\(\mathrm{3(x + 4) - 2(x - 1)}\) Which of the following is equivalent to the given expression? x + 10...

1. SIMPLIFY by distributing coefficients

• Start with: \(3(\mathrm{x} + 4) - 2(\mathrm{x} - 1)\)

• Distribute the first term: \(3(\mathrm{x} + 4) = 3\mathrm{x} + 12\)

• Distribute the second term carefully: \(-2(\mathrm{x} - 1) = -2\mathrm{x} + 2\)
  • Remember: \(-2 \times (-1) = +2\)

• Expression becomes: \(3\mathrm{x} + 12 - 2\mathrm{x} + 2\)

2. SIMPLIFY by combining like terms

• Group variable terms: \(3\mathrm{x} - 2\mathrm{x} = \mathrm{x}\)

• Group constant terms: \(12 + 2 = 14\)

• Final simplified form: \(\mathrm{x} + 14\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with sign handling: Students incorrectly distribute \(-2(\mathrm{x} - 1)\) as \(-2\mathrm{x} - 2\) instead of \(-2\mathrm{x} + 2\).

They forget that when distributing a negative coefficient, \(-2 \times (-1) = +2\), not \(-2\). This gives them \(3\mathrm{x} + 12 - 2\mathrm{x} - 2 = \mathrm{x} + 10\).

This leads them to select Choice A (\(\mathrm{x} + 10\)).

Second Most Common Error:

Poor SIMPLIFY technique in combining terms: Students correctly distribute but make arithmetic errors when combining the constant terms, getting \(12 + 2 = 16\) instead of 14, or they lose track of which terms are like terms.

This may lead them to select Choice C (\(\mathrm{x} + 16\)) or causes confusion and guessing.

The Bottom Line:

This problem tests careful execution of fundamental algebraic operations. The key challenge is maintaining accuracy with signs during distribution, especially when dealing with subtraction of a binomial expression.