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Question Number 164622 by mathocean1 last updated on 19/Jan/22
Show that ∀ a, b ∈ R,  1. ∣∣x∣−∣y∣∣≤∣x−y∣  2. 1+∣xy−1∣≤(1+∣x−1∣)(1+∣y−1∣).
Showthata,bR,1.∣∣xy∣∣⩽∣xy2.1+xy1∣⩽(1+x1)(1+y1).
Answered by hmrsh last updated on 19/Jan/22
  Answer of part(1):  ∗ ∣x∣ ≤ A ↔ −A ≤ x ≤ A  ∗∗ ∣x + y∣ ≤ ∣x∣ + ∣y∣      (triangle inequality)    ∣x∣ = ∣(x − y) + y∣  →^(∗∗)  ∣x∣ = ∣(x − y) + y∣ ≤ ∣x − y∣ + ∣y∣   (#)  also:  ∣y∣ = ∣(y−x)+x∣ ≤ ∣y−x∣+∣x∣=∣x−y∣+∣x∣  → ∣y∣ −∣x − y∣ ≤ ∣x∣    →^((#))  ∣y∣ −∣x − y∣ ≤ ∣x∣ ≤ ∣x − y∣ + ∣y∣  → −∣x − y∣ ≤ ∣x∣ − ∣y∣ ≤ ∣x − y∣  →^∗  ∣∣x∣ − ∣y∣∣ ≤ ∣x − y∣
Answerofpart(1):xAAxAx+yx+y(triangleinequality)x=(xy)+yYou can't use 'macro parameter character #' in math modealso:y=(yx)+xyx+x∣=∣xy+xyxyxYou can't use 'macro parameter character #' in math modexyxyxy∣∣xy∣∣xy
Commented by mathocean1 last updated on 19/Jan/22
thanks.
thanks.
Answered by hmrsh last updated on 20/Jan/22
Answer of part 2:  tip 1:  xy−1 = (1−x)(1−y)+(x−1)+(y−1)  tip 2:  (1+x_1 )(1+x_2 ) = 1+ x_1 +x_2 +x_1 x_2     using triangle inequality and tip 1:  ∣xy−1∣ = ∣(1−x)(1−y)+(x−1)+(y−1)∣  ≤ ∣(1−x)(1−y)∣+∣(x−1)+(y−1)∣  ≤ ∣(1−x)(1−y)∣+∣x−1∣+∣y−1∣  = ∣1−x∣∣1−y∣+∣x−1∣+∣y−1∣  = ∣x−1∣∣y−1∣+∣x−1∣+∣y−1∣    → ∣xy−1∣≤∣x−1∣∣y−1∣+∣x−1∣+∣y−1∣  →^(+1)  ∣xy−1∣+1 ≤∣x−1∣∣y−1∣+∣x−1∣+∣y−1∣+1  using tip 2 to RHS:  ∣xy−1∣+1≤(1+∣x−1∣)(1+∣y−1∣)
Answerofpart2:tip1:xy1=(1x)(1y)+(x1)+(y1)tip2:(1+x1)(1+x2)=1+x1+x2+x1x2usingtriangleinequalityandtip1:xy1=(1x)(1y)+(x1)+(y1)(1x)(1y)+(x1)+(y1)(1x)(1y)+x1+y1=1x∣∣1y+x1+y1=x1∣∣y1+x1+y1xy1∣⩽∣x1∣∣y1+x1+y1+1xy1+1⩽∣x1∣∣y1+x1+y1+1usingtip2toRHS:xy1+1(1+x1)(1+y1)
Commented by mathocean1 last updated on 22/Jan/22
thanks sir.
thankssir.

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