A team hosting an event to raise money for new uniforms plans to sell at least 140 tickets before this...

1. TRANSLATE the problem information

• Given information:
  • At least 140 tickets before the event
  • At least 220 tickets during the event
  • Want to raise at least $5,820 total
  • \(\mathrm{x}\) = price of ticket during the event
  • During-event price is $3 less than before-event price

• What this tells us: If during-event tickets cost \(\mathrm{x}\) dollars, then before-event tickets cost \(\mathrm{(x + 3)}\) dollars

2. TRANSLATE the revenue expressions

• Revenue from before-event tickets: \(\mathrm{140(x + 3)}\)
• Revenue from during-event tickets: \(\mathrm{220x}\)
• Total revenue: \(\mathrm{140(x + 3) + 220x}\)

3. INFER the correct inequality symbol

• The phrase "at least $5,820" means the total must be greater than or equal to $5,820
• This requires the \(\geq\) symbol

4. Write the complete inequality

\(\mathrm{140(x + 3) + 220x \geq 5{,}820}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the price relationship direction. They read "during-event price is $3 less than before-event price" and incorrectly think the before-event price is \(\mathrm{(x - 3)}\) instead of \(\mathrm{(x + 3)}\).

Their reasoning: "If during costs \(\mathrm{x}\) and it's $3 less, then before must cost \(\mathrm{x - 3}\)." This backwards thinking leads them to write \(\mathrm{140(x - 3) + 220x}\) for the total revenue.

This may lead them to select Choice C (\(\mathrm{140(x - 3) + 220x \leq 5{,}820}\)) or Choice D (\(\mathrm{140(x - 3) + 220x \geq 5{,}820}\)) depending on whether they also get the inequality symbol wrong.

Second Most Common Error:

Poor INFER reasoning: Students misinterpret "at least" as meaning "at most" and use \(\leq\) instead of \(\geq\). They think "at least $5,820" means they shouldn't exceed that amount rather than needing to reach or exceed it.

This may lead them to select Choice A (\(\mathrm{140(x + 3) + 220x \leq 5{,}820}\)) if they get the price relationship correct, or Choice C if they get both parts wrong.

The Bottom Line:

This problem tests your ability to carefully translate relationships between quantities. The key insight is paying attention to which variable represents which price, then building the correct mathematical relationship from there.