document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=110832>Question 110832</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I graph the inequality: x <= 5 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #80819 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=80819\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In order to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=5 ALIGN=MIDDLE  ALT=\"x%3C=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we need to graph the <b>equation</b> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just replace the inequality sign with an equal sign).<BR>\n" );
document.write( "So lets graph the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > (simply draw a vertical line through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C+1000%28x-5%29%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C+1000%28x-5%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Now lets pick a test point, say (0,0). Any point will work, (just make sure the point doesn't lie on the line) but this point is the easiest to work with. Now evaluate the inequality <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=5 ALIGN=MIDDLE  ALT=\"x%3C=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the test point\r<BR>\n" );
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document.write( "Substitute (0,0) into the inequality\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%29%3C=5 ALIGN=MIDDLE  ALT=\"%280%29%3C=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=5 ALIGN=MIDDLE  ALT=\"0%3C=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify\r<BR>\n" );
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document.write( "(note: for some reason, some of the following images do not display correctly in Internet Explorer. So I would recommend the use of <a  rel=nofollow HREF=http://www.mozilla.com/en-US/firefox/>Firefox</a> to see these images.)\r<BR>\n" );
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document.write( "Since this inequality is true, we simply shade the entire region that contains (0,0)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B1%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B2%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B3%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B4%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B5%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B6%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B7%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B8%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B9%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B10%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B11%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B12%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B13%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B14%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B15%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B16%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B17%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B18%29%29%2C%0D%0Agraph%28+500%2C+500%2C+-20%2C+20%2C+-20%2C+20%2C1000%28x-5%29%2C1000%28x-5%2B19%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3C=5 ALIGN=MIDDLE  ALT=\"x%3C=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the boundary (which is the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > in red) and the shaded region (in green) \r<BR>\n" );
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