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Question Number 182947 by myint last updated on 17/Dec/22

If  a< b<0,  then  ∣a−b∣ + ∣a+b∣ + ∣ab∣=

$$\mathrm{If}\:\:\mathrm{a}<\:\mathrm{b}<\mathrm{0},\:\:\mathrm{then}\:\:\mid\mathrm{a}−\mathrm{b}\mid\:+\:\mid\mathrm{a}+\mathrm{b}\mid\:+\:\mid\mathrm{ab}\mid= \\ $$

Commented bymr W last updated on 17/Dec/22

≠constant value!  could be any positive value!

$$\neq{constant}\:{value}! \\ $$ $${could}\:{be}\:{any}\:{positive}\:{value}! \\ $$

Answered by Frix last updated on 17/Dec/22

a<b<0 ⇒ a−b<0∧a+b<0∧ab>0 ⇒  ∣a−b∣+∣a+b∣+∣ab∣=  =−(a−b)−(a+b)+ab=  =−2a+ab=a(b−2)

$${a}<{b}<\mathrm{0}\:\Rightarrow\:{a}−{b}<\mathrm{0}\wedge{a}+{b}<\mathrm{0}\wedge{ab}>\mathrm{0}\:\Rightarrow \\ $$ $$\mid{a}−{b}\mid+\mid{a}+{b}\mid+\mid{ab}\mid= \\ $$ $$=−\left({a}−{b}\right)−\left({a}+{b}\right)+{ab}= \\ $$ $$=−\mathrm{2}{a}+{ab}={a}\left({b}−\mathrm{2}\right) \\ $$

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