6x^2/(x^2 - 2x - 8) - 2x/(x + 2) = 12/(x - 4)What value of x satisfies the equation above?

1. INFER the structure of denominators

• Looking at the denominators, I notice:
  • \(\mathrm{x^2 - 2x - 8}\) (needs factoring)
  • \(\mathrm{x + 2}\) (already factored)
  • \(\mathrm{x - 4}\) (already factored)

• SIMPLIFY by factoring the quadratic denominator:
  \(\mathrm{x^2 - 2x - 8 = (x - 4)(x + 2)}\)

2. INFER domain restrictions

• The denominators cannot equal zero:
  • \(\mathrm{(x - 4)(x + 2) \neq 0}\), so \(\mathrm{x \neq 4}\) and \(\mathrm{x \neq -2}\)
  • This means \(\mathrm{x = 4}\) and \(\mathrm{x = -2}\) are not valid solutions

3. INFER the LCD and solution strategy

• All denominators: \(\mathrm{(x - 4)(x + 2)}\), \(\mathrm{(x + 2)}\), \(\mathrm{(x - 4)}\)
• LCD = \(\mathrm{(x - 4)(x + 2)}\)
• Strategy: Multiply every term by the LCD to eliminate fractions

4. SIMPLIFY by multiplying through by LCD

• Multiplying each term by \(\mathrm{(x - 4)(x + 2)}\):
  \(\mathrm{6x^2 - 2x(x - 4) = 12(x + 2)}\)

• SIMPLIFY by expanding:
  \(\mathrm{6x^2 - 2x^2 + 8x = 12x + 24}\)
  \(\mathrm{4x^2 + 8x = 12x + 24}\)

• SIMPLIFY to standard form:
  \(\mathrm{4x^2 - 4x - 24 = 0}\)
  \(\mathrm{x^2 - x - 6 = 0}\)

5. SIMPLIFY by factoring the quadratic

• Looking for factors of -6 that add to -1: -3 and 2
• \(\mathrm{(x - 3)(x + 2) = 0}\)
• So \(\mathrm{x = 3}\) or \(\mathrm{x = -2}\)

6. APPLY CONSTRAINTS to eliminate extraneous solutions

• \(\mathrm{x = 3}\): Valid (not restricted)
• \(\mathrm{x = -2}\): Invalid (violates domain restriction \(\mathrm{x \neq -2}\))

Answer: D (3)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students often fail to recognize that \(\mathrm{x^2 - 2x - 8}\) needs to be factored first, or they don't identify the proper domain restrictions before solving.

Without factoring the quadratic denominator, students can't find the correct LCD or recognize that \(\mathrm{x = -2}\) is restricted. They might work with the wrong LCD or miss the domain restriction entirely, leading to accepting \(\mathrm{x = -2}\) as a valid solution. This may lead them to select Choice B (-2).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make algebraic errors when multiplying through by the LCD or when expanding and collecting terms.

Common mistakes include sign errors when expanding \(\mathrm{-2x(x - 4)}\) or arithmetic mistakes when collecting like terms. These errors lead to incorrect quadratic equations and wrong solutions. This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students can systematically work with rational equations by first analyzing structure (factoring, domain restrictions), then methodically eliminating fractions and solving the resulting quadratic. The key insight is that rational equations require checking solutions against domain restrictions - not all algebraic solutions are valid in the original equation.