A community theater plans to sell at least 120 adult tickets and at least 180 student tickets for an upcoming...

1. TRANSLATE the problem information

• Given information:
  • At least 120 adult tickets will be sold
  • At least 180 student tickets will be sold
  • Student ticket costs $5 less than adult ticket
  • Want to raise at least $6,300 total
  • \(\mathrm{x}\) = price of one adult ticket

• What this tells us:
  • Adult ticket price = \(\mathrm{x}\) dollars
  • Student ticket price = \(\mathrm{(x - 5)}\) dollars
  • Need an inequality for total revenue \(\geq\) $6,300

2. INFER the revenue structure

• We need to find total revenue from both ticket types
• Total revenue = (adult tickets sold × adult price) + (student tickets sold × student price)
• This must be at least $6,300, so we need a \(\geq\) inequality

3. TRANSLATE into mathematical expressions

• Revenue from adult tickets: \(\mathrm{120 \times x = 120x}\)
• Revenue from student tickets: \(\mathrm{180 \times (x - 5) = 180(x - 5)}\)
• Total revenue: \(\mathrm{120x + 180(x - 5)}\)
• Constraint: This total \(\geq\) 6,300

4. Write the complete inequality

• \(\mathrm{120x + 180(x - 5) \geq 6,300}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse which ticket costs more and incorrectly write the student ticket price as \(\mathrm{(x + 5)}\) instead of \(\mathrm{(x - 5)}\).

The problem states "student ticket costs $5 less than adult ticket," but students sometimes interpret this backwards, thinking the adult ticket costs $5 less. This leads them to use \(\mathrm{(x + 5)}\) for the student price.

This may lead them to select Choice C (\(\mathrm{120x + 180(x + 5) \geq 6,300}\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students use the wrong inequality direction, writing \(\leq\) instead of \(\geq\) because they don't properly interpret "at least."

"At least $6,300" means the amount should be greater than or equal to $6,300, but some students think of it as a maximum constraint and use \(\leq\).

This may lead them to select Choice D (\(\mathrm{120x + 180(x - 5) \leq 6,300}\)).

The Bottom Line:

This problem requires careful attention to the relationship between ticket prices and precise translation of constraint language. The key is recognizing that "student costs $5 less" means subtracting 5 from the adult price, and "at least" always means \(\geq\).