If 4x - 1/2 = -5, what is the value of 8x - 1?

1. TRANSLATE the problem information

• Given: \(\mathrm{4x - \frac{1}{2} = -5}\)
• Find: The value of \(\mathrm{8x - 1}\)

2. INFER the most efficient approach

• Notice that the target expression is \(\mathrm{8x - 1}\)
• The given equation has \(\mathrm{4x - \frac{1}{2}}\) on the left side
• Key insight: If I multiply the entire equation by 2, the left side becomes \(\mathrm{2(4x) - 2(\frac{1}{2}) = 8x - 1}\)
• This directly gives us what we're looking for!

3. SIMPLIFY by multiplying both sides by 2

• Start with: \(\mathrm{4x - \frac{1}{2} = -5}\)
• Multiply both sides by 2: \(\mathrm{2(4x - \frac{1}{2}) = 2(-5)}\)
• Apply distributive property: \(\mathrm{2(4x) - 2(\frac{1}{2}) = -10}\)
• Simplify: \(\mathrm{8x - 1 = -10}\)

Answer: D. -10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the strategic shortcut and instead solve for x first, then substitute.

While this method works (\(\mathrm{x = -\frac{9}{8}}\), so \(\mathrm{8x - 1 = -9 - 1 = -10}\)), it's much longer and creates more opportunities for arithmetic errors. Students might get lost in the fraction calculations or make sign errors when working with \(\mathrm{-\frac{9}{8}}\).

This approach can lead to calculation mistakes that result in selecting incorrect answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when handling fractions.

For example, when solving \(\mathrm{4x = -5 + \frac{1}{2}}\), they might incorrectly calculate \(\mathrm{-5 + \frac{1}{2} = -4.5}\) instead of \(\mathrm{-4.5 = -\frac{9}{2}}\), or make errors when converting between fractions and decimals. These errors propagate through to the final calculation.

This may lead them to select Choice B (\(\mathrm{-\frac{9}{8}}\)) if they confuse their x-value with the final answer, or other incorrect choices based on their arithmetic mistakes.

The Bottom Line:

The key insight is recognizing that multiplying by 2 transforms the given equation directly into the form we need. Students who miss this strategic approach make the problem much harder than necessary.