Question:\(10(2\mathrm{x} - 3) = 35(\mathrm{x} + 2)(2\mathrm{x} - 3)\)What is the sum of the solutions to the given equation?

1. INFER the strategic approach

Looking at the equation \(10(2\mathrm{x} - 3) = 35(\mathrm{x} + 2)(2\mathrm{x} - 3)\), notice that both sides contain the factor \((2\mathrm{x} - 3)\). This suggests we should rearrange to factor form rather than expanding everything out, which would create unnecessary complexity.

2. SIMPLIFY by rearranging to factor form

• Move everything to one side:
  \(10(2\mathrm{x} - 3) - 35(\mathrm{x} + 2)(2\mathrm{x} - 3) = 0\)

• Factor out the common term \((2\mathrm{x} - 3)\):
  \((2\mathrm{x} - 3)[10 - 35(\mathrm{x} + 2)] = 0\)

3. SIMPLIFY the second factor

• Expand:
  \(10 - 35(\mathrm{x} + 2) = 10 - 35\mathrm{x} - 70 = -35\mathrm{x} - 60\)
• Factor out -5:
  \(-35\mathrm{x} - 60 = -5(7\mathrm{x} + 12)\)
• The equation becomes:
  \((2\mathrm{x} - 3)(-5)(7\mathrm{x} + 12) = 0\)

4. INFER that zero product property applies

Since we have a product equal to zero, either factor can be zero:

• \(2\mathrm{x} - 3 = 0\) OR \(7\mathrm{x} + 12 = 0\)

5. SIMPLIFY by solving each linear equation

• From \(2\mathrm{x} - 3 = 0\):
  \(\mathrm{x} = \frac{3}{2}\)
• From \(7\mathrm{x} + 12 = 0\):
  \(\mathrm{x} = -\frac{12}{7}\)

6. SIMPLIFY to find the sum

• Convert to common denominator:
  \(\frac{3}{2} = \frac{21}{14}\)
  \(-\frac{12}{7} = -\frac{24}{14}\)
• Sum:
  \(\frac{21}{14} + (-\frac{24}{14}) = -\frac{3}{14}\)

Answer: \(-\frac{3}{14}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students expand both sides instead of recognizing the factoring opportunity, leading to a complex quadratic equation: \(20\mathrm{x} - 30 = 70\mathrm{x}^2 + 70\mathrm{x} - 105\mathrm{x} - 210\). This creates a much more difficult problem with multiple algebraic manipulation opportunities for error, often causing students to get stuck and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the factoring approach but make algebraic errors when simplifying \(10 - 35(\mathrm{x} + 2)\), such as getting \(10 - 35\mathrm{x} + 70\) instead of \(10 - 35\mathrm{x} - 70\), or make fraction arithmetic errors when computing the final sum.

The Bottom Line:

This problem rewards strategic thinking - recognizing patterns and choosing efficient solution paths over brute force expansion. The key insight is seeing the common factor \((2\mathrm{x} - 3)\) immediately and leveraging it.