SOLUTION: Solve the inequality and express the solution in interval notation: x^2 - 6x + 11 <18

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Question 39031: Solve the inequality and express the solution in interval notation:
x^2 - 6x + 11 <18
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Solve the inequality and express the solution in interval notation:
x^2 - 6x + 11 <18
Look at the EQUALITY:
x^2 - 6x + 11 =18
x^2-6x-7=0
(x-7)(x+1)=0
x=7 or x=-1
So look at the interval (-1,7)
Zero is in this interval.
Look at 0^2-6(0)+11<18
This is true.
So the interval of all real numbers
between -1 and 7 is the solution-interval
for the inequality.
This is what it looks like:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-6x%2B-7+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-6%29%5E2-4%2A1%2A-7=64.

Discriminant d=64 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--6%2B-sqrt%28+64+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+64+%29%29%2F2%5C1+=+7
x%5B2%5D+=+%28-%28-6%29-sqrt%28+64+%29%29%2F2%5C1+=+-1

Quadratic expression 1x%5E2%2B-6x%2B-7 can be factored:
1x%5E2%2B-6x%2B-7+=+1%28x-7%29%2A%28x--1%29
Again, the answer is: 7, -1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-6%2Ax%2B-7+%29

Cheers,
Stan H.