\((2\mathrm{x} - 9) \times 3^{(\mathrm{x} - 4)} = 27(2\mathrm{x} - 9)\) What is the sum of all solutions to the...

1. TRANSLATE the equation structure

• Given: \((2x - 9) \times 3^{(x - 4)} = 27(2x - 9)\)
• What this tells us: We have a product equation where both sides contain the term \((2x - 9)\)

2. INFER the solution strategy

• Key insight: Since \((2x - 9)\) appears on both sides, we can rearrange to factor it out
• Strategy: Move everything to one side, then factor out the common term to create a zero-product situation

3. SIMPLIFY by rearranging and factoring

• Move all terms to the left side:
  \((2x - 9) \times 3^{(x - 4)} - 27(2x - 9) = 0\)
• Factor out \((2x - 9)\):
  \((2x - 9) \times [3^{(x - 4)} - 27] = 0\)

4. INFER using zero-product property

• Since we have a product equal to zero, at least one factor must be zero
• This gives us two separate cases to solve

5. SIMPLIFY Case 1: First factor equals zero

• \(2x - 9 = 0\)
• \(2x = 9\)
• \(x = \frac{9}{2}\)

6. SIMPLIFY Case 2: Second factor equals zero

• \(3^{(x - 4)} - 27 = 0\)
• \(3^{(x - 4)} = 27\)

7. INFER exponential equation solution method

• Since \(27 = 3^3\), rewrite with the same base:
  \(3^{(x - 4)} = 3^3\)
• When bases are equal, exponents must be equal:
  \(x - 4 = 3\)
• \(x = 7\)

8. SIMPLIFY finding the sum

• Sum = \(\frac{9}{2} + 7\)
  \(= \frac{9}{2} + \frac{14}{2}\)
  \(= \frac{23}{2}\)

Answer: 23/2 (or 11.5)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the factoring opportunity and instead try to solve the original equation directly by dividing both sides by \((2x - 9)\).

When they divide both sides by \((2x - 9)\), they get \(3^{(x - 4)} = 27\), which gives them \(x = 7\). However, they lose the solution \(x = \frac{9}{2}\) because dividing by \((2x - 9)\) eliminates the case where \(2x - 9 = 0\). This leads them to find only one solution instead of two, giving an incomplete sum.

Second Most Common Error:

Missing conceptual knowledge: Students don't recognize that \(27 = 3^3\), so they can't solve the exponential equation \(3^{(x - 4)} = 27\).

Without this connection, they get stuck on the second case and either guess or only use the solution from the first case (\(x = \frac{9}{2}\)), leading to an incorrect sum of just \(\frac{9}{2}\).

The Bottom Line:

This problem requires recognizing that factoring preserves all solutions, while direct algebraic manipulation might eliminate solutions. The key insight is seeing the common factor structure rather than immediately trying to isolate variables.