A number x is at most 17 less than 5 times the value of y. If the value of y...

1. TRANSLATE the problem information

• Given information:
  • x is at most 17 less than 5 times the value of y
  • \(\mathrm{y = 3}\)
  • Need to find the greatest possible value of x

• TRANSLATE the key phrase: "at most 17 less than 5 times y"
  • "at most" means \(\leq\)
  • "17 less than 5y" means \(\mathrm{5y - 17}\)
  • Therefore: \(\mathrm{x \leq 5y - 17}\)

2. SIMPLIFY by substituting the known value

• Substitute \(\mathrm{y = 3}\) into the inequality:
  \(\mathrm{x \leq 5(3) - 17}\)
  \(\mathrm{x \leq 15 - 17}\)
  \(\mathrm{x \leq -2}\)

3. INFER the greatest possible value

• Since \(\mathrm{x \leq -2}\), x can be any value less than or equal to -2
• The greatest value that satisfies this condition is exactly -2
• Values like -1, 0, or 1 would violate the inequality

Answer: -2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "17 less than 5 times y"

Students often translate this as \(\mathrm{17 - 5y}\) instead of \(\mathrm{5y - 17}\), thinking "17 less than" means "17 minus something." This leads to the inequality \(\mathrm{x \leq 17 - 5y}\). With \(\mathrm{y = 3}\), they get \(\mathrm{x \leq 17 - 15 = 2}\), giving an incorrect greatest value of 2.

Second Most Common Error:

Poor TRANSLATE reasoning: Confusing "at most" with "at least"

Some students translate "at most" as \(\geq\) instead of \(\leq\), creating \(\mathrm{x \geq 5y - 17}\). This gives \(\mathrm{x \geq -2}\), making them think there's no greatest value or that any large positive number works. This leads to confusion and guessing.

The Bottom Line:

Success depends on careful translation of compound phrases. "At most 17 less than 5y" requires parsing two separate pieces: the direction of the inequality ("at most" = \(\leq\)) and the expression being compared to ("17 less than 5y" = \(\mathrm{5y - 17}\)).