\((2\mathrm{x} + 3) - (\mathrm{x} - 7)\) Which of the following is equivalent to the given expression?...

1. INFER the approach needed

• We have an expression with subtraction of a binomial: \((2\mathrm{x} + 3) - (\mathrm{x} - 7)\)
• Strategy: First distribute the negative sign, then combine like terms

2. SIMPLIFY by distributing the negative sign

• \((2\mathrm{x} + 3) - (\mathrm{x} - 7)\) becomes \((2\mathrm{x} + 3) - \mathrm{x} + 7\)
• Key insight: The negative sign in front of \((\mathrm{x} - 7)\) affects both terms inside the parentheses
• So \(-(\mathrm{x} - 7) = -\mathrm{x} - (-7) = -\mathrm{x} + 7\)

3. SIMPLIFY by rearranging and combining like terms

• Rearrange: \((2\mathrm{x} + 3) - \mathrm{x} + 7 = 2\mathrm{x} - \mathrm{x} + 3 + 7\)
• Combine x terms: \(2\mathrm{x} - \mathrm{x} = \mathrm{x}\)
• Combine constant terms: \(3 + 7 = 10\)
• Final result: \(\mathrm{x} + 10\)

Answer: C. x + 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly distribute the negative sign, treating \(-(\mathrm{x} - 7)\) as just \(-\mathrm{x} - 7\) instead of \(-\mathrm{x} + 7\).

When they write \((2\mathrm{x} + 3) - (\mathrm{x} - 7)\) as \((2\mathrm{x} + 3) - \mathrm{x} - 7\), they get:

\(2\mathrm{x} - \mathrm{x} + 3 - 7 = \mathrm{x} - 4\)

This leads them to select Choice A (x - 4).

Second Most Common Error:

Poor INFER reasoning: Students misinterpret the subtraction and add the second expression instead of subtracting it.

They treat the problem as \((2\mathrm{x} + 3) + (\mathrm{x} - 7)\), giving them:

\(2\mathrm{x} + \mathrm{x} + 3 - 7 = 3\mathrm{x} - 4\)

This causes them to select Choice B (3x - 4).

The Bottom Line:

The key challenge is correctly handling the negative sign when subtracting a binomial. Students must remember that subtracting \((\mathrm{x} - 7)\) means adding its opposite, which is \(-\mathrm{x} + 7\), not \(-\mathrm{x} - 7\).