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Solve the inequality x + 3 < x^2 + 2x + 14.  Give your answer in interval notation.

 Jun 11, 2022
 #1
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x + 3 < x^2 + 2x + 14       rearrange as

 

x^2 + x - 11  >  0

 

Let's solve  this

 

x^2  + x - 11 =  0

 

x^2 + x  =  11                complete the square on x

 

x^2  + x + 1/4=    11 + 1/4

 

(x + 1/2)^2  =  45/4       take both roots

 

x + 1/2  =  sqrt (45) / 2      or   x  + 1/2  = -sqrt (45)  / 2             {  sqrt (45)  =  3sqrt (5)  }

 

x = [   3 sqrt (5) - 1  ]  / 2     or       x = [  -3sqrt (5)  - 1 ]  / 2

 

These two answers are the x intercepts  of  a parabola that turns upward

 

Every real number less than the second answer will be in  the  solution and every  real number greater than the first answer will be in the  solutin ....so.....

 

x =  ( -inf ,  [-3 sqrt 5 - 1 ] / 2 )  U    (  [ 3sqrt (5) - 1 ]  / 2  , inf ) 

 

 

cool cool cool

 Jun 11, 2022

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