Question 13·Easy·Linear Inequalities in One or Two Variables
A reading club charges a one-time membership fee of $8 and $12 for each book purchased. If Maya has at most $80 to spend on the membership and books, what is the greatest number of books she can purchase?
For money-and-budget inequality questions, start by defining a variable for the unknown quantity (here, number of items), then write a total cost expression and connect it to the budget with the correct inequality sign ("at most" → ≤, "at least" → ≥). Solve the inequality algebraically, and finally interpret the result in context: the answer must be a whole number if you're counting items, so choose the largest whole number that satisfies the inequality when the question asks for the maximum.
Hints
Represent the situation with a variable
Let stand for the number of books Maya buys. How can you write an expression for the total cost using ?
Set up an inequality, not an equation
The phrase "at most $80" means the total cost is less than or equal to 80. How would you write that as an inequality using your total-cost expression?
Solve step by step and think about whole numbers
After you write the inequality, isolate using inverse operations (subtract, then divide). Once you have a value for , remember that must be a whole number of books. Which is the largest whole number that still fits the inequality?
Desmos Guide
Use Desmos to solve the inequality numerically
In Desmos, type the expression (80 - 8) / 12 and look at the output. This value is the upper limit for how many books she can afford; remember she must buy a whole number of books, so take the largest whole number that does not exceed this value.
Step-by-step Explanation
Define the variable and write the inequality
Let be the number of books Maya buys.
The total cost is the membership fee plus the cost of all books:
- Membership: $8
- Books: $12 per book, so
Because she can spend at most $80, the total cost satisfies
This inequality represents the situation.
Solve the inequality for the number of books
Solve the inequality step by step.
Start with
Subtract 8 from both sides:
Now divide both sides by 12:
Thus, . Interpret this inequality in context in the next step.
Interpret the result and choose the greatest possible number
The inequality means any whole number of books from 0 through 6 will keep the total cost at or below $80.
Since the question asks for the greatest number of books she can purchase, we choose the largest whole number that satisfies the inequality, which is 6.