Important Formulas for CAT Inequalities

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 aⁿ > bⁿ and 1/aⁿ < 1/bⁿ, 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
⇒ (a₁+a₂+ ….+a_n)/n ≥(a₁.a₂. …..a_n)^1/n
If a and b are positive quantities, then

If a,b,c,d are positive quantities, then
⇒
⇒ a⁴ + b⁴ + c⁴ + d⁴ ≥ 4abcd
If a,b,c …. k are n positive quantities and m is a natural number, then

EduRev's Tip:

• 
  
• For any positive integer n, 2 ≤
• a^mbⁿc^p……..will be greatest when
• If a > b and both are natural numbers, then
  ⇒ a^b < b^a {Except 3² > 2³ & 4² = 2⁴}
• (n!)² ≥ nⁿ
•  If the sum of two or more positive quantities is constant, their product is greatest when they are equal and if their product is constant then their sum is the least when the numbers are equal.
  ⇒ If x + y = k, then xy is greatest when x = y
  ⇒ If xy = k, then x + y is least when x = y