Is-necessary-and-suficient-for-two-inequalities-to-be-equivalent-If-a-gt-b-Are-A-gt-B-and-A-a-gt-B-b-equivalent-

Question Number 1937 by Rasheed Soomro last updated on 25/Oct/15

∙ I s^{'} \Leftrightarrow^{'} n e c e s s a r y a n d s u f i c i e n t f o r t w o

i n e q u a l i t i e s t o b e e q u i v a l e n t ?

∙ I f a > b :

A r e A > B a n d A + a > B + b e q u i v a l e n t ?

Answered by 123456 last updated on 25/Oct/15

(a, b, A, B) \in R^{4}

a > b \land A > B \Rightarrow A + a > B + b

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a > b

A + a > A + b (+ A)

B + a > B + b (+ B)

A > B

A + a > B + a (+ a)

A + b > B + b (+ b)

A + a > B + a > B + b

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a > b \land A + a > B + b ⇏ A > B

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A = B = 0 \Rightarrow a > b \Rightarrow A + a = a > b = B + b

Commented by Rasheed Soomro last updated on 25/Oct/15

T h a t m e a n s a d d i n g same - sense i n e q u a l i t y t o g i v e n

i n e q u a l i t y {d o e s n}^{'} t y e i l d e q u i v a l e n t i n e q u a l i t y .

E q u i v a l e n t i n e q u a l i t y m a y b e a c h i e v e d only b y :

∙ A d d i n g (S u b t r a c t i n g) a n \underline{e q u a t i o n} t o (f r o m) b o t h s i d e s o f a n i n e q u a l i t y .

∙ M u l t i p l y i n g / D i v i d i n g a n \underline{e q u a t i o n} t o b o t h s i d e s o f a n i n e q u a l i t y .

A m I c o r r e c t ?

Commented by prakash jain last updated on 25/Oct/15

Multiplying and dividing an equatily by

same value may reverse the sign and equality .

Commented by Rasheed Soomro last updated on 28/Oct/15

O f c o u r s e s i r!