document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=187773>Question 187773</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need your help on solving an inequality. I know how to solve it but I don't understand the test points. For example:<BR>\n" );
document.write( "x^2+6x<16<BR>\n" );
document.write( "I know that the problem gets put in standard form<BR>\n" );
document.write( "x^2+6x-16<0<BR>\n" );
document.write( "and then it is factored<BR>\n" );
document.write( "(x+8)(x-2)<BR>\n" );
document.write( "then you set the factors equal to zero for the x intercepts<BR>\n" );
document.write( "x+8=0 x=-8 and x-2=0 x=2\r<BR>\n" );
document.write( "\n" );
document.write( "but I don't understand the test points<BR>\n" );
document.write( "for example<BR>\n" );
document.write( "(-2,8)<BR>\n" );
document.write( "(-8,2)<BR>\n" );
document.write( "(-8,-2)U (8,8)<BR>\n" );
document.write( "(-8,-8)U (2,8)\r<BR>\n" );
document.write( "\n" );
document.write( "please explain </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #140780 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=140780\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You're off to a great start. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now construct a number line with the critical values plotted on the line:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/number_line.png\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice how there are three regions: \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Region #1: To the left of the first critical value -8. The region in interval notation is <font size=\"6\">(</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -\infty,-8\"><font size=\"6\">)</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Region #2: In between the critical values -8 and 2. The region in interval notation is <font size=\"6\">(</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -8,2\"><font size=\"6\">)</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Region #3: To the right of the second value 2. The region in interval notation is <font size=\"6\">(</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2,\infty\"><font size=\"6\">)</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For <i>any</i> region, the graph is either above the x-axis or below the x-axis (no region is a mix of the two or on the x-axis). So all we have to do is plug in test points that represent the three regions to find the solution set.\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( "\n" );
document.write( "Let's see if the first region is part of the solution set.\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=%28x%2B8%29%28x-2%29%3C0 ALIGN=MIDDLE  ALT=\"%28x%2B8%29%28x-2%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the factored expression\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=%28-9%2B8%29%28-9-2%29%3C0 ALIGN=MIDDLE  ALT=\"%28-9%2B8%29%28-9-2%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-9 ALIGN=MIDDLE  ALT=\"x=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > (this value is less than -8 which means that it lies in the first region)\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=%28-1%29%28-11%29%3C0 ALIGN=MIDDLE  ALT=\"%28-1%29%28-11%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\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=11%3C0 ALIGN=MIDDLE  ALT=\"11%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since this inequality is FALSE, this means that ANY x value in this region does NOT satisfy <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x%3C16 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x%3C16\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So we can ignore this region.\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( "Let's see if the second region is part of the solution set.\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=%28x%2B8%29%28x-2%29%3C0 ALIGN=MIDDLE  ALT=\"%28x%2B8%29%28x-2%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the factored expression\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=%28-7%2B8%29%28-7-2%29%3C0 ALIGN=MIDDLE  ALT=\"%28-7%2B8%29%28-7-2%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-7 ALIGN=MIDDLE  ALT=\"x=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > (this value is greater than -8 and less than 2)\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=%281%29%28-9%29%3C0 ALIGN=MIDDLE  ALT=\"%281%29%28-9%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\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=-9%3C0 ALIGN=MIDDLE  ALT=\"-9%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since this inequality is TRUE, this means that ANY x value in this region satisfies <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x%3C16 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x%3C16\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the interval <font size=\"6\">(</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -8,2\"><font size=\"6\">)</font> is part of the solution set.\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( "Let's see if the third region is part of the solution set.\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=%28x%2B8%29%28x-2%29%3C0 ALIGN=MIDDLE  ALT=\"%28x%2B8%29%28x-2%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the factored expression\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=%283%2B8%29%283-2%29%3C0 ALIGN=MIDDLE  ALT=\"%283%2B8%29%283-2%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (this value is greater than 2)\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=%2811%29%281%29%3C0 ALIGN=MIDDLE  ALT=\"%2811%29%281%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\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=11%3C0 ALIGN=MIDDLE  ALT=\"11%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since this inequality is FALSE, this means that ANY x value in this region does NOT satisfy <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x%3C16 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x%3C16\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So we can ignore this region.\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( "Answer: \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the solution set to the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x%3C16 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x%3C16\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <font size=\"6\">(</font><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -8,2\"><font size=\"6\">)</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If the above did not make any sense, then take a look at the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x%5E2%2B6x-16 ALIGN=MIDDLE  ALT=\"y=x%5E2%2B6x-16\"   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=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/Algebra%20dot%20com/graph2.png\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice that the x-intercepts are -8 and 2. The portion of the graph to the left of -8 is ALL above the x-axis. The portion in between -8 and 2 is ALL below the x-axis. Finally, the piece from 2 on to infinity is ALL above the x-axis.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the only portion that satisfies <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B6x-16%3C0 ALIGN=MIDDLE  ALT=\"x%5E2%2B6x-16%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is from -8 to 2. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );