SOLUTION: Solve {{{(3x^2+13x+4)/(x+6)>0}}}

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Question 135458: Solve %283x%5E2%2B13x%2B4%29%2F%28x%2B6%29%3E0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let's graph y=%283x%5E2%2B13x%2B4%29%2F%28x%2B6%29

+graph%28+500%2C+500%2C+-60%2C+60%2C+-60%2C+60%2C+%283x%5E2%2B13x%2B4%29%2F%28x%2B6%29%29+



So we can see that everything to the right of x=-6 is in the solution set. So part of our solution is x%3E-6



%283x%5E2%2B13x%2B4%29%2F%28x%2B6%29%3E0 Start with the given inequality

3x%5E2%2B13x%2B4=0 Set the numerator equal to zero



%28x%2B4%29%283x%2B1%29=0 Factor the left side (note: if you need help with factoring, check out this solver)



Now set each factor equal to zero:
x%2B4=0 or 3x%2B1=0

x=-4 or x=-1%2F3 Now solve for x in each case


So the critical values are

x=-4 or x=-1%2F3




Now plot the critical values on a number line

number_line%28+600%2C+-10%2C+10%2C+-4%2C-1%2F3+%29



So let's evaluate a point that is to the left of (which is the left most endpoint). Let's evaluate the value x=-5

Start with the given inequality

Plug in x=-5

Evaluate and simplify



Since is true, this means that one part of the solution in interval notation is
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Now let's test a point that is in between the critical values and


Start with the given inequality

Plug in x=-2

Evaluate and simplify


Since is false, this means that the interval does not work. So that means that this interval is not in the solution set and can be ignored.
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So let's evaluate a point that is to the right of (which is the right most endpoint). Let's evaluate the value x=1

Start with the given inequality

Plug in x=1

Evaluate and simplify



Since is true, this means that one part of the solution in interval notation is
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So the answer in interval notation is