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Determine the set of all real x satisfying \($$(x^2+3x-1)^2<9.$$\) Enter your answer in interval notation.

waffles  Nov 14, 2017
 #1
avatar+87604 
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(x^2 + 3x - 1)^2  < 9

 

Let us solve this

 

(x^2 + 3x - 1)^2  = 9

 

For this to be true, either

 

x^2 + 3x - 1  =  3              or            x^2 + 3x - 1  =  -3          so

 

x^2 + 3x - 4  = 0                              x^2 + 3x + 2  = 0

 

(x + 4) ( x - 1)  = 0                            (x + 1) ( x + 2)  = 0

 

x = -4, x= 1                                          x = -1, x = -2

 

So  we have  5  possiible intervals to test

 

(-inf ,  -4)(-4, -2) , (-2, - 1), (-1, 1) and (1, inf) 

 

Picking a test point in each interval, the intervals highlighted in red are the solution intervals

 

Here's the graph to prove this : https://www.desmos.com/calculator/znkgxcywlo

 

 

cool cool cool

CPhill  Nov 14, 2017

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