Inequations – Based on Absolute Value (Modulus) on GMAT

Quick Summary

Absolute value inequalities split into two cases. For |x – a| > b, solve both greater-than and less-than branches. For |x – a| < b, confine the variable within a range, –b b to +b. Always check both directions to avoid incomplete or incorrect solutions.

Overview

Modulus-based or absolute-value inequalities are a recurring theme on the GMAT, and they often catch students off guard because they require careful interpretation. The key lies in remembering that the definition of modulus always branches into two possibilities: the positive case and the negative case. If you ignore either, you risk losing half the solution set.

For example, if you are solving |x – 2| > 3, you must consider both x – 2 > 3 and x – 2 < -3. Many test takers miss the second case, which is where careless errors creep in. Similarly, for inequalities such as |x + 5| < 10, the logic flips and the variable must lie between two values.

By mastering this dual-case approach, you will solve modulus inequalities quickly and confidently. You can build depth with our on-demand GMAT prep course and test the approach under timed conditions in the GMAT mock test series.

Solving Absolute Value Inequalities on the GMAT

Absolute value inequalities demand precision because the modulus can open into two directions. Let us look at this step by step.

Example 1: |x – 2| > 3

When modulus is greater than a value, we must consider both greater-than and less-than cases:

x – 2 > 3 → x > 5

x – 2 < –3 → x < –1

Therefore, the complete solution is x > 5 or x < –1. Missing any one of the two cases would lead to an incomplete and incorrect answer.

Example 2: |x + 5| < 10

When modulus is less than a value, the expression must lie within a range:

–10 < x + 5 < 10

Solving gives –15 < x < 5. This captures all possible values of x.

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Key Learning

When solving modulus inequalities:

• For “greater than” (|expression| > value), split into two branches: considering the “greater than” as well as “less than” case.
• For “less than” (|expression| < value), confine within a range.

By internalizing these patterns, you will approach GMAT modulus questions with clarity and accuracy. For deeper guidance, explore our GMAT crash course, where these methods are reinforced with advanced applications.

Parting Thought

Absolute value reminds us to measure distance from a clear center. Inequalities ask us to look in both directions, to test boundaries with care. In GMAT preparation, that means writing both cases, checking endpoints, and choosing ranges that truly satisfy the conditions. In the MBA application journey, it means holding core values steady while weighing varied evidence from essays, recommendations, and interviews. In life, it means balancing conviction with openness, so decisions honor principle and context. When uncertainty rises, state the cases, reason quietly, and accept only what fits. Clarity comes from disciplined distance, balanced judgment, and respect for limits.