Question 6·Hard·Linear Inequalities in One or Two Variables
A fund-raising committee is selling two types of coupon books: standard and deluxe. Each standard book brings in $12 in revenue, and each deluxe book brings in $18 in revenue. The committee must raise at least $1,260 to cover its costs but can print no more than 100 books in total. If is the number of standard books printed and is the number of deluxe books printed, which system of inequalities correctly represents these constraints?
For word problems that ask you to write a system of inequalities, first define what each variable represents, then write a simple algebraic expression for each quantity mentioned (like total revenue or total items). Next, translate key phrases such as "at least" (≥), "no more than" (≤), "at most" (≤), and "more than" (>) into inequality symbols, being careful to attach the right number to the right expression. Finally, scan the answer choices to eliminate any that flip the inequality signs or swap the constants, and select the one that matches both your expressions and directions.
Hints
Identify the expressions
First, write an expression for the total revenue in terms of and , and another for the total number of books printed.
Match words to inequality symbols
Think about how the phrases "at least" and "no more than" translate into inequality symbols. Does "at least" suggest greater than or equal to, or less than or equal to? What about "no more than"?
Check numbers and inequality directions
Once you know the correct expressions and inequality directions, scan the choices for the one that uses the right numbers (1,260 and 100) with the right signs for each condition.
Desmos Guide
Graph the boundary lines
In Desmos, type 12x + 18y = 1260 on one line and x + y = 100 on another line to see the two boundary lines that correspond to the revenue requirement and the printing limit.
Use test points to understand the verbal conditions
Pick a simple test point, such as , and plug it into and to see the revenue and total books. Decide whether this point meets the phrases "at least 1260" and "no more than 100"; then look on the graph to see which side of each boundary line contains points that actually satisfy those verbal conditions (you can also test other points like or ).
Check each answer choice in Desmos
For each answer choice, type its two inequalities directly into Desmos; Desmos will shade the region that satisfies them. The correct choice will produce a shaded region that keeps all the points that meet both verbal conditions (enough revenue and not too many books) and excludes points that do not.
Step-by-step Explanation
Translate the revenue condition
Each standard book brings in $12, so standard books bring in dollars. Each deluxe book brings in $18, so deluxe books bring in dollars.
So, the total revenue is .
The problem says the committee must raise at least $1,260. In inequality language, "at least" means the expression should be greater than or equal to $1,260. So the revenue inequality uses on the left and a sign.
Translate the printing limit
The total number of books printed is the sum of standard and deluxe books, so that is .
The problem says they can print no more than 100 books. In inequality language, "no more than" means the quantity should be less than or equal to that number. So the book-count inequality uses on the left and a sign.
Combine both inequalities and match the answer choice
Now put both conditions together:
- Revenue: must be at least $1,260, so use .
- Total books: must be no more than 100, so use .
This gives the system
Looking at the answer options, this matches choice D.