SOLUTION: Solve the inequality (x + 3)(x + 1)(x � 5) < 0 and write the solution set both in interval notation and in set notation. Show work/explanation.

Question 149234: Solve the inequality (x + 3)(x + 1)(x � 5) < 0 and write the solution set both in interval notation and in set notation. Show work/explanation.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Start with the given inequality

Set the left side equal to zero

Set each individual factor equal to zero:

, or

Solve for x 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):

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 :

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 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 in between and :

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 greater than (notice how it's to the right of the rightmost endpoint):

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 the interval does not work. So this interval is not in our solution set and we can ignore it.

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

Summary:

So the solution in interval notation is:

(

)

(

)

Also, the answer in set notation is

Here's a graph to visually verify our answer:

Graph of

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