The maximum value of variable y is 5 more than half of another number z. Which of the following inequalities...

1. TRANSLATE the key phrase about maximum value

• Given information: "The maximum value of variable y is 5 more than half of another number z"
• What this tells us: If y has a maximum value, then y cannot be greater than that maximum, so \(\mathrm{y \leq maximum\ value}\)

2. TRANSLATE the expression for this maximum value

• "Half of another number z" = \(\mathrm{\frac{1}{2}z}\)
• "5 more than" this quantity = \(\mathrm{\frac{1}{2}z + 5}\)
• So the maximum value is \(\mathrm{\frac{1}{2}z + 5}\)

3. INFER the complete inequality relationship

• Since y cannot exceed its maximum value: \(\mathrm{y \leq \frac{1}{2}z + 5}\)
• This matches choice (C)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor TRANSLATE reasoning: Students misinterpret what "maximum value" means in mathematical terms. They think "maximum value of y is..." means y ≥ the expression, confusing the direction of the inequality.

Their reasoning: "If the maximum value of y is \(\mathrm{\frac{1}{2}z + 5}\), then y must be greater than or equal to this expression."

This leads them to select Choice (D) (\(\mathrm{y \geq \frac{1}{2}z + 5}\)).

Second Most Common Error:

Weak TRANSLATE skill: Students correctly understand that maximum means y ≤ something, but they incorrectly translate "5 more than half of z" as \(\mathrm{\frac{1}{2}z - 5}\) instead of \(\mathrm{\frac{1}{2}z + 5}\), mixing up "more than" with subtraction.

This may lead them to select Choice (A) (\(\mathrm{y \leq \frac{1}{2}z - 5}\)).

The Bottom Line:

This problem tests whether students can correctly translate both the verbal mathematical expression AND understand what "maximum value" means for inequality direction. The key insight is that maximum creates an upper limit, not a lower limit.