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Question Number 193080 by Mastermind last updated on 03/Jun/23
Show that for a,b∈R,  ∣a−b∣≥∣a∣−∣b∣    (Hint: write a=(a−b)+b
Showthatfora,bR,ab∣⩾∣ab(Hint:writea=(ab)+b
Answered by Subhi last updated on 03/Jun/23
∣a∣−∣b∣ = ∣(a−b)+b)∣−∣b∣  note that: ∣a+b∣≤∣a∣+∣b∣  ∣(a−b)+b∣−∣b∣≤∣(a−b)∣+∣b∣−∣b∣=∣a−b∣  ∴ ∣a−b∣≥∣a∣−∣b∣
ab=(ab)+b)bnotethat:a+b∣⩽∣a+b(ab)+bb∣⩽∣(ab)+bb∣=∣abab∣⩾∣ab

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