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Question Number 112280 by bemath last updated on 07/Sep/20

If 15 ≤ ∣10−(1/3)a∣ < 20 find a ∈ R

If15⩽∣10−13a∣<20 finda∈R

Answered by john santu last updated on 07/Sep/20

⇒ ∣10−(1/3)a∣ ≥15 ∧ ∣10−(1/3)a∣ < 20 (1) ∣10−(1/3)a∣ ≥ 15 10−(1/3)a ≤ −15 ∪ 10−(1/3)a ≥15 −(1/3)a ≤ −25 ∪ −(1/3)a ≥ 5 a ≥ 75 ∪ a ≤−15 (2) ∣10−(1/3)a∣ < 20 −20 < 10−(1/3)a < 20 −30 < −(1/3)a < 10 −30<a<90 solution (1)∩(2) −30< a≤ −15 ∪ 75 ≤ a < 90

⇒∣10−13a∣⩾15∧∣10−13a∣<20 (1)∣10−13a∣⩾15 10−13a⩽−15∪10−13a⩾15 −13a⩽−25∪−13a⩾5 a⩾75∪a⩽−15 (2)∣10−13a∣<20 −20<10−13a<20 −30<−13a<10 −30<a<90 solution(1)∩(2) −30<a⩽−15∪75⩽a<90

Answered by ajfour last updated on 07/Sep/20

(a−30)^2 ≥45^2 & (a−30)^2 <60^2 ⇒ 45≤ a−30 < 60 or −60 < a−30 ≤ −45

(a−30)2⩾452&(a−30)2<602 ⇒45⩽a−30<60 or−60<a−30⩽−45

Commented bybemath last updated on 07/Sep/20

(a−30)^2 ≥ 45^2

(a−30)2⩾452

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