SOLUTION: Solve the inequality {{{3x^2+13x+4>0}}}

Algebra ->  Graphs -> SOLUTION: Solve the inequality {{{3x^2+13x+4>0}}}      Log On


   

Question 138675: Solve the inequality 3x%5E2%2B13x%2B4%3E0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
3x%5E2%2B13x%2B4%3E0 Start with the given inequality

%28x%2B4%29%283x%2B1%29%3E0 Factor the left side (note: if you need help with factoring, check out this solver)




%28x%2B4%29%283x%2B1%29=0 Set the left side equal to zero


Set each individual factor equal to zero:

x%2B4=0 or 3x%2B1=0

Solve for x in each case:

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


So our critical values are x=-4 and x=-1%2F3

Now set up a number line and plot the critical values on the number line

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



So let's pick some test points that are near the critical values and evaluate them.


Let's pick a test value that is less than -4 (notice how it's to the left of the leftmost endpoint):

So let's pick x=-5

%28x%2B4%29%283x%2B1%29%3E0 Start with the given inequality


%28-5%2B4%29%283%28-5%29%2B1%29%3E+0 Plug in x=-5


14%3E+0 Evaluate and simplify the left side

Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.
So part our solution in interval notation is ()





---------------------------------------------------------------------------------------------



Let's pick a test value that is in between -4 and -1%2F3:

So let's pick x=-2

%28x%2B4%29%283x%2B1%29%3E0 Start with the given inequality


%28-2%2B4%29%283%28-2%29%2B1%29%3E+0 Plug in x=-2


-10%3E+0 Evaluate and simplify the left side

Since the inequality is false, this means that the interval does not work. So this interval is not in our solution set and we can ignore it.


---------------------------------------------------------------------------------------------



Let's pick a test value that is greater than -1%2F3 (notice how it's to the right of the rightmost endpoint):

So let's pick x=1

%28x%2B4%29%283x%2B1%29%3E0 Start with the given inequality


%281%2B4%29%283%281%29%2B1%29%3E+0 Plug in x=1


20%3E+0 Evaluate and simplify the left side

Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.
So part our solution in interval notation is ()





---------------------------------------------------------------------------------------------





Summary:

So the solution in interval notation is:


() ()