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log-x-2-10-3x-lt-2-

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Question Number 136256 by EDWIN88 last updated on 20/Mar/21
log _((x−2)) (10−3x) < 2
log(x−2)(10−3x)<2
Answered by liberty last updated on 20/Mar/21
(1) 10−3x > 0 ; 3x−10<0 x < ((10)/3) (2) log _((x−2)) (10−3x) < log _((x−2)) (x−2)^2 (x−2−1)(10−3x−(x−2)^2 ) < 0 ⇔ (x−3)(10−3x−x^2 +4x−4)< 0 (x−3)(−x^2 +x+6)< 0 ⇔(x−3)(x^2 −x−6)>0 (x−3)(x−3)(x+2)>0 (x−3)^2 (x+2)>0 ⇒−2<x<3 ∪ x>3 (3) x−2>0, x>2 The solution set is (2,3)∪(3,((10)/3))
(1)10−3x>0;3x−10<0x<103(2)log(x−2)(10−3x)<log(x−2)(x−2)2(x−2−1)(10−3x−(x−2)2)<0⇔(x−3)(10−3x−x2+4x−4)<0(x−3)(−x2+x+6)<0⇔(x−3)(x2−x−6)>0(x−3)(x−3)(x+2)>0(x−3)2(x+2)>0⇒−2<x<3∪x>3(3)x−2>0,x>2Thesolutionsetis(2,3)∪(3,103)

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