Inequalities – Based on -1 to 1 on GMAT

Quick Summary

Expressions behave differently in the −1 to 1 range. For example: squaring numbers in this range makes values smaller (while outside this range, squaring makes values larger). Interesting GMAT inequalities questions draw on such unintuitive behavior in the −1 to 1 range. Learn to recognize and navigate such problems.

Overview

Many GMAT questions involving inequalities and functions demand a very sharp awareness of how numbers behave across different ranges. What seems obvious for large values can completely reverse when you test numbers between -1 and 1. For example, while we usually expect 1/x to shrink when x grows, the exact opposite happens when x lies between 0 and 1. Similarly, squaring a small fraction makes it even smaller, and taking its square root makes it larger. The same applies to negatives, where behavior flips again. These subtle shifts form the foundation of several higher-level inequality problems. Without mastering these ranges, students often fall into traps, especially in advanced questions. A single oversight in this area can derail even a strong attempt. This is why careful practice with varied ranges is essential, using GMAT mocks as well as inequations based exercises.

 

Understanding Behavior of Functions in Different Ranges

On the GMAT, questions involving inequalities often test not just arithmetic, but your ability to understand how expressions behave across ranges. A critical learning is that the range from -1 to +1 behaves very differently from other intervals, and ignoring this can cost valuable points.

Case of 1/x

When x > 1, 1/x is clearly smaller than x. But when x lies between 0 and 1, 1/x becomes larger than x. For example, 1/0.5 = 2, which is greater than 0.5. For negative numbers, the same surprise occurs. While 1/–2 is greater than –2, in the interval –1 < x < 0, the reciprocal becomes an even bigger negative, such as 1/–0.5 = –2, which is less than –0.5.

Case of x² and x³

For large positives, both x² and x³ exceed x. But for small fractions, squaring or cubing makes the number even smaller. For example, 0.5² = 0.25, which is less than 0.5. With negatives, cubing preserves the negative sign, often making it smaller than the original, while squaring turns it positive.

Case of √x

For values greater than 1, square roots reduce size, such as √4 = 2 < 4. But for numbers between 0 and 1, the root increases the value, such as √0.5 ≈ 0.7 > 0.5. This is a key reversal that students must remember.

Comparison Table Across Ranges

 

Expression	x > 1	0 < x < 1	–1 < x < 0	x < –1
x²	Greater than x	Smaller than x	Positive, greater than x	Positive, greater than x
x³	Greater than x	Smaller than x	Greater than x (less negative)	Smaller than x (more negative)
√x	Smaller than x	Greater than x	Not real	Not real

Application in Inequalities

The best GMAT prep practice is to test four distinct ranges: a large positive, a small positive between 0 and 1, a small negative between –1 and 0, and a large negative. Only by checking all these cases can you reach sound conclusions. This disciplined habit prevents errors in “always correct” type questions.

A High Difficulty Problem

Which of the following is always true?

If 1/x is greater than x, x is greater than x².

A. If x is greater than 1/x, 2x is greater than x.

B. If x is greater than 2x, 1/x is greater than x.

C. If x² is greater than x, x³ is greater than x².

D. If x is greater than 1/x, x² is greater than 1/x.

We test each implication over the usual four ranges: x>1, 0<x<1, −1<x<0, x<−1.

A. If 1/x > x, then x > x².

• 1/x > x holds for 0<x<1 and for x<−1.
• Take x=−2: 1/x=−0.5 > −2 (condition true), but x>x² is −2>4 (false).
  Not always true.

B. If x > 1/x, then 2x > x.

• x > 1/x holds for x>1 and −1<x<0.
• Take x=−0.5: condition true (−0.5>−2), but 2x> x is −1>−0.5 (false).
  Not always true.

C. If x > 2x, then 1/x > x.

• x > 2x ⇔ x<0.
• Take x=−0.5: 1/x=−2 > −0.5 is false.
  Not always true.

D. If x² > x, then x³ > x².

• x² > x holds for x>1 or x<0.
• Take x=−2: x³=−8 > x²=4 is false.
  Not always true.

E. If x > 1/x, then x² > 1/x.

• For x>1: multiply both sides by x>0 → x³>1, hence x²>1/x.
• For −1<x<0: 1/x<−1 while x²∈(0,1); a positive number is greater than a negative number.
  Always true.

Correct answer: E.

Key Takeaway

Numbers do not behave uniformly across ranges. The region between –1 and +1 requires special attention, and overlooking it almost guarantees mistakes. By building the habit of checking all ranges, you can master even the trickiest inequality questions. For deeper practice, strengthen your fundamentals with GMAT simulation.

Parting Thought

Between −1 and 1, the world behaves differently. Small steps can invert outcomes: reciprocals grow, squaring shrinks, roots enlarge. Inequalities teach respect for context. In GMAT preparation, check every range and verify patiently; quiet proof outruns instinct. In the MBA admissions workflow, sound judgment knows when a modest, truthful improvement changes the whole profile, and when a loud claim does not. In life, pause at thresholds, test both directions, and choose actions that truly satisfy your own conditions. Let discipline frame courage. Precision is not rigidity; it is patient clarity that protects momentum and turns careful steps into steady progress.