Question Number 100666 by bobhans last updated on 28/Jun/20

find solution set of inequality  (log _2 x −2)^(3x−1)  < (log _2 x−2)^(3−x)

Commented byRasheed.Sindhi last updated on 28/Jun/20

(log _2 x −2)^(3x−1)  < (log _2 x−2)^(3−x)   log _2 x −2≠0,1⇒3x−1<3−x  ⇒4x<4⇒x<1

Commented bybramlex last updated on 28/Jun/20

i think not correct sir

Commented byRasheed.Sindhi last updated on 28/Jun/20

You′re right sir!

Answered by bramlex last updated on 28/Jun/20

⇔(log _2 x−2−1)(3x−1−(3−x))<0  (log _2 x−3)(4x−4)<0   4(log _2 x−3)(x−1) <0  case 1 ⇒ log _2 x−3<0 ∧ x−1>0  x<8 ∧x>1 ⇒1<x<8  case 2 ⇒ log _2 x−3>0 ∧ x−1<0  x >8 ∧ x<1 ⇒x = ∅  solution (1)∪(2) ⇒ 1 < x < 8

Commented bybemath last updated on 28/Jun/20

sir bramlex it should be (log _2 x−2−1)  not (log _2 x−2+1)

Commented bybramlex last updated on 28/Jun/20

oo yes..your are right

Answered by 1549442205 last updated on 28/Jun/20

  the condition for the given inequality   is defined as { ((x>0)),((log_2 x−2>0)) :}  ⇔x>4.Then  The given inequality is equivalent to  [log_2 x−2−1)[(3x−1)−(3−x)]<0  ⇔(log_2 x−3)(4x−4)<0⇔(log_2 x−3)(x−1)<0  ⇔[_( { ((log_2 x−3<0)),((x−1>0)) :}  ⇔ { ((4<x<8)),((x>1)) :}   ⇔4<x<8) ^( { ((log_2 x−3>0)),((x−1<0)) :}  ⇔ { ((x>8)),((x<1)) :}   ⇒has no solutions)   Thus,solution set of the inequality is  the interval (4,8)