Inequalities

# Inequalities | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Table of contents Inequalities Important Inequality 1. Arithmetico-Geometric and Harmonic Mean Inequality 2. Cauchy – Schwartz’s inequality 3. Tchebychef’s Inequality 4. Weierstrass Inequality 5. Logarithm Inequality Application of Inequalities to Find the Greatest and Least Values

## Inequalities

Let a and b be real numbers. If a – b is negative, we say that a is less than b (a < b) and if a – b is positive, then a is greater than b (a > b).

### Important Points to be Remembered

(i) If a > b and b > c, then a > c. Generally, if a1 > a2, a2 > a3,…., an – 1 > an, then a1 > an.

(vii) If a < x < b and a, b are positive real numbers then a2 < x2 < b2

## 1. Arithmetico-Geometric and Harmonic Mean Inequality

(i) If a, b > 0 and a ≠ b, then

(ii) if ai > 0, where i = 1,2,3,…,n, then

(iii) If a1, a2,…, an are n positive real numbers and m1, m2,…,mn are n positive rational numbers, then

i.e., Weighted AM > Weighted GM

(iv) If a1, a2,…, an are n positive distinct real numbers, then

(a)

(b)

(c) If a1, a2,…, an and b1, b2,…, bn are rational numbers and M is a rational number, then

(d)

(v) If a1, a2, a3,…, an are distinct positive real numbers and p, ,q, r are natural numbers, then

## 2. Cauchy – Schwartz’s inequality

If a1, a2,…, an and b1, b2,…, bn are real numbers, such that

(a1b1 + a2b2 + …+ anbn)2 ≤ (a12 + a22 + …, an2) * (b12 + b22 + …, bn2)

Equality holds, iff a1 / b1 = a2 / b2 = an / bn

## 3. Tchebychef’s Inequality

Let a1, a2,…, an and b1, b2,…, bn are real numbers, such that

(i) If a1 ≤ a2 ≤ a3 ≤… ≤ an and b1 ≤ b2 ≤ b3 ≤… ≤ bn, then

n(a1b1 + a2b2 + a3b3 + …+ anbn) ≥ (a1 + a2 + …+ an) (b1 + b2 + …+ bn)

(ii) If If a1 ≥ a2 ≥ a3 ≥… ≥ an and b1 ≥ b2 ≥ b3 ≥… ≥ bn, then

n(a1b1 + a2b2 + a3b3 + …+ anbn) ≤ (a1 + a2 + …+ an) (b1 + b2 + …+ bn)

## 4. Weierstrass Inequality

(i) If a1, a2,…, an are real positive numbers, then for n ≥ 2

(1 + a1)(1 + a2) … (1 + an) > 1 + a1 + a2 + … + an

(ii) If a1, a2,…, an are real positive numbers, then

(1 – a1)(1 – a2) … (1 – an) > 1 – a1 – a2 – … – an

## 5. Logarithm Inequality

(i) (a) When y > 1 and logy x > z ⇒ x > yz

(b) When y > 1 and logy x < z ⇒ 0 < x < yz

(ii) (a) When 0 < y < 1 and logy x > z ⇒ 0 < x < yz

(b) hen 0 < y < 1 and logy x < z ⇒ x > yz

## Application of Inequalities to Find the Greatest and Least Values

(i) If xl,x2,…,xn are n positive variables such that xl + x2 +…+ xn = c (constant), then the product xl * x2 *….* xn is greatest when x1 = x2 =… =xn = c/n and the greatest value is (c/n)n.

(ii) If xl,x2,…,xn are positive variables such that xl,x2,…,xn = c (constant), then the sum xl + x2 +….+ xn is least when x1 = x2 =… =xn = c1/n and the least value of the sum is n (c1/n).

(iii) If xl,x2,…,xn are variables and ml,m2,…,mn are positive real number such that xl + x2 +….+ xn = c (constant), then xlml * x2m2 *… * xnmn is greatest, when

xl / ml = x2 / m2 =…= xn / mn

= xl + x2 +….+ xn / ml + m2 +….+ mn

The document Inequalities | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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## Mathematics (Maths) for JEE Main & Advanced

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## FAQs on Inequalities - Mathematics (Maths) for JEE Main & Advanced

 1. What are inequalities in mathematics?
Answer: In mathematics, inequalities are mathematical statements that compare two quantities using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). They represent a relationship between two values where one is smaller or larger than the other.
 2. How are inequalities solved?
Answer: Inequalities can be solved by following similar rules as equations, with some additional considerations. The goal is to isolate the variable on one side of the inequality sign. However, if you multiply or divide both sides of the inequality by a negative number, you need to reverse the inequality sign. After solving, the solution can be represented on a number line or interval notation.
 3. What is the difference between an inequality and an equation?
Answer: In mathematics, an inequality is a statement that compares two quantities, indicating that one is greater than, less than, or equal to the other. On the other hand, an equation is a statement that asserts the equality of two expressions. Inequalities contain inequality symbols (<, >, ≤, ≥), while equations use an equal sign (=).
 4. Can inequalities be represented graphically?
Answer: Yes, inequalities can be represented graphically on a coordinate plane. For linear inequalities with one variable, the graph is a number line divided into two regions: the solution set and the non-solution set. On the other hand, linear inequalities with two variables can be graphed as shaded regions in the coordinate plane to represent all the possible solutions.
 5. How are inequalities used in real-life situations?
Answer: Inequalities are commonly used in real-life situations to model and solve various problems. For example, they can be used to represent budget constraints, determine the range of possible values for a given situation, analyze market trends, or solve optimization problems. Inequalities provide a mathematical framework for understanding and making decisions based on conditions and limitations in real-world scenarios.

## Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

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