A number x is at most 2 less than 3 times the value of y. If the value of y...

1. TRANSLATE the problem information

• Given information:
  • x is at most 2 less than 3 times the value of y
  • y = -4
  • Need to find the greatest possible value of x

• TRANSLATE the English phrase piece by piece:
  • '3 times the value of y' → \(\mathrm{3y}\)
  • '2 less than 3 times the value of y' → \(\mathrm{3y - 2}\)
  • 'x is at most...' → \(\mathrm{x \leq (3y - 2)}\)

This gives us the inequality: \(\mathrm{x \leq 3y - 2}\)

2. SIMPLIFY by substituting the known value

• Substitute \(\mathrm{y = -4}\) into the inequality:
  \(\mathrm{x \leq 3(-4) - 2}\)
  \(\mathrm{x \leq -12 - 2}\)
  \(\mathrm{x \leq -14}\)

3. INFER the meaning for 'greatest possible value'

• Since \(\mathrm{x \leq -14}\), x can be any number less than or equal to -14
• The greatest value that satisfies \(\mathrm{x \leq -14}\) is exactly -14
• Values like -15, -20, etc. are smaller and therefore not the greatest possible

Answer: -14

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often struggle with the phrase 'at most 2 less than' and may incorrectly translate this as \(\mathrm{x \geq 3y - 2}\) (reversing the inequality) or \(\mathrm{x = 3y - 2}\) (missing the 'at most' constraint entirely).

When they use \(\mathrm{x \geq 3y - 2}\), they get \(\mathrm{x \geq -14}\), leading them to think there's no greatest possible value or to select an answer that doesn't make sense in the context.

Second Most Common Error:

Weak INFER reasoning: Students correctly translate to \(\mathrm{x \leq 3y - 2}\) and substitute to get \(\mathrm{x \leq -14}\), but then don't understand what 'greatest possible value' means in the context of an inequality. They might think the answer is 'all numbers less than or equal to -14' rather than recognizing that -14 itself is the greatest value that satisfies the constraint.

This leads to confusion and guessing among available answer choices.

The Bottom Line:

This problem tests whether students can accurately translate complex English phrases involving inequalities and then interpret what 'greatest possible' means when dealing with upper bounds. The key insight is that 'at most' creates an upper limit, and the greatest value satisfying that limit is the limit itself.