Question 19·Easy·Linear Inequalities in One or Two Variables
At a craft fair, Sarah is selling bracelets for $5 each and necklaces for $8 each. She wants to earn at least $120 in total sales. Which inequality represents this situation, where is the number of bracelets sold and is the number of necklaces sold?
For word problems with inequalities, first write a single expression for the quantity the problem cares about (here, total money: price × quantity for each item, then add). Next, carefully translate key phrases: "at least" usually means , "at most" means , "more than" means , and "less than" means . Finally, match both the expression and the inequality symbol to the answer choices, watching for swapped coefficients or reversed inequality signs.
Hints
Find the total earnings expression
How do you express the money from bracelets and necklaces separately, and then combine them into one expression for total sales?
Focus on the words "at least"
Does "at least $120" mean the total can be less than 120, more than 120, or 120 and above? Which inequality symbol matches that idea?
Match your inequality to the choices
Once you know the expression for total earnings and which inequality symbol to use for "at least," look for the option that uses both correctly.
Desmos Guide
Represent the variables in Desmos
In Desmos, use for the number of bracelets () and for the number of necklaces ().
Enter each answer choice as an inequality
For each option A–D, type the corresponding inequality into Desmos, replacing with and with (for example, an option like becomes 5x + 8y >= 120). Observe the shaded region and its boundary line for each choice.
Use test points to check the meaning
Pick an easy combination of items and compute its total sales, such as (0 bracelets, 15 necklaces gives exactly $120) and (20 bracelets gives $100). For each graphed inequality, see whether these points are in the shaded region. The correct inequality will include points with total sales of 120 or more and exclude points with total sales less than 120.
Step-by-step Explanation
Write an expression for total earnings
Each bracelet brings in $5, so selling bracelets earns dollars.
Each necklace brings in $8, so selling necklaces earns dollars.
To get Sarah’s total sales, add these amounts:
Translate the phrase "at least $120"
The problem says Sarah wants to earn at least $120.
- "At least" means no smaller than a number.
- In inequality symbols, "at least 120" means greater than or equal to 120, written as .
So we want Sarah’s total sales to be .
Form the inequality and match the choice
Combine the total sales expression with the inequality from Step 2:
This matches answer choice B) .