If 4x - 28 = -24, what is the value of x - 7?

1. TRANSLATE the problem information

• Given equation: \(4\mathrm{x} - 28 = -24\)
• Need to find: \(\mathrm{x} - 7\)

2. INFER the most efficient approach

• Notice that if we divide the entire equation by 4, we get:
  • \(\frac{4\mathrm{x}}{4} = \mathrm{x}\)
  • \(\frac{-28}{4} = -7\)
  • \(\frac{-24}{4} = -6\)
• This means dividing by 4 will give us \(\mathrm{x} - 7\) on the left side directly!

3. SIMPLIFY by dividing every term by 4

• Start with: \(4\mathrm{x} - 28 = -24\)
• Divide each term by 4: \(\frac{4\mathrm{x}}{4} - \frac{28}{4} = \frac{-24}{4}\)
• This gives us: \(\mathrm{x} - 7 = -6\)

Answer: C. -6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students immediately try to isolate x by adding 28 to both sides, then dividing by 4. While this works, they get \(\mathrm{x} = 1\), then must remember to calculate \(\mathrm{x} - 7 = 1 - 7 = -6\). The extra step creates opportunity for sign errors or forgetting what the question actually asks for.

This may lead them to select Choice D (-1) if they find \(\mathrm{x} = 1\) and mistakenly think that's the final answer.

Second Most Common Error:

Poor SIMPLIFY execution: When dividing by 4, students make sign errors, particularly with \(\frac{-28}{4} = -7\) or \(\frac{-24}{4} = -6\). Negative number division is a common stumbling point.

This may lead them to select Choice A (-24) if they confuse the original right-side value with their final answer, or Choice B (-22) from calculation errors.

The Bottom Line:

This problem rewards strategic thinking - recognizing that you don't always need to solve for the variable explicitly if the question asks for an expression involving that variable. The most elegant approach sidesteps potential errors by going directly to what's asked.