SOLUTION: Can someone please help me solve the inequality. p^3 - 16p (less than or equal to) 0

Algebra ->  Inequalities -> SOLUTION: Can someone please help me solve the inequality. p^3 - 16p (less than or equal to) 0      Log On


   

Question 201219: Can someone please help me solve the inequality.
p^3 - 16p (less than or equal to) 0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

p%5E3-16p%3C=0 Start with the given inequality


p%28p-4%29%28p%2B4%29%3C=0 Factor the left side


p%28p-4%29%28p%2B4%29=0 Set the left side equal to zero


Set each individual factor equal to zero:

p=0, p-4=0 or p%2B4=0

Solve for p in each case:

p=0, p=4 or p=-4


So our critical values are p=0, p=4 and p=-4

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

number_line%28+600%2C+-10%2C+10%2C0%2C4%2C-4%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). This will determine if the interval (] is part of the solution set.


So let's pick p=-5


p%28p-4%29%28p%2B4%29%3C=0 Start with the given inequality


-5%28-5-4%29%28-5%2B4%29%3C=+0 Plug in p=-5


-45%3C=+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 0. This will determine if the interval [] is part of the solution set.

So let's pick p=-2

p%28p-4%29%28p%2B4%29%3C=0 Start with the given inequality


-2%28-2-4%29%28-2%2B4%29%3C=+0 Plug in p=0


24%3C=+0 Evaluate and simplify the left side


Since the inequality is false, this means that we can ignore this interval.

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



Now let's pick a test value that is in between 0 and 4 to see if the interval [] is in the solution set

So let's pick p=2

p%28p-4%29%28p%2B4%29%3C=0 Start with the given inequality


2%282-4%29%282%2B4%29%3C=+0 Plug in p=5


-24%3C=+0 Evaluate and simplify the left side

Since the inequality is true, this means that the interval [] is in the solution set.





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



Now let's pick a test value that is greater than 4 to see if the interval [) is in the solution set

So let's pick p=5

p%28p-4%29%28p%2B4%29%3C=0 Start with the given inequality


5%285-4%29%285%2B4%29%3C=+0 Plug in p=5


45%3C=+0 Evaluate and simplify the left side

Since the inequality is false, this means that we can ignore this interval.

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





Summary:

So the solution in interval notation is:


(] []