#21 Measure Your Calibre
∣ ∣ ∣ ∣ x − 1 2 x − 1 ∣ ∣ ∣ ∣ > 2
If the set of x satisfy the inequality above is ( a , ∞ ) − { 1 } , where a is in the form d c with c , d are coprime positive integers, find the value of c + d .
More problems: Check your Calibre
The answer is 7.
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1 solution
@Chew-Seong Cheong But at x = 1 , it is not defined...
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x → 1 − lim ∣ ∣ ∣ ∣ x − 1 2 x − 1 ∣ ∣ ∣ ∣ = x → 1 + lim ∣ ∣ ∣ ∣ x − 1 2 x − 1 ∣ ∣ ∣ ∣ = x → 1 lim ∣ ∣ ∣ ∣ x − 1 2 x − 1 ∣ ∣ ∣ ∣ = ∞ > 2
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Thanks for the explanation.
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@Ankit Kumar Jain – Yes that is why I didnt excluded 1 from the set
No, when x = 1 , LHS is undefined, not infinity. This has nothing to do with limits.
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@Pi Han Goh – Thanks. I get it.
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@Chew-Seong Cheong – So shall I edit the problem?
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@Md Zuhair – Wait for the staff to solve it.
For x < 2 1 , ⟹ ∣ ∣ ∣ ∣ x − 1 2 x − 1 ∣ ∣ ∣ ∣ = 1 − x 1 − 2 x = 1 − x 2 − 2 x − 1 = 2 − 1 − x 1 < 2 .
For 2 1 ≤ x < 1 , ⟹ ∣ ∣ ∣ ∣ x − 1 2 x − 1 ∣ ∣ ∣ ∣ = 1 − x 2 x − 1
∣ ∣ ∣ ∣ x − 1 2 x − 1 ∣ ∣ ∣ ∣ ⟹ 1 − x 2 x − 1 2 x − 1 ⟹ x > 2 > 2 > 2 − 2 x > 4 3
For x ≥ 1 , ⟹ ∣ ∣ ∣ ∣ x − 1 2 x − 1 ∣ ∣ ∣ ∣ = x − 1 2 x − 1 = x − 1 2 x − 2 + 1 = 2 + x − 1 1 > 2
Therefore, ∣ ∣ ∣ ∣ x − 1 2 x − 1 ∣ ∣ ∣ ∣ > 2 for x ∈ ( 4 3 , ∞ ) . ⟹ c + d = 3 + 4 = 7 .