Question 96·Hard·Linear Inequalities in One or Two Variables
A food bank is ordering boxes of granola bars and pouches of fruit snacks from a wholesaler. The wholesaler will load no more than 1,000 ounces of product onto the delivery van. Each box of granola bars weighs 14.8 ounces, and each pouch of fruit snacks weighs 3.7 ounces.
To meet demand, the food bank needs at least three times as many pouches of fruit snacks as boxes of granola bars.
Let be the number of boxes of granola bars and be the number of pouches of fruit snacks, where and are non-negative integers.
Which system of inequalities best models this situation?
For word problems that ask you to write a system of inequalities, first clearly define each variable. Then translate each sentence piece by piece: write an expression for any total (like total weight or cost), and match key phrases to inequality symbols: "no more than" or "at most" means , "at least" or "no less than" means . For "times as many" statements, decide which quantity should be larger (e.g., if there are at least three times as many as , then ). Finally, compare your inequalities—both the expressions and the inequality directions—with each answer choice to find the exact match.
Hints
Focus on the weight limit
How do you express the total weight of boxes (14.8 ounces each) and pouches (3.7 ounces each)? Then, which inequality symbol matches the phrase "no more than 1,000 ounces"?
Interpret "at least three times as many"
If there must be at least three times as many pouches as boxes, which quantity should be bigger: or ? Should be greater than or less than ?
Match the inequalities to the answer choices
Once you know the correct form of the weight inequality and the correct form of the "at least three times" inequality, look for the choice that uses both of these correctly (including the right inequality directions and which variable is multiplied by 3).
Desmos Guide
Set up variables in Desmos
In Desmos, let the horizontal axis represent the number of boxes (use ) and the vertical axis represent the number of pouches (use ).
Graph the weight constraint
Type the inequality for the weight limit using and :
14.8x + 3.7y <= 1000
Make sure you use <= to match the phrase "no more than 1000 ounces". The shaded region represents all combinations of boxes and pouches that satisfy the van's weight limit.
Graph the demand constraint
Now type the inequality that represents at least three times as many pouches as boxes:
y >= 3x
This shaded region shows all combinations where the number of pouches is at least three times the number of boxes.
Use the graph to confirm the algebraic system
Look at the overlap of the two shaded regions: that is the solution set to the system. The two inequalities you typed into Desmos should match one of the answer choices exactly; choose the option whose system uses the same two inequalities.
Step-by-step Explanation
Translate the variables and weight information
We are told:
- = number of boxes of granola bars
- = number of pouches of fruit snacks.
Each box weighs ounces and each pouch weighs ounces, so the total weight of the order is .
The wholesaler will load no more than 1{,}000 ounces. "No more than" means the total weight is less than or equal to , so the weight condition becomes an inequality of the form
Translate the demand requirement
The food bank needs at least three times as many pouches of fruit snacks as boxes of granola bars.
- "At least" means greater than or equal to.
- "Three times as many pouches as boxes" means the number of pouches is three times the number of boxes.
So the number of pouches must be greater than or equal to three times the number of boxes , giving an inequality of the form
Write the full system and match it to a choice
Putting both conditions together, the system that models the situation is
This matches choice D.