We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
124
2
avatar

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

 Jul 26, 2019
 #1
avatar+8778 
+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
avatar
+1

thanks hectictar

 Jul 27, 2019

32 Online Users

avatar
avatar