document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=118165>Question 118165</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solving inequalities with two variables\r<BR>\n" );
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document.write( "I need an example of this and a way to solve it.<BR>\n" );
document.write( "thank you<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #86368 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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/cgi-bin/show-face.mpl?user=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=86368\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "For example, look at the graph of following equality:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B3y=6 ALIGN=MIDDLE  ALT=\"2x+%2B3y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.	\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=Plot_a_graph_of_functions.solver>PLOT a graph of a function</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Plotting a graph:<BR><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+600%2C+400%2C+-15%2C+15%2C+-10%2C+15%2C+%28-2%2F3%29x%2B2+%29 ALIGN=MIDDLE  ALT=\"graph%28+600%2C+400%2C+-15%2C+15%2C+-10%2C+15%2C+%28-2%2F3%29x%2B2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  This solver uses <A  rel=nofollow HREF=/services/rendering/>formula rendering system</A> and here's the actual formula that was plotted:<BR>\n" );
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document.write( "  graph( 600, 400, -15, 15, -10, 15, (-2/3)x+2 )<BR>\n" );
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document.write( "Recall that when a line is drawn in a plane, the line divides the plane into three sets of points:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=two ALIGN=MIDDLE  ALT=\"two\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+half-planes ALIGN=MIDDLE  ALT=\"+half-planes\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=line ALIGN=MIDDLE  ALT=\"line\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+itself ALIGN=MIDDLE  ALT=\"+itself\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
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document.write( "	Now, consider the solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B3y%3C=6 ALIGN=MIDDLE  ALT=\"2x+%2B3y%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.. <BR>\n" );
document.write( "Recall that the solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=one ALIGN=MIDDLE  ALT=\"one\"   DISABLED_style= \"max-width:90%; height:auto;\" > of these <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=half-planes ALIGN=MIDDLE  ALT=\"half-planes\"   DISABLED_style= \"max-width:90%; height:auto;\" > along with the<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+line ALIGN=MIDDLE  ALT=\"+line\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Note that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+%2B3y%3C6 ALIGN=MIDDLE  ALT=\"2x+%2B3y%3C6\"   DISABLED_style= \"max-width:90%; height:auto;\" > would result in the same half-plane;<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+but ALIGN=MIDDLE  ALT=\"+but\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+not ALIGN=MIDDLE  ALT=\"+not\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=the ALIGN=MIDDLE  ALT=\"the\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+line ALIGN=MIDDLE  ALT=\"+line\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "One method for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=determining ALIGN=MIDDLE  ALT=\"determining\"   DISABLED_style= \"max-width:90%; height:auto;\" > the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=correct ALIGN=MIDDLE  ALT=\"correct\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+half-plane ALIGN=MIDDLE  ALT=\"+half-plane\"   DISABLED_style= \"max-width:90%; height:auto;\" > is to \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=test ALIGN=MIDDLE  ALT=\"test\"   DISABLED_style= \"max-width:90%; height:auto;\" >\" the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=inequality ALIGN=MIDDLE  ALT=\"inequality\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the coordinates of a point that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NOT ALIGN=MIDDLE  ALT=\"NOT\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the line. \r<BR>\n" );
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document.write( "If the coordinates <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=satisfy ALIGN=MIDDLE  ALT=\"satisfy\"   DISABLED_style= \"max-width:90%; height:auto;\" > the inequality, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=that ALIGN=MIDDLE  ALT=\"that\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+point ALIGN=MIDDLE  ALT=\"+point\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+and ALIGN=MIDDLE  ALT=\"+and\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+all+ ALIGN=MIDDLE  ALT=\"+all+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=points ALIGN=MIDDLE  ALT=\"points\"   DISABLED_style= \"max-width:90%; height:auto;\" > in that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=half-plane ALIGN=MIDDLE  ALT=\"half-plane\"   DISABLED_style= \"max-width:90%; height:auto;\" > are solutions. \r<BR>\n" );
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document.write( "We then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=shade ALIGN=MIDDLE  ALT=\"shade\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+that ALIGN=MIDDLE  ALT=\"+that\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+half-plane ALIGN=MIDDLE  ALT=\"+half-plane\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "Usually the point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O ALIGN=MIDDLE  ALT=\"O\"   DISABLED_style= \"max-width:90%; height:auto;\" >=(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is the simplest <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=test+ ALIGN=MIDDLE  ALT=\"test+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=point ALIGN=MIDDLE  ALT=\"point\"   DISABLED_style= \"max-width:90%; height:auto;\" >-provided the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=does ALIGN=MIDDLE  ALT=\"does\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+not+ ALIGN=MIDDLE  ALT=\"+not+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pass ALIGN=MIDDLE  ALT=\"pass\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+origin ALIGN=MIDDLE  ALT=\"+origin\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In this example, the (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) satisfies the inequality and gives us the solution:\r<BR>\n" );
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document.write( "On graph above, shade the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=line ALIGN=MIDDLE  ALT=\"line\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+half-plane ALIGN=MIDDLE  ALT=\"+half-plane\"   DISABLED_style= \"max-width:90%; height:auto;\" > which lies on the left from the line.<BR>\n" );
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