document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=138678>Question 138678</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The profit of a company is modeled by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-0.01x%5E2%2B5x-400 ALIGN=MIDDLE  ALT=\"y=-0.01x%5E2%2B5x-400\"   DISABLED_style= \"max-width:90%; height:auto;\" > how many items must they sell to make a positive profit? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #101191 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=101191\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-0.01x%5E2%2B5x-400 ALIGN=MIDDLE  ALT=\"y=-0.01x%5E2%2B5x-400\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=-0.01x%5E2%2B5x-400 ALIGN=MIDDLE  ALT=\"0=-0.01x%5E2%2B5x-400\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the equation equal to zero\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=100 ALIGN=MIDDLE  ALT=\"x=100\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=400 ALIGN=MIDDLE  ALT=\"x=400\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for x (if you need help with this step, let me know). So the critical values are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=100 ALIGN=MIDDLE  ALT=\"x=100\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=400 ALIGN=MIDDLE  ALT=\"x=400\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.01x%5E2%2B5x-400%3E0 ALIGN=MIDDLE  ALT=\"-0.01x%5E2%2B5x-400%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Set the expression greater than zero\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now set up a number line and plot the critical values on the number line\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=number_line%28+600%2C+80%2C+420%2C100%2C400%29 ALIGN=MIDDLE  ALT=\"number_line%28+600%2C+80%2C+420%2C100%2C400%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So let's pick some test points that are near the critical values and evaluate them.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's pick a test value that is less than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100 ALIGN=MIDDLE  ALT=\"100\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice how it's to the left of the leftmost endpoint):\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So let's pick <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=99 ALIGN=MIDDLE  ALT=\"x=99\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.01x%5E2%2B5x-400%3E0 ALIGN=MIDDLE  ALT=\"-0.01x%5E2%2B5x-400%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given inequality\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.01%2899%29%5E2%2B5%2899%29-400%3E+0 ALIGN=MIDDLE  ALT=\"-0.01%2899%29%5E2%2B5%2899%29-400%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=99 ALIGN=MIDDLE  ALT=\"x=99\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3.01%3E+0 ALIGN=MIDDLE  ALT=\"-3.01%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate and simplify the left side\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "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.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's pick a test value that is in between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100 ALIGN=MIDDLE  ALT=\"100\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400 ALIGN=MIDDLE  ALT=\"400\"   DISABLED_style= \"max-width:90%; height:auto;\" >:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So let's pick <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=250 ALIGN=MIDDLE  ALT=\"x=250\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.01x%5E2%2B5x-400%3E0 ALIGN=MIDDLE  ALT=\"-0.01x%5E2%2B5x-400%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given inequality\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.01%28250%29%5E2%2B5%28250%29-400%3E+0 ALIGN=MIDDLE  ALT=\"-0.01%28250%29%5E2%2B5%28250%29-400%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=250 ALIGN=MIDDLE  ALT=\"x=250\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=225%3E+0 ALIGN=MIDDLE  ALT=\"225%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate and simplify the left side\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.\r<BR>\n" );
document.write( "\n" );
document.write( "   So part our solution in interval notation is <font size=\"8\">(</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 100,400\"><font size=\"8\">)</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "   \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's pick a test value that is greater than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400 ALIGN=MIDDLE  ALT=\"400\"   DISABLED_style= \"max-width:90%; height:auto;\" > (notice how it's to the right of the rightmost endpoint):\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So let's pick <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=401 ALIGN=MIDDLE  ALT=\"x=401\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.01x%5E2%2B5x-400%3E0 ALIGN=MIDDLE  ALT=\"-0.01x%5E2%2B5x-400%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given inequality\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.01%28401%29%5E2%2B5%28401%29-400%3E+0 ALIGN=MIDDLE  ALT=\"-0.01%28401%29%5E2%2B5%28401%29-400%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=401 ALIGN=MIDDLE  ALT=\"x=401\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3.01%3E+0 ALIGN=MIDDLE  ALT=\"-3.01%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate and simplify the left side\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "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.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Summary:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the solution in interval notation is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<font size=\"8\">(</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 100,400\"><font size=\"8\">)</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the company needs to sell at least 100 and at most 400 widgets to get a positive profit. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );