How many solutions does the equation (12x - 8)/(3x - 2) = 4 have?Exactly oneExactly twoInfinitely manyZero

1. INFER the domain restrictions

• The denominator \(3\mathrm{x} - 2\) cannot equal zero
• Setting \(3\mathrm{x} - 2 = 0\) gives us \(\mathrm{x} = \frac{2}{3}\)
• Therefore, the domain is all real numbers except \(\mathrm{x} = \frac{2}{3}\)

2. SIMPLIFY by eliminating the fraction

• Multiply both sides by \((3\mathrm{x} - 2)\):
  \(\frac{12\mathrm{x} - 8}{3\mathrm{x} - 2} \cdot (3\mathrm{x} - 2) = 4 \cdot (3\mathrm{x} - 2)\)
• This gives us: \(12\mathrm{x} - 8 = 4(3\mathrm{x} - 2)\)

3. SIMPLIFY the right side

• Distribute: \(12\mathrm{x} - 8 = 12\mathrm{x} - 8\)

4. INFER what this equation tells us

• Subtracting \(12\mathrm{x}\) from both sides: \(-8 = -8\)
• This statement is always true!
• When an equation reduces to an always-true statement, the original equation is satisfied by every value in its domain

5. INFER the final answer

• The equation works for all real numbers except \(\mathrm{x} = \frac{2}{3}\) (our domain restriction)
• Since there are infinitely many real numbers in this set, we have infinitely many solutions

Answer: C (Infinitely many)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding what it means when an equation reduces to an always-true statement like \(-8 = -8\).

Students often think this means "no solution" because they expect to find a specific value of x. They don't realize that when both sides become identical after simplification, it means the original equation is true for all values in the domain. This confusion may lead them to select Choice D (Zero).

Second Most Common Error:

Conceptual confusion about domain restrictions: Thinking that because \(\mathrm{x} = \frac{2}{3}\) is excluded from the domain, the equation has zero solutions.

Students may correctly find that \(\mathrm{x} \neq \frac{2}{3}\), but then incorrectly conclude this means no solutions exist. They miss the key insight that infinitely many other values still work. This may lead them to select Choice D (Zero).

The Bottom Line:

This problem tests whether students understand the difference between "no solutions" and "infinitely many solutions" when working with rational equations. The key insight is recognizing that an equation reducing to a true statement indicates infinitely many solutions within the domain.