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Find all $x$ that satisfy the inequality $(2x+10)(x+3)<(3x+1)(x+6)$. Express your answer in interval notation.

 Mar 6, 2024
 #1
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Simplify as

 

2x^2 + 16x + 30 < 3x^2 + 19x + 6

 

Rearrange as

 

x^2 + 3x - 24 > 0     (1)

 

Express as an equality

 

x^2 + 3x - 24 =  0   

 

x^2 + 3x  + 9/4   =  24 + 9/4

 

(x + 3/2)^2   = 105/4        take  both roots

 

x + 3/2  = sqrt (105) /2                                  x + 3/2   =   -sqrt(105)/2

x = [ -3 + sqrt (105) ] / 2   ≈  3.6                      x  = [ -3 -sqrt (105) ]  / 2 ≈  -6.6

 

Our answer  comes from  either  ( ≈ -6.6 , ≈ 3.6)     or  (-inf, ≈ -6.6) U ( ≈ 3.6 , inf)

 

Testing x =0 in (1)  makes it false

 

So

 

The solution comes  from  (-inf, [-3-sqrt (105) ] / 2)  U [ -3 + sqrt (105) ]  / 2  ,  inf)

 

 

cool cool cool

 Mar 6, 2024

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