Let y be a real number. Consider the equation |2y - 6| = 2|y - 3|. How many real solutions...

1. TRANSLATE the problem information

• Given equation: \(|2y - 6| = 2|y - 3|\)
• Need to find: Number of real solutions

2. INFER the best approach

• Rather than immediately doing case analysis, look for algebraic patterns
• Notice that the left side might factor nicely

3. SIMPLIFY the left side through factoring

• Factor out 2 from the expression inside: \(|2y - 6| = |2(y - 3)|\)
• Apply absolute value property \(|ab| = |a|\cdot|b|\): \(|2(y - 3)| = |2|\cdot|y - 3| = 2|y - 3|\)

4. INFER what the simplified equation tells us

• The equation becomes: \(2|y - 3| = 2|y - 3|\)
• This is an identity - both sides are exactly the same!
• An identity is true for every value in its domain

5. APPLY CONSTRAINTS to determine the solution set

• Since absolute value is defined for all real numbers, the domain is all real numbers
• Therefore, the equation has infinitely many solutions

Answer: D (Infinitely many)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students immediately jump to case analysis without first trying to simplify algebraically.

They set up cases like "when \(y \geq 3\)" and "when \(y \lt 3\)" and work through the algebra in each case, but they do this inefficiently and may make computational errors in the process. Even if they get the right answer, they miss the elegant insight that reveals why the answer is what it is.

This approach works but is unnecessarily complicated and prone to arithmetic mistakes.

Second Most Common Error:

Poor INFER reasoning: Students correctly simplify to get \(2|y - 3| = 2|y - 3|\) but don't recognize what this means.

They might think "this looks wrong" or "I made an error somewhere" and go back to redo their work, or they might think the equation has no solutions because "it doesn't give me a specific value for y." They don't understand that identical expressions on both sides means the equation is always true.

This leads to confusion and guessing among answer choices A, B, or C.

The Bottom Line:

This problem tests whether students can recognize when algebraic simplification reveals a fundamental structure (an identity) rather than just mechanically applying case-by-case analysis. The key insight is that sometimes the most elegant approach comes from algebraic manipulation before considering cases.