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).
(i) If a > b and b > c, then a > c. Generally, if a_{1} > a_{2}, a_{2} > a_{3},…., a_{n – 1} > a_{n}, then a_{1} > a_{n}.
(vii) If a < x < b and a, b are positive real numbers then a^{2} < x^{2} < b^{2}
(i) If a, b > 0 and a ≠ b, then
(ii) if a_{i} > 0, where i = 1,2,3,…,n, then
(iii) If a_{1}, a_{2},…, a_{n} are n positive real numbers and m_{1}, m_{2},…,m_{n} are n positive rational numbers, then
i.e., Weighted AM > Weighted GM
(iv) If a_{1}, a_{2},…, a_{n} are n positive distinct real numbers, then
(a)
(b)
(c) If a_{1}, a_{2},…, a_{n} and b_{1}, b_{2},…, b_{n} are rational numbers and M is a rational number, then
(d)
(v) If a_{1}, a_{2}, a_{3},…, a_{n} are distinct positive real numbers and p, ,q, r are natural numbers, then
If a_{1}, a_{2},…, a_{n} and b_{1}, b_{2},…, b_{n} are real numbers, such that
(a_{1}b_{1} + a_{2}b_{2} + …+ a_{n}b_{n})^{2} ≤ (a_{1}^{2} + a_{2}^{2} + …, a_{n}^{2}) * (b_{1}^{2} + b_{2}^{2} + …, b_{n}^{2})
Equality holds, iff a_{1} / b_{1} = a_{2} / b_{2} = a_{n} / b_{n}
Let a_{1}, a_{2},…, a_{n} and b_{1}, b_{2},…, b_{n} are real numbers, such that
(i) If a_{1} ≤ a_{2} ≤ a_{3} ≤… ≤ a_{n} and b_{1} ≤ b_{2} ≤ b_{3} ≤… ≤ b_{n}, then
n(a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} + …+ a_{n}b_{n}) ≥ (a_{1} + a_{2} + …+ a_{n}) (b_{1} + b_{2} + …+ b_{n})
(ii) If If a_{1} ≥ a_{2} ≥ a_{3} ≥… ≥ a_{n} and b_{1} ≥ b_{2} ≥ b_{3} ≥… ≥ b_{n}, then
n(a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} + …+ a_{n}b_{n}) ≤ (a_{1} + a_{2} + …+ a_{n}) (b_{1} + b_{2} + …+ b_{n})
(i) If a_{1}, a_{2},…, a_{n} are real positive numbers, then for n ≥ 2
(1 + a_{1})(1 + a_{2}) … (1 + a_{n}) > 1 + a_{1} + a_{2} + … + a_{n}
(ii) If a_{1}, a_{2},…, a_{n} are real positive numbers, then
(1 – a_{1})(1 – a_{2}) … (1 – a_{n}) > 1 – a_{1} – a_{2} – … – a_{n}
(i) (a) When y > 1 and log_{y} x > z ⇒ x > y^{z}
(b) When y > 1 and log_{y} x < z ⇒ 0 < x < y^{z}
(ii) (a) When 0 < y < 1 and log_{y} x > z ⇒ 0 < x < y^{z}
(b) hen 0 < y < 1 and log_{y} x < z ⇒ x > y^{z}
(i) If x_{l},x_{2},…,x_{n} are n positive variables such that x_{l} + x_{2} +…+ x_{n} = c (constant), then the product x_{l} * x_{2} *….* x_{n} is greatest when x_{1} = x_{2} =… =x_{n} = c/n and the greatest value is (c/n)^{n}.
(ii) If x_{l},x_{2},…,x_{n} are positive variables such that x_{l},x_{2},…,x_{n} = c (constant), then the sum x_{l} + x_{2} +….+ x_{n} is least when x_{1} = x_{2} =… =x_{n} = c^{1/n} and the least value of the sum is n (c^{1/n}).
(iii) If x_{l},x_{2},…,x_{n} are variables and m_{l},m_{2},…,m_{n} are positive real number such that x_{l} + x_{2} +….+ x_{n} = c (constant), then x_{l}^{m}_{l} * x_{2}^{m}_{2} *… * x_{n}^{m}_{n} is greatest, when
x_{l} / m_{l} = x_{2} / m_{2} =…= x_{n} / m_{n}
= x_{l} + x_{2} +….+ x_{n} / m_{l} + m_{2} +….+ m_{n}
209 videos443 docs143 tests

1. What are inequalities in mathematics? 
2. How are inequalities solved? 
3. What is the difference between an inequality and an equation? 
4. Can inequalities be represented graphically? 
5. How are inequalities used in reallife situations? 
209 videos443 docs143 tests


Explore Courses for JEE exam
