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Question Number 1543 by 112358 last updated on 17/Aug/15

Let a, b∈R. Show that ∣∣a∣−∣b∣∣≤∣a−b∣.

Leta,b∈R.Showthat∣∣a∣−∣b∣∣⩽∣a−b∣.

Answered by Rasheed Soomro last updated on 18/Aug/15

Alternative way , technically simpler. ∣∣a∣−∣b∣∣≤∣a−b∣......................(I) Let ∣a∣ = x and ∣b∣ = y , Clearly x,y≥0 ∣a∣=x ⇒ a=±x and ∣b∣=±y By substituting in (I) we have four simpler inequalities: ∣ x−y ∣≤ ∣ (+x)−(+y) ∣= ∣ x−y ∣......................(II) ≤ ∣ (+x)−(−y) ∣=∣ x+y ∣.......(III) ≤∣ (−x)−(+y) ∣=∣−(x+y)∣= ∣ x+y ∣..........(IV) ≤∣ (−x)−(−y) ∣=∣ −(x−y) ∣= ∣ x−y ∣ .....(V) (II) and (V) are obvious.For (III) and (IV) we proceed as follow x−y≤x+y for x and y are both +ve. ∴ ∣ x−y ∣ ≤ ∣x + y∣ Since (II) , (III) , (IV) and (V) are true. It implies that ∣∣a∣−∣b∣∣≤∣a−b∣

Alternativeway,technicallysimpler.∣∣a∣−∣b∣∣⩽∣a−b∣......................(I)Let∣a∣=xand∣b∣=y,Clearlyx,y⩾0∣a∣=x⇒a=±xand∣b∣=±yBysubstitutingin(I)wehavefoursimplerinequalities:∣x−y∣⩽∣(+x)−(+y)∣=∣x−y∣......................(II)⩽∣(+x)−(−y)∣=∣x+y∣.......(III)⩽∣(−x)−(+y)∣=∣−(x+y)∣=∣x+y∣..........(IV)⩽∣(−x)−(−y)∣=∣−(x−y)∣=∣x−y∣.....(V)(II)and(V)areobvious.For(III)and(IV)weproceedasfollowx−y⩽x+yforxandyareboth+ve.∴∣x−y∣⩽∣x+y∣Since(II),(III),(IV)and(V)aretrue.Itimpliesthat∣∣a∣−∣b∣∣⩽∣a−b∣

Answered by 123456 last updated on 17/Aug/15

we have that 0≤∣a+b∣≤∣a∣+∣b∣ 0≤∣a+(b−a)∣≤∣a∣+∣b−a∣⇒−∣a∣≤∣b−a∣ 0≤∣b∣≤∣a∣+∣b−a∣⇒∣a∣≤∣a+b∣≤∣a∣+∣b∣≤2∣a∣+∣b−a∣ −∣a∣≤∣b∣−∣a∣≤∣b−a∣ we have −∣a∣≤∣a∣≤∣b−a∣⇒−∣b−a∣≤−∣a∣≤∣a∣ −∣b−a∣≤−∣a∣≤∣b∣−∣a∣≤∣b−a∣ ∣∣b∣−∣a∣∣≤∣b−a∣

wehavethat0⩽∣a+b∣⩽∣a∣+∣b∣0⩽∣a+(b−a)∣⩽∣a∣+∣b−a∣⇒−∣a∣⩽∣b−a∣0⩽∣b∣⩽∣a∣+∣b−a∣⇒∣a∣⩽∣a+b∣⩽∣a∣+∣b∣⩽2∣a∣+∣b−a∣−∣a∣⩽∣b∣−∣a∣⩽∣b−a∣wehave−∣a∣⩽∣a∣⩽∣b−a∣⇒−∣b−a∣⩽−∣a∣⩽∣a∣−∣b−a∣⩽−∣a∣⩽∣b∣−∣a∣⩽∣b−a∣∣∣b∣−∣a∣∣⩽∣b−a∣

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