In a set of four consecutive odd integers, where the integers are ordered from least to greatest, the first integer...

1. TRANSLATE the problem information

• Given information:
  • Four consecutive odd integers ordered from least to greatest
  • First integer is represented by x
  • Product of 12 and the fourth odd integer ≤ 26 less than sum of first and third integers

2. INFER the pattern for consecutive odd integers

• Since we're dealing with consecutive odd integers, each integer is 2 more than the previous one
• This gives us: \(\mathrm{x, x+2, x+4, x+6}\)

3. TRANSLATE each part of the condition

• "Product of 12 and the fourth odd integer" = \(\mathrm{12(x + 6)}\)
• "Sum of the first and third odd integers" = \(\mathrm{x + (x + 4)}\)
• "26 less than" this sum = \(\mathrm{x + (x + 4) - 26}\)
• "at most" means we use ≤

4. TRANSLATE the complete inequality

• The product is at most 26 less than the sum:
  \(\mathrm{12(x + 6) \leq x + (x + 4) - 26}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skills: Students misinterpret "at most 26 less than" and flip the inequality direction or rearrange the expression incorrectly.

Instead of recognizing that "\(\mathrm{Product \leq Sum - 26}\)," they might think "\(\mathrm{Product \geq 26 - Sum}\)" and rearrange terms. This leads to setting up the inequality as \(\mathrm{12(x + 6) \geq 26 - (x + (x + 4))}\).

This may lead them to select Choice B (\(\mathrm{12(x + 6) \geq 26 - (x + (x + 4))}\))

Second Most Common Error:

Missing conceptual knowledge: Students forget that consecutive odd integers differ by 2, not 1.

They might think the integers are \(\mathrm{x, x+1, x+2, x+3}\), but then realize these aren't all odd and get confused about the correct pattern. This confusion can cause them to abandon systematic solution and start guessing.

The Bottom Line:

This problem requires careful translation of complex language involving inequalities and a solid understanding of integer sequences. Students must parse multiple layers of mathematical language while keeping track of which integer is which.