Question:If (6x + 30)/4 - x = (3x + 60)/4, what is the value of 6x?Format: [FIB]

1. INFER the solution strategy

• Given: \(\frac{6\mathrm{x} + 30}{4} - \mathrm{x} = \frac{3\mathrm{x} + 60}{4}\)
• Strategy: Clear fractions first by multiplying both sides by the common denominator (4), then solve systematically

2. SIMPLIFY by eliminating fractions

• Multiply both sides by 4:
  \(4 \cdot [\frac{6\mathrm{x} + 30}{4} - \mathrm{x}] = 4 \cdot [\frac{3\mathrm{x} + 60}{4}]\)
• Be careful with the distributive property on the left side:
  \((6\mathrm{x} + 30) - 4\mathrm{x} = 3\mathrm{x} + 60\)

3. SIMPLIFY by combining like terms

• Left side: \(6\mathrm{x} - 4\mathrm{x} + 30 = 2\mathrm{x} + 30\)
• Equation becomes: \(2\mathrm{x} + 30 = 3\mathrm{x} + 60\)

4. SIMPLIFY to isolate the variable

• Subtract \(2\mathrm{x}\) from both sides: \(30 = \mathrm{x} + 60\)
• Subtract 60 from both sides: \(\mathrm{x} = -30\)

5. INFER that we need 6x, not just x

• The question asks for the value of \(6\mathrm{x}\)
• Calculate: \(6(-30) = -180\)

Answer: -180

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: When multiplying both sides by 4, students often forget to multiply the \(-\mathrm{x}\) term by 4, writing:
\(6\mathrm{x} + 30 - \mathrm{x} = 3\mathrm{x} + 60\) instead of \(6\mathrm{x} + 30 - 4\mathrm{x} = 3\mathrm{x} + 60\)

This leads to: \(5\mathrm{x} + 30 = 3\mathrm{x} + 60\), so \(2\mathrm{x} = 30\), giving \(\mathrm{x} = 15\), and therefore \(6\mathrm{x} = 90\)
This causes them to select an incorrect positive answer instead of the correct negative value.

Second Most Common Error:

Incomplete solution: Students correctly solve for \(\mathrm{x} = -30\) but forget that the question asks for \(6\mathrm{x}\), not \(\mathrm{x}\).
They submit -30 as their final answer, missing the final multiplication step.

The Bottom Line:

This problem tests careful algebraic manipulation with fractions and attention to what the question actually asks for. Success requires both systematic equation-solving skills and careful reading of the final question.