Question
A local charity is organizing a fundraising event. They charge $15 per ticket and have fixed costs of $450 for
venue rental. If they want to raise at least $1200, the number of tickets they need to sell can be represented by
the inequality 15x−450 ≥ 1200 , where x is the number of tickets sold. Solve this inequality to find the minimum
number of tickets they need to sell: x ≥ ____
2. Emma is planning her monthly budget. She earns $2000 per month and wants to save at least $500. Her fixed
expenses (rent, utilities, etc.) total $1200. If x represents her variable expenses (food, entertainment, etc.),
which inequality correctly represents her budget constraints?
A.
2000 − 1200 − x ≥ 500
C.
x + 1200 + 500 ≤ 2000
B.
2000 − 1200 − x ≤ 500
D.
x + 1200 + 500 ≥ 2000
3. A small business produces custom t-shirts. The cost to produce each t-shirt is $8, and there’s a fixed overhead
cost of $200 per day for running the workshop. If they sell each t-shirt for $20, create a linear equation to
represent their daily profit (P) in terms of the number of t-shirts sold (x) . Then, determine how many t-shirts
they need to sell to break even (make zero profit). Show your work and explain your reasoning
venue rental. If they want to raise at least $1200, the number of tickets they need to sell can be represented by
the inequality 15x−450 ≥ 1200 , where x is the number of tickets sold. Solve this inequality to find the minimum
number of tickets they need to sell: x ≥ ____
2. Emma is planning her monthly budget. She earns $2000 per month and wants to save at least $500. Her fixed
expenses (rent, utilities, etc.) total $1200. If x represents her variable expenses (food, entertainment, etc.),
which inequality correctly represents her budget constraints?
A.
2000 − 1200 − x ≥ 500
C.
x + 1200 + 500 ≤ 2000
B.
2000 − 1200 − x ≤ 500
D.
x + 1200 + 500 ≥ 2000
3. A small business produces custom t-shirts. The cost to produce each t-shirt is $8, and there’s a fixed overhead
cost of $200 per day for running the workshop. If they sell each t-shirt for $20, create a linear equation to
represent their daily profit (P) in terms of the number of t-shirts sold (x) . Then, determine how many t-shirts
they need to sell to break even (make zero profit). Show your work and explain your reasoning
Answers
GPT-5 mini
1. Solve 15x − 450 ≥ 1200:
15x ≥ 1200 + 450 = 1650
x ≥ 1650/15 = 110
Minimum number of tickets: x ≥ 110
2. Correct inequality: A. 2000 − 1200 − x ≥ 500
(Equivalently this can be rearranged to option C: x + 1200 + 500 ≤ 2000. Solving gives 800 − x ≥ 500 ⇒ −x ≥ −300 ⇒ x ≤ 300.)
3. Profit model and break-even:
Revenue = 20x
Costs = fixed 200 + variable 8x
Profit P = revenue − costs = 20x − (8x + 200) = 12x − 200
Break-even when P = 0:
12x − 200 = 0 ⇒ 12x = 200 ⇒ x = 200/12 = 16 2/3
So they must sell at least 17 t-shirts to avoid a loss (since you can’t sell a fraction of a shirt).
15x ≥ 1200 + 450 = 1650
x ≥ 1650/15 = 110
Minimum number of tickets: x ≥ 110
2. Correct inequality: A. 2000 − 1200 − x ≥ 500
(Equivalently this can be rearranged to option C: x + 1200 + 500 ≤ 2000. Solving gives 800 − x ≥ 500 ⇒ −x ≥ −300 ⇒ x ≤ 300.)
3. Profit model and break-even:
Revenue = 20x
Costs = fixed 200 + variable 8x
Profit P = revenue − costs = 20x − (8x + 200) = 12x − 200
Break-even when P = 0:
12x − 200 = 0 ⇒ 12x = 200 ⇒ x = 200/12 = 16 2/3
So they must sell at least 17 t-shirts to avoid a loss (since you can’t sell a fraction of a shirt).