Modulus & Inequality — Comprehensive Concept Explanation

In algebra and analytical reasoning, Modulus (or Absolute Value) and Inequalities describe relationships between quantities involving magnitude, sign, and comparison. These concepts appear in GMAT, GRE, and quantitative aptitude reasoning questions requiring logical understanding of number behavior.

I. Modulus (Absolute Value)

The modulus or absolute value of a real number \(x\) represents its distance from zero on the number line — always nonnegative.

\[ |x| = \begin{cases} x, & x \ge 0\\[4pt] -x, & x < 0 \end{cases} \]

1. Basic Properties

  • \(|x| \ge 0\) for all real \(x\).
  • \(|x| = 0 \iff x = 0\).
  • \(|ab| = |a|\cdot|b|\)
  • \(|a/b| = |a|/|b|, \text{ where } b \ne 0\)
  • \(|a+b| \le |a| + |b|\) (Triangle Inequality)
  • \(|a-b| \ge \big||a| – |b|\big|\)

2. Converting Absolute Equations

Absolute value equations and inequalities can be converted into standard linear equations by considering both positive and negative cases.

Form Equivalent Conditions
\(|x| = a\) \(x = a\) or \(x = -a\)
\(|x| < a\) \(-a < x < a\)
\(|x| > a\) \(x < -a\) or \(x > a\)
\(|x| \le a\) \(-a \le x \le a\)
\(|x| \ge a\) \(x \le -a\) or \(x \ge a\)

3. Examples

Example 1:

Solve \( |x – 2| = 5 \).

\[ x – 2 = 5 \text{ or } x – 2 = -5 \Rightarrow x = 7 \text{ or } x = -3. \]

Example 2:

Solve \( |x – 2| < 1 \).

\[ -1 < x – 2 < 1 \Rightarrow 1 < x < 3. \]

Example 3:

If \( |a – b| = 0 \Rightarrow a = b.\)

Example 4:

If \( |a – b| = |b – a|,\) both sides are equal — modulus removes direction.

Example 5:

\( |x| = -x \Rightarrow x \text{ is negative.}\)

4. Graphical Insight

The graph of \(y = |x|\) is a “V”-shaped curve symmetric about the y-axis:

  • For \(x \ge 0:\ y = x\)
  • For \(x < 0:\ y = -x\)
  • Vertex at (0,0); slope changes from -1 to +1.

Translating \( |x – h| = k \) shifts the “V” right by \(h\) units and up by \(k\).

II. Inequalities

An inequality compares two quantities, showing one is greater or less than the other. The principles governing inequality manipulations are essential for solving range problems and logical data sufficiency questions.

1. Basic Inequality Signs

\[ a < b,\quad a > b,\quad a \le b,\quad a \ge b \]

  • Adding or subtracting the same number on both sides does not change the inequality.
  • Multiplying or dividing by a positive number keeps the direction unchanged.
  • Multiplying or dividing by a negative number reverses the inequality sign.
Example: If \(3x – 5 < 7 \Rightarrow 3x < 12 \Rightarrow x < 4.\) If both sides are multiplied by -2: \(-6x > -24 \Rightarrow x < 4\) becomes \(x > 4.\)

2. Compound Inequalities

Expressions bounded by two inequalities can be solved as a combined range.

\[ a < x < b \Rightarrow x \text{ lies between } a \text{ and } b. \]

Example:

Solve \(2 < 3x – 4 < 11.\)

\[ 6 < 3x < 15 \Rightarrow 2 < x < 5. \]

3. Product and Sign-Based Inequalities

Sign analysis determines variable relations when inequalities involve products.

  • If \(ab > 0,\) then \(a\) and \(b\) have the same sign.
  • If \(ab < 0,\) then \(a\) and \(b\) have opposite signs.
  • If \(a-b > a+b \Rightarrow -b > b \Rightarrow b < 0.\)
  • If \((a-b)c < 0,\) then \((a-b)\) and \(c\) have opposite signs.
Example: If \(a,b>0\) and \(a>b,\) then \(1/a < 1/b.\)

4. Squared and Absolute Inequalities

  • If \(x^2 < a^2,\) then \(-a < x < a.\)
  • If \(x^2 > a^2,\) then \(x < -a\) or \(x > a.\)
  • Absolute inequalities follow identical logic: \(|x| < a \Leftrightarrow -a < x < a.\)

5. Inequalities Involving Fractions

When denominators contain variables, first determine sign constraints to avoid invalid domain regions (e.g., denominators ≠ 0). Multiply across only after confirming positivity or negativity of each side.

Example: Solve \( \frac{2}{x} < 1.\)
Case 1: \(x>0 \Rightarrow 2 < x \Rightarrow x>2.\)
Case 2: \(x<0 \Rightarrow\) inequality reverses \(2 > x \Rightarrow x<0.\)
Solution set: \(x<0\) or \(x>2.\)

6. Range Interpretation in Data Sufficiency

In GMAT/GRE-style Data Sufficiency, inequalities are tested for range sufficiency rather than numeric values.

  • Check whether sign or magnitude can be uniquely determined.
  • When both statements lead to overlapping but incomplete ranges → data insufficient.
  • When both independently define same sign or value range → either sufficient.

Example: If \(|x| > 3,\) statement (1) \(x>2\) insufficient; statement (2) \(x<-4\) insufficient; together, \(x>2\) or \(x<-4\) satisfies \(|x|>3\) ⇒ together sufficient.

III. Concept Summary

  • Modulus gives distance; always nonnegative.
  • For \(|x| = a,\) always consider both positive and negative cases.
  • When solving \(|x| < a\) or \(|x| > a,\) rewrite as compound inequalities.
  • In inequalities, sign reversal occurs when multiplied/divided by negative numbers.
  • Analyze product inequalities via sign charts.
  • When working with fractions or variables in denominators, handle domain carefully before cross-multiplying.

Source reference: Based on “GMAT Algebra Topics 6” compiled by Manoj K. Singh (Modulus & Inequality section). Expanded with logical reasoning, property proofs, and GMAT-style examples for conceptual mastery.