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

•Is   ′⇔′  necessary and suficient for two  inequalities to be equivalent?  •If  a>b :  Are  A>B and A+a > B+b equivalent?

$$\bullet{Is}\:\:\:'\Leftrightarrow'\:\:{necessary}\:{and}\:{suficient}\:{for}\:{two} \\ $$ $${inequalities}\:{to}\:{be}\:{equivalent}? \\ $$ $$\bullet{If}\:\:\boldsymbol{\mathrm{a}}>\boldsymbol{\mathrm{b}}\:: \\ $$ $${Are}\:\:\boldsymbol{\mathrm{A}}>\boldsymbol{\mathrm{B}}\:{and}\:\boldsymbol{\mathrm{A}}+\boldsymbol{\mathrm{a}}\:>\:\boldsymbol{\mathrm{B}}+\boldsymbol{\mathrm{b}}\:{equivalent}? \\ $$

Answered by 123456 last updated on 25/Oct/15

(a,b,A,B)∈R^4   a>b∧A>B⇒A+a>B+b  −−−−−−−−−−  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  −−−−−  a>b∧A+a>B+b⇏A>B  −−−−−−  A=B=0⇒a>b⇒A+a=a>b=B+b

$$\left({a},{b},\mathrm{A},\mathrm{B}\right)\in\mathbb{R}^{\mathrm{4}} \\ $$ $${a}>{b}\wedge\mathrm{A}>\mathrm{B}\Rightarrow\mathrm{A}+{a}>\mathrm{B}+{b} \\ $$ $$−−−−−−−−−− \\ $$ $${a}>{b}\: \\ $$ $$\mathrm{A}+{a}>\mathrm{A}+{b}\:\left(+\mathrm{A}\right) \\ $$ $$\mathrm{B}+{a}>\mathrm{B}+{b}\:\left(+\mathrm{B}\right) \\ $$ $$\mathrm{A}>\mathrm{B} \\ $$ $$\mathrm{A}+{a}>\mathrm{B}+{a}\:\left(+{a}\right) \\ $$ $$\mathrm{A}+{b}>\mathrm{B}+{b}\:\left(+{b}\right) \\ $$ $$\mathrm{A}+{a}>\mathrm{B}+{a}>\mathrm{B}+{b} \\ $$ $$−−−−− \\ $$ $${a}>{b}\wedge\mathrm{A}+{a}>\mathrm{B}+{b}\nRightarrow\mathrm{A}>\mathrm{B} \\ $$ $$−−−−−− \\ $$ $$\mathrm{A}=\mathrm{B}=\mathrm{0}\Rightarrow{a}>{b}\Rightarrow\mathrm{A}+{a}={a}>{b}=\mathrm{B}+{b} \\ $$

Commented byRasheed Soomro last updated on 25/Oct/15

That means adding same−sense inequality to  given  inequality doesn′t yeild equivalent inequality.    Equivalent inequality may be achieved only  by:  •Adding (Subtracting)an equation_(−)  to(from) both sides of an inequality.  •Multiplying/Dividing  an equation_(−)  to both sides of an inequality.  Am I correct?

$${That}\:{means}\:{adding}\:\boldsymbol{\mathrm{same}}−\boldsymbol{\mathrm{sense}}\:{inequality}\:{to}\:\:{given} \\ $$ $${inequality}\:{doesn}'{t}\:{yeild}\:{equivalent}\:{inequality}. \\ $$ $$ \\ $$ $${Equivalent}\:{inequality}\:{may}\:{be}\:{achieved}\:\boldsymbol{\mathrm{only}}\:\:{by}: \\ $$ $$\bullet{Adding}\:\left({Subtracting}\right){an}\:\underset{−} {{equation}}\:{to}\left({from}\right)\:{both}\:{sides}\:{of}\:{an}\:{inequality}. \\ $$ $$\bullet{Multiplying}/{Dividing}\:\:{an}\:\underset{−} {{equation}}\:{to}\:{both}\:{sides}\:{of}\:{an}\:{inequality}. \\ $$ $${Am}\:{I}\:{correct}? \\ $$

Commented byprakash jain last updated on 25/Oct/15

Multiplying and dividing an equatily by  same value may reverse the sign and equality.

$$\mathrm{Multiplying}\:\mathrm{and}\:\mathrm{dividing}\:\mathrm{an}\:\mathrm{equatily}\:\mathrm{by} \\ $$ $$\mathrm{same}\:\mathrm{value}\:\mathrm{may}\:\mathrm{reverse}\:\mathrm{the}\:\mathrm{sign}\:\mathrm{and}\:\mathrm{equality}. \\ $$

Commented byRasheed Soomro last updated on 28/Oct/15

Of course sir!

$${Of}\:{course}\:{sir}! \\ $$

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