The length of a rectangle is 50 inches and the width is x inches. The perimeter is at most 210...

1. TRANSLATE the problem information

• Given information:
  • Length = 50 inches
  • Width = x inches
  • Perimeter is at most 210 inches

• What this tells us: We need to write an inequality for the perimeter constraint.

2. INFER what formula to use

• For any rectangle, perimeter = \(2 \times \mathrm{length} + 2 \times \mathrm{width}\)
• We'll substitute our known values into this formula

3. TRANSLATE the constraint into math notation

• Substitute into perimeter formula: \(\mathrm{P} = 2(50) + 2(x) = 100 + 2x = 2x + 100\)
• "At most 210 inches" means the perimeter can be 210 or less
• This translates to: \(2x + 100 \leq 210\)

Answer: A. \(2x + 100 \leq 210\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "at most" as meaning "at least" and use \(\geq\) instead of \(\leq\).

The reasoning goes: "At most sounds like it should be greater than or equal to because 'most' sounds like 'more.'" This leads to the inequality \(2x + 100 \geq 210\).

This may lead them to select Choice B (\(2x + 100 \geq 210\)).

Second Most Common Error:

Poor TRANSLATE reasoning: Students incorrectly calculate the perimeter expression, getting \(2x + 50\) instead of \(2x + 100\).

This happens when they forget to multiply the length by 2, thinking: "Perimeter = length + 2 × width = \(50 + 2x\)." They miss that perimeter involves adding ALL four sides.

This may lead them to select Choice C (\(2x + 50 \leq 210\)) or Choice D (\(2x + 50 \geq 210\)).

The Bottom Line:

This problem tests whether students can accurately translate verbal constraints into mathematical inequalities and correctly apply the rectangle perimeter formula. The key challenge is precision in translation - both the arithmetic setup and the inequality direction must be correct.