f(x)=log_(x-2)((2x+3)/(7-x))<1

Elias Heath 2022-09-27 Answered
  f ( x ) = log x 2 ( 2 x + 3 7 x ) < 1
Which is the solution of   f ( x ) < 1 ? I did:
  2 x + 3 7 x > 0
  x < 7 results in   x > 3 2 whereas   x > 7 results in   x < 3 2 so x=(   3 2 ,7) and also   x 2 > 0 so   x > 2
  x = ( 2 , 7 ) { 3 }
So after finding out where f is defined, I did this:
log x 2 ( 2 x + 3 7 x ) < log x 2 ( x 2 )
  2 x + 3 7 x < x 2
  x 2 7 x + 17 < 0
And I figured I must have done a mistake somewhere as   d < 0 and   x 2 's coefficient is   > 0 so   f would be   > 0. Could you let me know where is my mistake? The correct answer is   x = ( 2 , 3 )
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Answers (2)

Emaidedip6g
Answered 2022-09-28 Author has 11 answers
log ( x 2 ) ( 2 x + 3 7 x ) < 1
Hint
1 ) x 2 > 0 and x 2 1
2 ) 2 x + 3 7 x > 0
2 ) If x 2 > 1 then
2 x + 3 7 x < x 2
3 ) If 0 < x 2 < 1 then
2 x + 3 7 x > x 2
Now you can solve it.
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tarjetaroja2t
Answered 2022-09-29 Author has 2 answers
The domain gives 2 < x < 7, x 3 and since 2 x + 3 7 x x 2
we get the answer without any cases immediately by the intervals method:
( 2 , 3 ) .
Done!
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