Inequalities – Based on Signs on GMAT

Quick Summary

Sign rules often drive inequality logic on GMAT. Remember: (+)(+) and (−)(−) are positive; (+)(−) is negative; even powers nonnegative, odd powers keep the sign. To verify options, test cases in four ranges : x>1, 0<x<1, −1<x<0, x<−1, beware of edge cases, and catch exceptions.

Overview

Understanding how positive and negative numbers behave under multiplication and powers is one of the most fundamental ideas in mathematics. On the GMAT quant section, this knowledge plays a decisive role in inequality questions and number property problems. A single misstep with a sign can completely change the outcome, making these questions tricky. The rules may appear simple at first glance: positive times positive is positive, negative times negative is positive, and so on. Yet, when powers and inequalities combine, the situations become layered. For example, the impact of raising a negative number to an odd power versus an even power is drastically different. A strong grasp of these basics, in your GMAT prep course, not only ensures accuracy in straightforward questions but also provides the confidence to handle advanced problem types tested in GMAT mock tests. Let us now carefully walk through these rules and their applications with examples.

Positive and Negative Multiplication Rules

The first principle is simple yet powerful:

• Positive x Positive = Positive
• Negative x Negative = Positive
• Positive x Negative = Negative
• Positive^Even = Positive
• Positive^Odd = Positive
• Negative^Even = Positive
• Negative^Odd = Negative

These core rules form the foundation for inequality questions.

Inequalities Based on Signs: An Explanatory Question

Which of the following may be correct?

I: x < x³ < x²  

II: x < x³ < x² < x⁴

III: x³ < x² < x⁴

IV: x³ < x < x² < x⁴

1. I only
2. I and III only
3. I, II, and IV only
4. I, III, and IV only
5. I, II, III, and IV

• Statement I is possible. For instance, if x = -0.5, then x³ = -0.125 and x² = 0.25. Here, x² is the greatest, making the condition true.
• Statement II is not possible. Checking across ranges (x > 1, 0 < x < 1, -1 < x < 0, and x < -1), the condition fails in all cases.
• Statement III is possible. For x = -2, we get x³ = -8, x² = 4, and x⁴ = 16. The pattern aligns with the condition.
• Statement IV is also possible. For x = -2, the comparison of values again holds true.

Thus, Statements I, III, and IV are correct, making option D the correct answer.

Key Learning for GMAT Students

The real takeaway is not just memorizing the rules but practicing them in diverse contexts. For inequality questions, always test numbers in four ranges: x > 1, 0 < x < 1, -1 < x < 0, and x < -1. This ensures that no case is overlooked. With consistent practice, recognizing these patterns becomes intuitive, saving valuable time on the test. Strong command over such number properties is a skill that directly strengthens performance in GMAT simulation and builds confidence for the most challenging quant problems.

Parting Thought

Signs teach a quiet ethic: direction matters as much as magnitude. A negative raised to an odd power stays true to its sign; to an even power it becomes nonnegative. In GMAT preparation, cultivate the habit of checking both directions, testing cases calmly, and letting patterns speak before pride. In the MBA admissions process, weigh praise and critique with equal care, transform raw feedback into improvement, and keep your core values intact. In life, choose actions that align with your principles, then amplify them through steady repetition. When you attend to signs with patience and rigor, your decisions carry clarity, balance, and strength.