Question 127·Easy·Linear Inequalities in One or Two Variables
Jordan is buying notebooks and pens for a school club. Each notebook costs $4, and each pen costs $2. The club can spend no more than $120 on these supplies. Which inequality represents this situation, where is the number of notebooks and is the number of pens?
For linear inequality word problems, first write an expression for the quantity being limited (often money, distance, or time) by multiplying each item's cost or rate by its quantity and adding them. Then translate key phrases like "no more than" (use ), "at least" (use ), and "less than" (use ) into inequality symbols. Finally, check that each variable’s coefficient matches the correct price or rate and that the inequality direction matches the wording; this lets you quickly eliminate choices that flip prices or use the wrong inequality sign.
Hints
Identify the total cost expression
How do you write the total cost if notebooks cost $4 each and pens cost $2 each? Think: (cost per notebook)(number of notebooks) plus (cost per pen)(number of pens).
Focus on the phrase "no more than"
Does "no more than $120" mean the total cost must be greater than, less than, or less than or equal to $120? Which inequality symbol matches that idea?
Check both coefficients and inequality sign
Make sure the coefficient of matches the notebook price, the coefficient of matches the pen price, and the inequality symbol matches "no more than."
Quick elimination using meaning
Any inequality that forces the total cost to be at least $120 cannot represent a situation where the club is allowed to spend less than that amount. Use this to cross out some choices.
Desmos Guide
Model the total cost in Desmos
In Desmos, type 4*x + 2*y to represent the total cost in dollars. You can temporarily replace x and y with specific numbers (like 4*10 + 2*20) to see the total cost for certain combinations of notebooks and pens.
Test a combination that should be allowed
Pick numbers that give a total cost under $120, such as notebooks and pens. In Desmos, compute 4*10 + 2*20 to get the cost, and then for each answer choice, type the corresponding inequality with those numbers (for example, 2*10 + 4*20 <= 120) to see whether it is true or false. The correct inequality will be satisfied (true) for a combination that stays within the budget.
Test a combination that should NOT be allowed
Now pick a combination that goes over the budget, like notebooks and pens. Compute the cost in Desmos with 4*40 + 2*10, then plug those numbers into each choice’s inequality. The correct inequality will be false for this over-budget combination, while at least some incorrect inequalities may still appear true.
Step-by-step Explanation
Write an expression for the total cost
Each notebook costs $4 and there are notebooks, so the notebook cost is .
Each pen costs $2 and there are pens, so the pen cost is .
So, the total cost of all items is the sum of these:
Translate "no more than $120" into an inequality
The phrase "no more than" means the amount can be less than or equal to a number.
So, if the total cost is no more than $120, that means:
- The total cost is allowed to be less than $120
- Or it can be exactly $120
In inequality form, that means we want something of the form:
Combine the cost expression with the inequality idea
From Step 1, the total cost is .
From Step 2, the inequality should say that this total cost is less than or equal to .
So we want an inequality where:
- The left side is
- The inequality symbol is
- The right side is
Now compare this structure to the answer choices.
Match with the correct answer choice
Using the structure from Step 3, the correct inequality is:
This matches choice D.