An elementary school teacher is ordering x workbooks and y sets of flash cards for a math class. The teacher...

1. TRANSLATE the problem constraints

• Given information:
  • x = number of workbooks, y = number of flash card sets
  • Workbooks cost $3 each, flash cards cost $4 per set
  • Must order at least 20 items total
  • Total cost cannot exceed $80

• What this tells us: We need two separate inequalities - one for quantity, one for cost

2. TRANSLATE the quantity constraint

• 'Must order at least 20 items'
• Total items = \(\mathrm{x + y}\)
• 'At least 20' means the total can be 20 or more
• Mathematical translation: \(\mathrm{x + y \geq 20}\)

3. TRANSLATE the cost constraint

• Total cost = (cost per workbook × number of workbooks) + (cost per flash card set × number of sets)
• Total cost = \(\mathrm{3x + 4y}\)
• 'Must not be over $80' means the cost can be $80 or less
• Mathematical translation: \(\mathrm{3x + 4y \leq 80}\)

4. INFER which answer choice matches

• Our system: \(\mathrm{x + y \geq 20}\) and \(\mathrm{3x + 4y \leq 80}\)
• Scanning choices: Only Choice A has both inequalities with correct directions

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing inequality directions, especially 'at least' vs 'at most' language

Students often mix up:

• 'At least 20' with 'at most 20' → writing \(\mathrm{x + y \leq 20}\) instead of \(\mathrm{x + y \geq 20}\)
• 'Not over $80' with 'not under $80' → writing \(\mathrm{3x + 4y \geq 80}\) instead of \(\mathrm{3x + 4y \leq 80}\)

This may lead them to select Choice D (\(\mathrm{x + y \leq 20}\), \(\mathrm{3x + 4y \geq 80}\)) which has both inequalities backwards.

Second Most Common Error:

Poor TRANSLATE reasoning: Switching which expression goes with which constraint

Students sometimes get confused about what represents quantity vs. cost, writing something like 'the cost constraint limits items' or 'the quantity constraint limits money.'

This may lead them to select Choice C (\(\mathrm{3x + 4y \leq 20}\), \(\mathrm{x + y \geq 80}\)) where the algebraic expressions are matched to the wrong constraints.

The Bottom Line:

This problem tests your ability to systematically translate constraint language into mathematical symbols. The key is to handle each constraint separately and double-check that inequality directions match the English meaning.