Question 151505
{{{2-x-x^2>0}}} Start with the given inequality.



{{{-x^2-x+2>0}}} Rearrange the terms.



{{{-x^2-x+2>0}}} Rearrange the terms.



{{{-(x+2)(x-1)>0}}}  Factor the left side (note: if you need help with factoring, check out this <a href=http://www.algebra.com/algebra/homework/playground/change-this-name4450.solver>solver</a>)




{{{-(x+2)(x-1)=0}}} Set the left side equal to zero



Set each individual factor equal to zero:


{{{x+2=0}}} or  {{{x-1=0}}} 


Solve for x in each case:


{{{x=-2}}} or  {{{x=1}}} 



So our critical values are {{{x=-2}}} and  {{{x=1}}} 


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


{{{number_line( 600, -10, 10,-2,1)}}}




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 {{{-2}}} (notice how it's to the left of the leftmost endpoint):


So the interval that we are testing is <font size="8">(</font>*[Tex \LARGE -\infty,-2]<font size="8">)</font>



So let's pick {{{x=-3}}} to test the interval


{{{-x^2-x+2>0}}} Start with the given inequality



{{{-(-3+2)(-3-1)> 0}}} Plug in {{{x=-3}}}



{{{-4> 0}}} Evaluate and simplify the left side


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



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Let's pick a test value that is in between {{{-2}}} and {{{1}}}:



So the interval that we are testing is <font size="8">(</font>*[Tex \LARGE -2,1]<font size="8">)</font>


So let's pick {{{x=-0}}} to test the interval


{{{-x^2-x+2>0}}} Start with the given inequality



{{{-(-0+2)(-0-1)> 0}}} Plug in {{{x=-0}}}



{{{2> 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 <font size="8">(</font>*[Tex \LARGE -2,1]<font size="8">)</font>



   



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



So the interval that we are testing is <font size="8">(</font>*[Tex \LARGE 1,\infty]<font size="8">)</font>



So let's pick {{{x=2}}}


{{{-x^2-x+2>0}}} Start with the given inequality



{{{-(2+2)(2-1)> 0}}} Plug in {{{x=2}}}



{{{-4> 0}}} Evaluate and simplify the left side


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



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Summary:


So the solution in interval notation is:



<font size="8">(</font>*[Tex \LARGE -2,1]<font size="8">)</font>






Here's a graph to visually verify the answer:


{{{ graph( 500, 500, -10, 10, -10, 10, -(x+2)(x-1)) }}}