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Find all values of $x$ that satisfy
\[5x - 1 < (x + 1)^2 + 7x - 3.\]

 Aug 18, 2023
 #1
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Simplify this as

 

5x - 1  <  x^2 + 2x + 1 + 7x - 3

 

5x - 1  < x^2 + 9x - 2

 

0 <  x^2 - 4x - 1

 

So

 

x^2 - 4x - 1  >   0     

 

This is a  parabola that turns  upward......using the  Quadratic Formula, we can find  where the parabola intersects  the  x axis......the solution will lie on  (-inf , root 1) U (root 2, inf)

 

So

 

Root 1 =    [ 4 - sqrt ( (-4)^2 - 4 (1) (-1) } / 2  =  [  4 - sqrt (20) ] / 2 =   [4 - sqrt (4 * 5) ] / 2  =

[4 - 2sqrt (5) ] / 2 =  2 - sqrt 5

 

And by the conjugate property, Root 2 =  2 + sqrt 5

 

So.....the solution  is (  inf, 2 -sqrt 5)  U  ( 2 + sqrt 5, inf) 

 

 

cool cool cool

 Aug 18, 2023

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