Which of the following is equivalent to \(3(\mathrm{x} + 5) - 6\)?

1. TRANSLATE the problem information

• We need to find an expression equivalent to \(\mathrm{3(x + 5) - 6}\)
• This means we need to simplify the given expression using algebraic rules

2. SIMPLIFY using the distributive property

• Apply the distributive property to \(\mathrm{3(x + 5)}\):
  • \(\mathrm{3(x + 5) = 3·x + 3·5 = 3x + 15}\)
• Now our expression becomes: \(\mathrm{3x + 15 - 6}\)

3. SIMPLIFY by combining like terms

• Combine the constant terms: \(\mathrm{15 - 6 = 9}\)
• Final simplified expression: \(\mathrm{3x + 9}\)

Answer: C. \(\mathrm{3x + 9}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly group terms before applying the distributive property, rewriting \(\mathrm{3(x + 5) - 6}\) as \(\mathrm{3(x + 5 - 6) = 3(x - 1) = 3x - 3}\).

This misapplies the order of operations - you can't move the "-6" inside the parentheses without dividing by 3 first. This leads them to select Choice A (\(\mathrm{3x - 3}\)).

Second Most Common Error:

Poor SIMPLIFY reasoning: Students forget to apply the distributive property entirely, instead treating the expression as \(\mathrm{(3x + 5) - 6}\), which gives \(\mathrm{3x + 5 - 6 = 3x - 1}\).

This shows confusion about what the parentheses mean and when to apply the distributive property. This may lead them to select Choice B (\(\mathrm{3x - 1}\)).

The Bottom Line:

This problem tests whether students can correctly apply the distributive property while maintaining proper order of operations. The key insight is recognizing that you must distribute the 3 to everything inside the parentheses BEFORE dealing with the "-6" outside.