Question 115·Easy·Linear Inequalities in One or Two Variables
A school club is selling tickets to a talent show to raise money for a field trip. An adult ticket costs $20 and a student ticket costs $10. The club needs to collect at least $500 from ticket sales. Which inequality represents this situation, where is the number of adult tickets sold and is the number of student tickets sold?
For word problems that ask you to write an inequality, first translate each part of the situation into algebra: turn the quantities (like numbers of tickets) into expressions using their prices, then combine them to get a total. Next, focus on key phrases such as “at least,” “at most,” “no more than,” and “no less than” to choose the correct inequality symbol (≥ or ≤). Finally, match your algebraic expression and inequality symbol to the choice that exactly represents the total and the condition described.
Hints
Represent the money from each type of ticket
How can you write an expression for the dollars from adult tickets using and for the dollars from student tickets using ?
Combine to get the total amount of money
Once you have expressions for adult-ticket money and student-ticket money, what should you do with them to get the total amount raised?
Think about what “at least” means in math
Does “at least $500” mean the total must be less than, greater than, or equal to ? Which inequality symbol matches that idea?
Match your inequality to the answer choices
Use your expression for the total money and the correct inequality symbol for “at least” to see which choice matches.
Desmos Guide
Check total money for sample ticket numbers
In Desmos’s calculator, type an expression for the total money, such as 20*a + 10*s, then pick some sample values for and (for example, 15 adult tickets and 20 student tickets) by substituting them: type 20*15 + 10*20 and see the total. Compare this total to 500 to understand whether the expression should be greater than, less than, or equal to 500 in the inequality.
Compare with each option
For the same sample values of and , plug into each answer choice inequality (for example, check whether 20*15 + 10*20 >= 500 is true, whether 20*15 + 10*20 <= 500 is true, etc.). The correct inequality will be the one that matches the idea that the club must raise 500 dollars or more.
Step-by-step Explanation
Translate ticket counts into money expressions
Each adult ticket costs $20, and there are adult tickets. The total money from adult tickets is . Each student ticket costs $10, and there are student tickets, so the total from student tickets is .
Write an expression for the total money raised
To find the total amount of money raised from all ticket sales, add the money from adult tickets and student tickets: the total dollars collected is .
Interpret the phrase “at least $500”
The phrase “at least $500” means the amount can be 500 or more. In math, “500 or more” is written with the inequality symbol (greater than or equal to 500).
Combine the expression and the condition
We now connect the total money expression and the requirement:
- Total money collected:
- This must be at least $500:
So the inequality that represents the situation is .