A spending cap and a protein minimum form a system of inequalities: both limits must hold at once. Write one inequality for each limit, then graph them in Desmos with nonnegative packet counts. To minimize almond packets, find the leftmost point that meets every condition. A protein-rich combination is not enough if it exceeds the budget.
Hints
- Hint 1
Let count almond packets and count pretzel packets. Each packet adds its own cost, so an inequality for the budget adds the two costs and uses for at most.
- Hint 2
Write a second inequality for protein. Its sign goes the other way: at least means . Graph both inequalities in Desmos, then ask where their shaded regions can meet.
- Hint 3
Packet counts cannot be negative. After adding that restriction, look for the leftmost possible point. A solid boundary line counts because both limits allow equality.
Step-by-step
Approach 1: Graph the limits in Desmos
Step 1Turn the budget into an inequality
Let be almond packets and be pretzel packets. Their total cost is , so at most $280 means . Type that inequality in Desmos; it shades combinations within the budget.
- Step 2
Turn the protein minimum into an inequality
Each almond packet adds grams and each pretzel packet adds , so at least grams means . Add it in Desmos. The second shading shows combinations that meet the protein requirement.
- Step 3
Exclude negative packet counts
Add and , since you can't buy a negative number of packets. Desmos now shows only counts on the allowed side of each axis. The feasible region means the points that satisfy every limit; here, look closely at where the solid boundaries meet the -axis.
- Step 4
Read the meeting point
Click where the budget and protein boundary lines meet. Desmos shows . Both lines are solid, so that point meets the $280 cap and the -gram minimum exactly. To its left, meeting the protein minimum needs more pretzels than the budget allows; to its right, almond packets alone exceed the budget. The only combination meeting every limit is the meeting point. The minimum number of almond packets is . Grid in 400.
Approach 2: Find the cheapest way to supply the protein
Step 1Find how many pretzels the protein requires
For a chosen , find the fewest pretzel packets that supply enough protein. Divide the protein inequality by :
Subtract : . This is a lower bound on : fewer pretzels would leave the boxes short of protein.
- Step 2
Find when the cheapest combination fits
Using the fewest pretzels gives a cost of . Type ; Desmos reports under PARAMETERS. Each almond replaces three pretzels, which cost $1.35, so fewer almonds make this minimum cost higher than $280. At , the pretzel bound gives , and the protein and budget limits are both met. The minimum is 400 almond packets. Grid in 400.