Solve a system of inequalities { <mtable columnalign="left left" rowspacing=".2em" colu

tinydancer27br

tinydancer27br

Answered question

2022-05-25

Solve a system of inequalities
{ log 2 2 ( log 2 x ) + log 2 log 2 2 x 3 4 | x 2 1 | 3 1 x 2 1
What I've tried:Make substitution t = x 2 1 and solve second inequality:
4 | t | 3 1 t <=> 4 | t | 3 1 t 0
Where I got interval for t: 1 4 t < 0
And then for x: x ( 1 ; 3 2 ] [ 3 2 ; 1 )
But I have no thoughts how to solve first inequality.

Answer & Explanation

Brennen Bishop

Brennen Bishop

Beginner2022-05-26Added 6 answers

log 2 2 ( log 2 x ) + log 2 log 2 2 x 3
log 2 2 ( log 2 x ) + 2 log 2 ( log 2 x ) 3
Determine domain of LHS:
log 2 ( x ) > 0
log 2 ( x ) < 0
x < 1
Also:
log 2 ( x ) > 0
So domain of this inequality is: x ϕ. Inequality doesn't have any solutions.
Avah Knapp

Avah Knapp

Beginner2022-05-27Added 6 answers

the first one seems to be invalid ,since if we set t = log 2 x: log 2 2 ( t ) + 2 log 2 t 3 we see that t 0 in order for ( log 2 2 ( t ) ) 2 to be defined,but this immediatley,disqualifies 2 log 2 t which works for t > 0 so obviously we have a contradiction

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