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Find all x such that \(x^2+5x<6\). Express your answer in interval notation.

 Jul 26, 2019
 #1
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+3

x2 + 5x  <  6

                            Subtract  6  from both sides of the inequality.

x2 + 5x - 6  <  0

                            Let's find what values of  x  make  x2 + 5x - 6  equal  0

x2 + 5x - 6  =  0

                            Factor the left side. What two numbers add to  5  and multiply to  -6 ?   -1  and  +6

(x - 1)(x + 6) = 0

                            Set each factor equal to zero and solve for  x

x - 1  =  0     or     x + 6  =  0

 

 

x  =  1                   x  =  -6

 

   

Since a graph of   y  =  x2 + 5x - 6  is a parabola, we can be sure that

 

 

the values of  x  that would make  y < 0  fall in one of these two intervals:

 

either   the interval  (-6, 1)   or   the interval  (-∞, -6) U (1, ∞)

 

 

   

To determine which interval is the solution set, let's test a number in both of them.

 

   
0  is a number in the interval  (-6, 1)

 

 

If   x  =  0   then   y  =  (0)2 + 5(0) - 6  =  -6

 

And   -6 < 0   so we know  0  should be included.

 

 

   
2  is a number in the interval  (-∞, -6) U (1, ∞)

 

 

If   x  =  2   then   y  =  (2)2 + 5(2) - 6  =  8

 

And  2  > 0   so we know  2  should not be included.

 

 

   
So we can be sure that   x2 + 5x - 6  <  0   if and only if  x  is in the interval  (-6, 1)  

 

Check: https://www.desmos.com/calculator/ovccl7zguc

 Jul 26, 2019
 #2
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+1

thanks hectictar

 Jul 27, 2019

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