A linear inequality, a comparison that limits a changing quantity, fits a purchase with a per-item price, a flat fee, and a spending cap. Multiply the price by the count, add the fee once, and use “at most” to include the cap. Use a Desmos regression to find where the cost reaches the cap, then choose the greatest allowed whole number.
Hints
- Hint 1
A flat fee is paid once per order, even if more items are bought. Let count cupcakes. What expression adds the single decoration charge to the cost of cupcakes?
- Hint 2
The words at most include the spending limit itself. Find where total cost equals that limit, then remember that adding cupcakes increases the cost. Could that boundary be the greatest allowed whole number?
Step-by-step
Find the spending cutoff
Step 1Write the cost limit
Let be the number of cupcakes. At $3 each, the cupcakes cost . Because the $8 fee is per order, add it once rather than multiplying it by . At most $50 means the total can equal $50 but cannot exceed it, so .
- Step 2
Find the greatest whole-number count
The cost increases by $3 with each cupcake, so find where it reaches the limit. Type in Desmos. The tells Desmos to find the that makes the sides match; under PARAMETERS, it shows . That is a whole number, and another cupcake would exceed the budget. Ella can order at most cupcakes. Grid in 14.