SOLUTION: Can someone please help me solve the inequality.
p^3 - 16p (less than or equal to) 0
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Question 201219: Can someone please help me solve the inequality.
p^3 - 16p (less than or equal to) 0
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given inequality
Factor the left side
Set the left side equal to zero
Set each individual factor equal to zero:
, or
Solve for p in each case:
, or
So our critical values are , and
Now set up a number line and plot the critical values on the number line
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 (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
Start with the given inequality
Plug in
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 and . This will determine if the interval [
] is part of the solution set.
So let's pick
Start with the given inequality
Plug in
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 and to see if the interval [
] is in the solution set
So let's pick
Start with the given inequality
Plug in
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 to see if the interval [
) is in the solution set
So let's pick
Start with the given inequality
Plug in
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:
(] 
[
]
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