55/(x + 6) = x What is the positive solution to the given equation?...

1. INFER the solution strategy

• Given: \(\frac{55}{x + 6} = x\)
• Key insight: Clear the denominator first by multiplying both sides by \((x + 6)\)
• This will transform the rational equation into a more manageable quadratic equation

2. SIMPLIFY by clearing the denominator

• Multiply both sides by \((x + 6)\):

\(\frac{55}{x + 6} \times (x + 6) = x \times (x + 6)\)

• Left side simplifies: \(55\)
• Right side: \(x(x + 6)\)
• Result: \(55 = x(x + 6)\)

3. SIMPLIFY using the distributive property

• Expand the right side: \(55 = x^2 + 6x\)
• Rearrange to standard quadratic form: \(x^2 + 6x - 55 = 0\)

4. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to \(-55\) and add to \(6\)
• Try factor pairs of \(55\): \(1\times55\), \(5\times11\)
• Since we need \(-55\), one must be negative: \(11\) and \(-5\) work perfectly
• Check: \((11)(-5) = -55\) ✓ and \(11 + (-5) = 6\) ✓
• Factor: \((x + 11)(x - 5) = 0\)

5. SIMPLIFY using zero product property

• If \((x + 11)(x - 5) = 0\), then either factor equals zero
• \(x + 11 = 0 \rightarrow x = -11\)
• \(x - 5 = 0 \rightarrow x = 5\)

6. APPLY CONSTRAINTS to select final answer

• The problem asks specifically for the positive solution
• Between \(x = -11\) and \(x = 5\), only \(x = 5\) is positive

Answer: 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when rearranging to standard form, often writing \(x^2 + 6x + 55 = 0\) instead of \(x^2 + 6x - 55 = 0\).

When they try to factor \(x^2 + 6x + 55 = 0\), they look for two numbers that multiply to \(+55\) and add to \(6\), but no such integer pair exists. This leads to confusion and either guessing or incorrectly concluding the equation has no solution.

Second Most Common Error:

Poor INFER reasoning: Students attempt to solve the rational equation through complex algebraic manipulation without first clearing the denominator, leading to unnecessarily complicated fractions and arithmetic errors.

They might try to isolate \(x\) directly from \(\frac{55}{x + 6} = x\), creating messy expressions that are prone to calculation mistakes. This causes them to get stuck and abandon systematic solution.

The Bottom Line:

The key insight is recognizing that multiplying both sides by the denominator transforms a challenging rational equation into a straightforward quadratic that can be factored using familiar techniques.