Question 92·Medium·Linear Inequalities in One or Two Variables
A school club is buying cookie boxes for a fundraiser. The bakery charges a one-time packaging fee of $22 and $3.90 per box. The club has a budget of $180.
What is the greatest number of boxes they can buy without exceeding the budget?
(Express the answer as an integer)
For SAT word problems about budgets and “at most” or “no more than,” turn the situation into a linear inequality: fixed fee plus (cost per item) times (number of items) is less than or equal to the budget. Solve the inequality step by step—first isolate the variable term, then divide by the coefficient—and finally interpret the result in context: if the variable counts objects (like boxes), choose the greatest whole number that satisfies the inequality without exceeding the given limit.
Hints
Translate words into an inequality
Try defining a variable for the number of boxes. How can you write an expression for the total cost using the $22 fee and the $3.90 per box?
Include the budget correctly
The club cannot spend more than $180. Should your inequality use , , , or when comparing the total cost to 180?
Solve step by step
Once you have an inequality, first move the 22 to the other side, then divide by 3.9. Pay attention to what the solution tells you about the possible values of your variable.
Think about whole numbers
If your solution for the variable is not a whole number, how do you decide which whole number of boxes is acceptable without going over the budget?
Desmos Guide
Enter the total cost inequality
In Desmos, type the inequality 22 + 3.9x <= 180. Desmos will shade the region of -values that satisfy this inequality.
Find the maximum whole-number solution
Look at where the boundary line intersects the -axis (or use a table for the expression 22 + 3.9x and find when it first goes above 180). Then determine the largest integer that still lies in the shaded (true) region and keeps the cost at or below 180.
Step-by-step Explanation
Define the variable and write the inequality
Let be the number of cookie boxes.
The bakery charges a one-time fee of $22 and $3.90 per box, so the total cost is:
The club cannot spend more than $180, so we write the inequality:
Isolate the term with x
Subtract 22 from both sides to get the part that depends on alone:
Solve for x
Now divide both sides of the inequality by :
Use division (or a calculator) to evaluate . You should get a value a little larger than 40.
Interpret the result in context
The inequality solution tells you that must be less than or equal to about .
Because represents a number of boxes, it must be a whole number, and it must not make the cost exceed $180. The largest whole number less than or equal to this value is 40, so the greatest number of boxes they can buy without exceeding the budget is 40.