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If a > b and c > 0,
⇒ a + c > b + c
⇒ a  c > b  c
⇒ ac > bc
⇒ a/c > b/c
If a, b ≥ 0, then a^{n} > b^{n} and 1/a^{n} < 1/b^{n}, where n is positive.
a < b and x > 0, then
a > b and x > 0, then
Modular Inequalities
x y = y  x
x. y = x . y
x+ y < x + y
x+ y > x  y
Quadratic Inequalities
(x – a) (x – b) > 0 {a < b}
⇒ (x < a) U (x > b)
(x – a) (x – b) < 0 {a > b}
⇒ a < x < b
For any set of positive numbers: AM≥GM≥HM
⇒ (a_{1}+a_{2}+ ….+a_{n})/n ≥(a_{1}.a_{2}. …..a_{n})^{1/n}
If a and b are positive quantities, then
If a,b,c,d are positive quantities, then
⇒
⇒ a^{4} + b^{4} + c^{4} + d^{4} ≥ 4abcd
If a,b,c …. k are n positive quantities and m is a natural number, then
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