Inequalities Cheat Sheet - IBPS PO (Probationary Officer) Prelims

Inequalities Cheat Sheet - IBPS PO Prelims Study Material

Mastering inequalities is essential for securing high marks in the Quantitative Aptitude and Reasoning Ability sections of the IBPS PO (Probationary Officer) Prelims exam. This comprehensive LibreTexts-style guide breaks down fundamental algebraic properties, absolute value rules, interval methods, classical mean inequalities, and advanced theorems required to solve memory-based questions and Previous Year Questions (PYQs).

Table of Contents

• 1. Fundamental Properties of Inequalities
• 2. Absolute Value Inequalities & Triangle Inequalities
• 3. Polynomial & Rational Inequalities (Wavy Line Method)
• 4. Classical Means & Inequalities (AM-GM-HM)
• 5. Cauchy-Schwarz Inequality & Engel Form
• 6. Bernoulli's Inequality
• 7. Rearrangement Inequality
• 8. IBPS PO Prelims Exam Analysis & PYQ Patterns
• 9. Frequently Asked Questions (People Also Ask)

1. Fundamental Properties of Inequalities

Let \(a, b, c, d \in \mathbb{R}\). The foundation of algebraic inequalities rests on these core operational properties:

• Trichotomy Property: For any real numbers \(a\) and \(b\), exactly one of the following relations holds: \[a < b, \quad a = b, \quad \text{or} \quad a > b\]
• Transitivity: If \(a < b\) and \(b < c\), then \(a < c\).
• Addition Property: If \(a < b\), then for any real number \(c\): \[a + c < b + c\]
• Multiplication by Positive Scalar: If \(a < b\) and \(c > 0\), then: \[ac < bc\]
• Multiplication by Negative Scalar: Multiplying or dividing by a negative number reverses the inequality direction. If \(a < b\) and \(c < 0\), then: \[ac > bc\]
• Reciprocal Property: If both numbers have the same sign such that \(0 < a < b\), then: \[\frac{1}{a} > \frac{1}{b}\]

2. Absolute Value Inequalities & Triangle Inequalities

Absolute value relations appear frequently in both quantitative and logical reasoning problems in banking exams. For any positive real constant \(c > 0\):

• \(|x| < c \iff -c < x < c\)
• \(|x| \le c \iff -c \le x \le c\)
• \(|x| > c \iff x < -c \text{ or } x > c\)
• \(|x| \ge c \iff x \le -c \text{ or } x \ge c\)

Triangle Inequalities

For any real numbers \(x\) and \(y\), the upper bound and lower bound of the absolute value sum are bounded as follows:

\[|x + y| \le |x| + |y|\] \[||x| - |y|| \le |x - y|\]

3. Polynomial & Rational Inequalities (Wavy Line Method)

The Method of Intervals (Wavy Line Method) is a key technique for solving higher-degree polynomial and rational inequalities of the form \(P(x) > 0\) or \(\frac{P(x)}{Q(x)} \le 0\).

Step-by-Step Execution Procedure

1. Standardization: Shift all terms to one side of the inequality sign so that the opposite side equals zero (e.g., \(P(x) > 0\)).
2. Factorization: Factor the polynomial expression completely into linear factors and irreducible quadratic factors with positive leading coefficients.
3. Identify Critical Points: Determine all real roots where \(P(x) = 0\) or where the expression is undefined (denominators equal zero).
4. Plotting: Mark all critical points on a real number line in strictly ascending order.
5. Sign Assignment: Starting from the rightmost interval, place a positive sign (\(+\)). Move left across each critical point, alternating the sign for roots of odd multiplicity, and keeping the sign unchanged for roots of even multiplicity.

4. Classical Means & Inequalities (AM-GM-HM)

For any set of non-negative real numbers \(x_1, x_2, \dots, x_n\), classical means provide powerful tools for optimization and finding extreme values.

Arithmetic Mean - Geometric Mean (AM-GM) Inequality

\[\frac{x_1 + x_2 + \dots + x_n}{n} \ge \sqrt[n]{x_1 x_2 \dots x_n}\]

Equality holds if and only if all variables are equal: \(x_1 = x_2 = \dots = x_n\).

Geometric Mean - Harmonic Mean (GM-HM) Inequality

\[\sqrt[n]{x_1 x_2 \dots x_n} \ge \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \dots + \frac{1}{x_n}}\]

Complete Means Chain Inequality

The Root Mean Square (RMS), Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) satisfy the universal chain:

\[\text{RMS} \ge \text{AM} \ge \text{GM} \ge \text{HM}\]

where \(\text{RMS} = \sqrt{\frac{x_1^2 + x_2^2 + \dots + x_n^2}{n}}\).

5. Cauchy-Schwarz Inequality & Engel Form

For any sequences of real numbers \(a_1, a_2, \dots, a_n\) and \(b_1, b_2, \dots, b_n\):

\[\Big(\sum_{i=1}^n a_i b_i\Big)^2 \le \Big(\sum_{i=1}^n a_i^2\Big) \Big(\sum_{i=1}^n b_i^2\Big)\]

Engel Form (Titu's Lemma)

When \(b_i > 0\) for all \(i\), the algebraic application of Cauchy-Schwarz takes the Engel form:

\[\frac{a_1^2}{b_1} + \frac{a_2^2}{b_2} + \dots + \frac{a_n^2}{b_n} \ge \frac{(a_1 + a_2 + \dots + a_n)^2}{b_1 + b_2 + \dots + b_n}\]

6. Bernoulli's Inequality

For any real number \(x \ge -1\) and non-negative integer \(n \ge 0\):

\[(1 + x)^n \ge 1 + nx\]

When \(n \ge 2\) and \(x \ge -1\) with \(x eq 0\), the inequality becomes strict:

\[(1 + x)^n > 1 + nx\]

7. Rearrangement Inequality

Let \(a_1 \le a_2 \le \dots \le a_n\) and \(b_1 \le b_2 \le \dots \le b_n\) be two ordered sequences of real numbers. For any permutation \((x_1, x_2, \dots, x_n)\) of \((b_1, b_2, \dots, b_n)\):

\[a_n b_1 + a_{n-1} b_2 + \dots + a_1 b_n \le \sum_{i=1}^n a_i x_i \le a_1 b_1 + a_2 b_2 + \dots + a_n b_n\]

The scalar product is maximized when the sequences are sorted in the same direction, and minimized when sorted in opposite directions.

8. IBPS PO Prelims Exam Analysis & PYQ Patterns

PYQ Analysis & Topic Weightage

In the IBPS PO Prelims exam pattern, inequalities are tested in two distinct areas:

• Reasoning Ability (Coded & Mathematical Inequalities): 3 to 5 direct questions testing relation symbols (\(>, <, \ge, \le, =\)).
• Quantitative Aptitude (Quadratic Comparison & Algebraic Inequalities): 5 questions involving comparing variable pairs \(x\) and \(y\) derived from quadratic or absolute value equations.

Exam Pattern Strategy

To maximize speed and accuracy during the 20-minute sectional timeframe of the IBPS PO Prelims exam, focus on recognizing critical sign combinations and applying standard algebraic properties directly without testing arbitrary numbers.

9. Frequently Asked Questions (People Also Ask)

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