4x + 5 = 165 What is the solution to the given equation?...

1. INFER the solving strategy

• Given: \(4\mathrm{x} + 5 = 165\)
• Goal: Find the value of x that makes this equation true
• Strategy: Use inverse operations to isolate x
• Key insight: Undo operations in reverse order - first undo the addition, then undo the multiplication

2. SIMPLIFY by removing the constant term

• Subtract 5 from both sides:

\(4\mathrm{x} + 5 - 5 = 165 - 5\)
\(4\mathrm{x} = 160\)

3. SIMPLIFY by removing the coefficient

• Divide both sides by 4:

\(4\mathrm{x} \div 4 = 160 \div 4\)
\(\mathrm{x} = 40\)

Answer: 40

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to divide by 4 first, before dealing with the +5

• Wrong reasoning: "I see 4x, so I should divide by 4 first"
• This leads to: \((4\mathrm{x} + 5) \div 4 = 165 \div 4\), giving \(\mathrm{x} + 1.25 = 41.25\), then \(\mathrm{x} = 40\)
• While this technically gives the right answer, it demonstrates poor understanding of systematic solving and can lead to errors in more complex problems

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic mistakes in basic operations

• Common errors: \(165 - 5 = 150\) (instead of 160) or \(160 \div 4 = 30\) (instead of 40)
• This leads to confusion and potentially guessing among similar-looking answer choices

The Bottom Line:

Success requires recognizing that equation solving follows a systematic process of "undoing" operations in reverse order, combined with careful arithmetic execution.