a single expression that represents Paula's total cost for the month?
Translate "at most" into math
When a word problem says someone can spend "at most" a certain amount, what inequality symbol should you use, and how should Paula's total cost be related to 100 dollars?
Desmos Guide
Graph the total cost as a function of visits
In Desmos, type y = 25 + 10x to represent Paula's total cost (y) when she makes visits in a month.
Graph the spending limit
On a new line, type y = 100 to represent Paula's maximum allowed spending as a horizontal line.
Use the graph to think about the inequality
Look at where the line is at or below the line ; those -values show the number of visits that keep the cost within the budget. Now write an inequality that says the total cost expression is not more than 100 dollars.
Step-by-step Explanation
Write an expression for the total cost
There is a fixed signup fee of 25 dollars that Paula pays no matter what, and she also pays 10 dollars for each visit. If is the number of visits, then the total cost is the fixed fee plus the per-visit cost: .
Interpret the phrase "at most 100 dollars"
The words "at most" mean the quantity can be less than or equal to a number, but not more. So if Paula can spend at most 100 dollars, her total cost must satisfy "total cost ".
Combine the cost expression with the inequality
Substitute the expression for total cost into the inequality from step 2: since the total cost is and it must be at most 100, we get . This matches choice B, so the correct inequality is .