Question 98·Medium·Linear Inequalities in One or Two Variables
A librarian has 3 hours available at the service desk today but must attend a 30-minute meeting. Processing a paperback return takes 4 minutes, and processing a hardcover return takes 10 minutes.
If is the number of paperbacks and is the number of hardcovers she processes today, which inequality represents all possible combinations of and she can complete within her available time?
For word problems that ask for an inequality with two variables, first convert all units so they match (usually convert hours to minutes or vice versa). Next, write an expression for the total quantity in terms of the variables (here, total time: minutes per item times number of items, added together). Then carefully translate the key phrase—such as “no more than,” “at least,” or “within”—into the correct inequality symbol (, , , or ). Finally, compare what you wrote to the answer choices, checking both the coefficients (do they match the situation?) and the right-hand side (does it use the correct total amount?).
Hints
Convert all time to minutes
First, change the 3 hours into minutes and then subtract the 30-minute meeting to find how many minutes she can actually spend processing returns.
Express the total time for x paperbacks and y hardcovers
Think: if each paperback takes 4 minutes, what expression gives the time for paperbacks? If each hardcover takes 10 minutes, what expression gives the time for hardcovers? Add those together.
Decide which inequality symbol to use
Her total processing time must fit within her available time (cannot go over it). Should the total time be less than, greater than, or equal to her available minutes—and which inequality symbol shows that?
Desmos Guide
Compute the available minutes
On your scratch paper (not in Desmos), convert 3 hours to minutes and subtract the 30-minute meeting to find that she has 150 minutes for processing returns.
Graph each answer choice in Desmos
In Desmos, enter each answer choice exactly as written (for example, type 4x + 10y >= 150, then on another line 10x + 4y <= 150, etc.). Each one will show a boundary line and a shaded region of combinations.
Match the shading to the story
For each graphed inequality, think about whether the shaded region represents combinations whose total time stays at or below 150 minutes. The correct inequality’s graph will include reasonable points like and any point where the total time is less than 150 minutes, and it will use 150 (not 180) as the limit.
Step-by-step Explanation
Find the librarian’s available working time in minutes
She has 3 hours at the service desk.
Convert hours to minutes:
She must attend a 30-minute meeting, so subtract that from 180:
Write an expression for total processing time
Each paperback takes 4 minutes, and she processes paperbacks, so time for paperbacks is minutes.
Each hardcover takes 10 minutes, and she processes hardcovers, so time for hardcovers is minutes.
Total processing time is the sum:
Translate “within her available time” into an inequality
"Within her available time" means the total processing time cannot be more than 150 minutes; it can be equal to or less than 150.
So the inequality comparing total time to available time is:
Now look at the answer choices and select the one that matches this inequality, which is choice D.