6x - 9y gt 12 Which of the following inequalities is equivalent to the inequality above?...

1. TRANSLATE the problem information

• Given inequality: \(6\mathrm{x} - 9\mathrm{y} \gt 12\)
• Need to find: Which answer choice represents an equivalent inequality
• What equivalent means: Same solution set (any point that satisfies one must satisfy the other)

2. INFER the solution approach

• Notice that 6, 9, and 12 all have a common factor
• Strategy: Simplify by dividing both sides by the greatest common factor
• This will create a simpler equivalent inequality

3. SIMPLIFY by factoring out the common factor

• Greatest common factor of 6, 9, and 12 is 3
• Divide both sides by 3:
  • \(6\mathrm{x} ÷ 3 = 2\mathrm{x}\)
  • \(9\mathrm{y} ÷ 3 = 3\mathrm{y}\)
  • \(12 ÷ 3 = 4\)
• Result: \(2\mathrm{x} - 3\mathrm{y} \gt 4\)

4. INFER the answer and verify

• This matches Choice B exactly
• To be thorough, test with a point like \((0, -1.5)\):
  • Original: \(6(0) - 9(-1.5) = 13.5 \gt 12\) ✓
  • Choice B: \(2(0) - 3(-1.5) = 4.5 \gt 4\) ✓
  • Both work, confirming equivalence

Answer: B. \(2\mathrm{x} - 3\mathrm{y} \gt 4\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic mistakes when dividing coefficients by 3, particularly with the middle term involving y.

For example, they might incorrectly calculate \(9\mathrm{y} ÷ 3\) as \(6\mathrm{y}\) instead of \(3\mathrm{y}\), leading to \(2\mathrm{x} - 6\mathrm{y} \gt 4\). Since this doesn't match any answer choice exactly, this leads to confusion and guessing.

Second Most Common Error:

Missing conceptual knowledge about equivalent inequalities: Students might think they need to solve for x or y, rather than recognizing they just need to simplify the existing inequality.

They might try to rearrange terms or solve for one variable, getting lost in unnecessary steps. This causes them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students can recognize when to simplify expressions and execute that simplification accurately. The key insight is seeing the common factor and knowing that dividing by a positive number preserves the inequality relationship.