Which of the following is equivalent to 4x + 6 = 12?

1. INFER what equivalent equations means

• Given: \(\mathrm{4x + 6 = 12}\)
• Find: Which equation is equivalent (has the same solutions)
• Key insight: Equivalent equations are created by performing the same operation on both sides

2. INFER the most promising approach

• Looking at the answer choices, most have smaller coefficients than the original
• This suggests dividing both sides by a common factor
• The coefficient 4 and constant 6 both divide evenly by 2

3. SIMPLIFY by dividing both sides by 2

• Original equation: \(\mathrm{4x + 6 = 12}\)
• Divide every term by 2: \(\mathrm{(4x + 6)/2 = 12/2}\)
• Distribute the division: \(\mathrm{4x/2 + 6/2 = 6}\)
• Simplify: \(\mathrm{2x + 3 = 6}\)

4. INFER which choice matches our result

• Our simplified equation: \(\mathrm{2x + 3 = 6}\)
• This exactly matches Choice D!

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors when dividing

Students might incorrectly distribute the division, getting something like \(\mathrm{2x + 4 = 6}\) instead of \(\mathrm{2x + 3 = 6}\). They forget that \(\mathrm{6/2 = 3}\), not 4.

This may lead them to select Choice A (\(\mathrm{2x + 4 = 6}\))

Second Most Common Error:

Poor INFER reasoning about equivalent equations: Students think equivalent means "looks similar" rather than "same solutions"

Students might focus on visual patterns instead of proper algebraic operations. They see smaller numbers in the choices and randomly pick one that "looks simpler" without performing any systematic operation.

This leads to confusion and guessing

The Bottom Line:

This problem tests whether students understand that equivalent equations must be created through valid algebraic operations, not just visual similarity. The key is systematically applying the same operation to both sides.