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# Inequalities Mathematical Formulas

## Inequalities Mathematical Formulas

One of the most important topic in Competitive math is Inequalities Mathematical Formulas. Few questions will definitely come from this topic.

The sign  shows that it is a greater than suppose 5 > 3 which means 5 is bigger than 3.

Ex. 5>4, 9> 7

<

The sign  shows that it is a lesser than suppose 3 < 5 which means 3 is lesser than 5.
Ex. 4<9, 7<9

>

The sign ≥ shows that both are equal also a is greater than b suppose a≥b .

Ex. a > b

<

The sign ≤ shows that both are equal also a is lesser than b suppose a≤b ..

Ex.a<b

Types of Inequalities :

• a ≤ b => -a≥b
• a ≤ b => a ± c ≤ b ± c

• a ≤ b, c > 0 => ac ≤ bc, a / c  ≤  b /  c;
a ≤ b , c < 0  => ac ≥ bc,  a  / c  ≥   b / c

• 0 < a  b ⇒ 1 / a > 1 / b > 0
• a < b <  0 ⇒ 0 > 1 / a > 1 / b
• a < 0 < b  ⇒ 1 / a < 0 < 1 / b

• a ≤ b ⇔  an ≤ bn,                     (n,a,b > 0)
• a ≤ b ⇔ a-n ≤ b-n
• a ≤ b ⇔ In a ≤  In b

• ex ≥ 1 + x
• xx  ≥ (1/e)1/e,                      x  ≥ 1
• Xxx ≥ x                                  x ≥ 1

• aa + bb  ≥   ab + ba  > 1,          a , b  > 0