Question:3/4x - 1/2y lt -3/2Which of the following inequalities is equivalent to the inequality above?x - 2y lt -63x -...

1. TRANSLATE the problem information

• Given inequality: \(\frac{3}{4}\mathrm{x} - \frac{1}{2}\mathrm{y} \lt -\frac{3}{2}\)
• Need to find: Which answer choice represents an equivalent inequality

2. INFER the solution strategy

• Key insight: The fractions make this inequality hard to work with
• Strategy: Eliminate fractions by multiplying all terms by a common denominator
• This creates an equivalent inequality that's easier to recognize

3. Find the common denominator

• Denominators present: 4, 2, and 2 (from \(-\frac{3}{2}\))
• LCM of 4 and 2 = 4

4. SIMPLIFY by multiplying each term by 4

• Left side: \(4 \times (\frac{3}{4}\mathrm{x}) = 3\mathrm{x}\)
• Middle: \(4 \times (\frac{1}{2}\mathrm{y}) = 2\mathrm{y}\)
• Right side: \(4 \times (-\frac{3}{2}) = -6\)
• Result: \(3\mathrm{x} - 2\mathrm{y} \lt -6\)

5. INFER the inequality direction

• Since we multiplied by positive 4, the inequality direction stays the same (\(\lt\))
• Important: Only multiplication/division by negative numbers flips the inequality sign

Answer: B (\(3\mathrm{x} - 2\mathrm{y} \lt -6\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students flip the inequality sign when multiplying by 4, confusing the rule about when to flip inequality directions.

They think: "I'm doing something to both sides, so maybe I need to flip the sign" and change \(\lt\) to \(\gt\), getting \(3\mathrm{x} - 2\mathrm{y} \gt -6\).

This leads them to select Choice C (\(3\mathrm{x} - 2\mathrm{y} \gt -6\))

Second Most Common Error:

Poor SIMPLIFY execution: Students don't multiply ALL terms by the same value, or make arithmetic errors in the multiplication process.

For example, they might multiply the x-term by 4 but forget to multiply the y-term by the same amount, or incorrectly calculate \(4 \times (-\frac{3}{2})\).

This causes calculation errors that don't match any answer choice, leading to confusion and guessing.

The Bottom Line:

This problem tests whether students can systematically clear fractions while maintaining equivalent relationships. The key insight is recognizing that multiplying by a common denominator simplifies the inequality without changing its meaning.