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

1. TRANSLATE the maximum value concept

• Given information: 'The maximum value of variable y is...'
• What this tells us: If y has a maximum value, then y cannot be greater than this value
• This means we need the inequality symbol \(\leq\) (less than or equal to)

2. TRANSLATE the expression for the maximum value

• The phrase: '5 more than half of another number z'
• Break it down:
  • 'Half of another number z' = \(\frac{1}{2}\mathrm{z}\)
  • '5 more than' this quantity = \(\frac{1}{2}\mathrm{z} + 5\)
• So the maximum value is \(\frac{1}{2}\mathrm{z} + 5\)

3. INFER the complete inequality relationship

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

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret '5 more than' as subtraction instead of addition, thinking it means '5 less than half of z' and writing \(\frac{1}{2}\mathrm{z} - 5\).

This leads them to select Choice A (\(\mathrm{y} \leq \frac{1}{2}\mathrm{z} - 5\)) because they correctly understand that maximum value requires \(\leq\), but they get the expression wrong.

Second Most Common Error:

Weak INFER skill: Students correctly translate the expression to \(\frac{1}{2}\mathrm{z} + 5\), but confuse what 'maximum value' means for the inequality direction. They think 'maximum' means y should be greater than or equal to this value.

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

The Bottom Line:

This problem tests both translation skills (converting words to math) and logical reasoning about what 'maximum value' means for constraining a variable. Students need to be careful with both the arithmetic translation and the inequality direction.