Question 138678
{{{y=-0.01x^2+5x-400}}} Start with the given equation



{{{0=-0.01x^2+5x-400}}} Set the equation equal to zero



{{{x=100}}} or {{{x=400}}} Solve for x (if you need help with this step, let me know). So the critical values are {{{x=100}}} and {{{x=400}}}




{{{-0.01x^2+5x-400>0}}} Set the expression greater than zero





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


{{{number_line( 600, 80, 420,100,400)}}}




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


So let's pick {{{x=99}}}


{{{-0.01x^2+5x-400>0}}} Start with the given inequality



{{{-0.01(99)^2+5(99)-400> 0}}} Plug in {{{x=99}}}



{{{-3.01> 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.



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




Let's pick a test value that is in between {{{100}}} and {{{400}}}:


So let's pick {{{x=250}}}


{{{-0.01x^2+5x-400>0}}} Start with the given inequality



{{{-0.01(250)^2+5(250)-400> 0}}} Plug in {{{x=250}}}



{{{225> 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 100,400]<font size="8">)</font>



   



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




Let's pick a test value that is greater than {{{400}}} (notice how it's to the right of the rightmost endpoint):


So let's pick {{{x=401}}}


{{{-0.01x^2+5x-400>0}}} Start with the given inequality



{{{-0.01(401)^2+5(401)-400> 0}}} Plug in {{{x=401}}}



{{{-3.01> 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.



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






Summary:


So the solution in interval notation is:



<font size="8">(</font>*[Tex \LARGE 100,400]<font size="8">)</font>




So the company needs to sell at least 100 and at most 400 widgets to get a positive profit.