The inequality \(-5(3\mathrm{y} - 2\mathrm{x}) \geq -20\) is given. Which of the following inequalities in terms of y is equivalent...
GMAT Advanced Math : (Adv_Math) Questions
The inequality \(-5(3\mathrm{y} - 2\mathrm{x}) \geq -20\) is given. Which of the following inequalities in terms of y is equivalent to the given inequality?
- \(\mathrm{y} \geq \frac{2}{3}\mathrm{x} + \frac{4}{3}\)
- \(\mathrm{y} \leq \frac{2}{3}\mathrm{x} + \frac{4}{3}\)
- \(\mathrm{y} \leq \frac{3}{2}\mathrm{x} + \frac{4}{3}\)
- \(\mathrm{y} \geq \frac{2}{3}\mathrm{x} - \frac{4}{3}\)
- \(\mathrm{y} \leq \frac{2}{3}\mathrm{x} - \frac{4}{3}\)
1. TRANSLATE the problem information
- Given: \(-5(3y - 2x) \geq -20\)
- Goal: Find equivalent inequality solved for y
2. SIMPLIFY by distributing the coefficient
- Apply distributive property to the left side:
\(-5(3y - 2x) = -5(3y) + -5(-2x) = -15y + 10x\)
- Our inequality becomes: \(-15y + 10x \geq -20\)
3. SIMPLIFY by isolating the y term
- Subtract 10x from both sides to get y terms alone:
\(-15y \geq -10x - 20\)
4. INFER the critical inequality rule and apply it
- We need to divide both sides by -15 to solve for y
- Key insight: When dividing an inequality by a negative number, we must flip the inequality sign
- Dividing by -15: \(y \leq \frac{-10x - 20}{-15}\)
5. SIMPLIFY the resulting fraction
- Separate the fraction: \(\frac{-10x - 20}{-15} = \frac{-10x}{-15} + \frac{-20}{-15}\)
- Simplify each part:
- \(\frac{-10x}{-15} = \frac{10x}{15} = \frac{2x}{3} = \frac{2}{3}x\)
- \(\frac{-20}{-15} = \frac{20}{15} = \frac{4}{3}\)
Answer: B. \(y \leq \frac{2}{3}x + \frac{4}{3}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Forgetting to flip the inequality sign when dividing by a negative number
Students correctly work through the distribution and isolation steps, getting to \(-15y \geq -10x - 20\). However, when they divide both sides by -15, they forget the critical rule about negative division and keep the same inequality direction: \(y \geq \frac{2}{3}x + \frac{4}{3}\).
This leads them to select Choice A (\(y \geq \frac{2}{3}x + \frac{4}{3}\)) instead of the correct answer.
Second Most Common Error:
Poor SIMPLIFY execution: Making sign errors during distribution
Some students struggle with the distribution step, particularly with -5(-2x), incorrectly getting -10x instead of +10x. This creates \(-15y - 10x \geq -20\), leading to \(y \leq \frac{-2}{3}x + \frac{4}{3}\) after proper manipulation.
This may lead them to select Choice E (\(y \leq \frac{2}{3}x - \frac{4}{3}\)) after making additional errors in the final simplification.
The Bottom Line:
The inequality flip rule when dividing by negatives is the most crucial concept here - it's easy to remember all the algebra steps but forget this one critical rule that completely changes the answer.