SOLUTION: Example 4 on page 352 shows how to do this problem, but I am having problems factoring once I get to the point of setting the 2-x-x^2>0 and then solving for x (factoring is one of

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2-x-x^2>0
-X^2-X+2>0
(-X+1)(X+2)=0
-X+1=0
-X=-1
X=1 ANSWER.
X+2=0
X=-2 ANSWER.

You can put this solution on YOUR website!
Start with the given inequality.


Rearrange the terms.


Rearrange the terms.


Factor the left side (note: if you need help with factoring, check out this solver)



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 the interval that we are testing is ()


So let's pick to test the interval

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.


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Let's pick a test value that is in between and :


So the interval that we are testing is ()

So let's pick to test the interval

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 ()





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


So the interval that we are testing is ()


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.


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Summary:

So the solution in interval notation is:


()





Here's a graph to visually verify the answer: