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

1. INFER the strategy needed

• Given: \(\frac{3}{4}x - \frac{1}{2}y \lt -\frac{3}{2}\)
• Goal: Find equivalent inequality without fractions
• Strategy: Multiply everything by the least common multiple of all denominators

2. INFER the appropriate multiplier

• Denominators present: 4, 2, and 2
• LCM of 4 and 2 is 4
• We'll multiply every term by 4

3. SIMPLIFY by multiplying each term

• \(4(\frac{3}{4}x) = 4 \times \frac{3x}{4} = 3x\)
• \(4(\frac{1}{2}y) = 4 \times \frac{y}{2} = 2y\)
• \(4(-\frac{3}{2}) = 4 \times \frac{-3}{2} = -6\)

4. SIMPLIFY to write the equivalent inequality

• Original: \(\frac{3}{4}x - \frac{1}{2}y \lt -\frac{3}{2}\)
• After multiplying by 4: \(3x - 2y \lt -6\)
• Important: Since we multiplied by positive 4, the inequality direction stays the same

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

Why Students Usually Falter on This Problem

Most Common Error Path:

Conceptual confusion about inequality properties: Students know they sometimes need to flip inequality signs, but incorrectly apply this rule when multiplying by positive numbers. They think "whenever I multiply an inequality, I flip the sign" and change < to >.

This leads them to get \(3x - 2y \gt -6\) and select Choice C (\(3x - 2y \gt -6\)).

Second Most Common Error:

Weak SIMPLIFY execution: Students make arithmetic errors when multiplying fractions, particularly with the coefficient manipulation. They might incorrectly calculate \(4(\frac{3}{4}x)\) as \(x\) instead of \(3x\), or mix up which coefficients go with which variables.

This leads to incorrect coefficients and may cause them to select Choice A (\(x - 2y \lt -6\)) or Choice D (\(2x - 3y \lt -6\)).

The Bottom Line:

This problem tests whether students can systematically eliminate fractions while correctly applying inequality properties. The key insight is remembering that inequality direction only changes when multiplying or dividing by negative numbers, not positive ones.