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Question Number 56775 by zambolly19 last updated on 23/Mar/19
if x,y∍ℜ,show that ∣x+y∣=∣x∣+∣y∣ iff xy≥0
ifx,y∍ℜ,showthat∣x+y∣=∣x∣+∣y∣iffxy⩾0
Commented by maxmathsup by imad last updated on 23/Mar/19
we have ∣x+y∣^2 −(∣x∣+∣y∣)^2 =x^2 +2xy+y^2 −x^2 −2∣xy∣−y^2 =2(xy−∣xy∣) =0 because xy≥0 ⇒∣x+y∣=∣x∣+∣y∣ .
wehave∣x+y∣2−(∣x∣+∣y∣)2=x2+2xy+y2−x2−2∣xy∣−y2=2(xy−∣xy∣)=0becausexy⩾0⇒∣x+y∣=∣x∣+∣y∣.

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