document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=83028>Question 83028</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following system of linear inequalities by graphing.<BR>\n" );
document.write( "x – y < 3<BR>\n" );
document.write( "x + 2y >6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #59584 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=59584\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Lets graph the equations first\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y=3 ALIGN=MIDDLE  ALT=\"x-y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simply replace the inequality sign with an equal sign<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y=6 ALIGN=MIDDLE  ALT=\"x%2B2y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So lets graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y=3 ALIGN=MIDDLE  ALT=\"x-y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > first\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=graphing_linear_equations.solver>Graphing Linear Equations</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax-1%2Ay=3 ALIGN=MIDDLE  ALT=\"1%2Ax-1%2Ay=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >Start with the given equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ay=3-1%2Ax ALIGN=MIDDLE  ALT=\"-1%2Ay=3-1%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax ALIGN=MIDDLE  ALT=\"1%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%29%283-1%2Ax%29 ALIGN=MIDDLE  ALT=\"y=%28-1%29%283-1%2Ax%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%29%283%29%2B%281%29%281%29x%29 ALIGN=MIDDLE  ALT=\"y=%28-1%29%283%29%2B%281%29%281%29x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-3%2B%281%29x ALIGN=MIDDLE  ALT=\"y=-3%2B%281%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2Ax-3 ALIGN=MIDDLE  ALT=\"y=1%2Ax-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2Ax-3 ALIGN=MIDDLE  ALT=\"y=1%2Ax-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce any fractions<BR>\n" );
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document.write( "  So the equation is now in slope-intercept form (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx%2Bb ALIGN=MIDDLE  ALT=\"y=mx%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >) where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=1 ALIGN=MIDDLE  ALT=\"m=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the slope) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-3 ALIGN=MIDDLE  ALT=\"b=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the y-intercept)<BR>\n" );
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document.write( "  So to graph this equation lets plug in some points<BR>\n" );
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document.write( "  Plug in x=-6<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2A%28-6%29-3 ALIGN=MIDDLE  ALT=\"y=1%2A%28-6%29-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-6-3 ALIGN=MIDDLE  ALT=\"y=-6-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-9 ALIGN=MIDDLE  ALT=\"y=-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add<BR>\n" );
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document.write( "  So here's one point (-6,-9)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-16%2C5%2C+-19%2C+2%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++blue%28+circle%28+-6%2C-9%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-6%2C-9%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now lets find another point <BR>\n" );
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document.write( "  Plug in x=-5<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2A%28-5%29-3 ALIGN=MIDDLE  ALT=\"y=1%2A%28-5%29-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5-3 ALIGN=MIDDLE  ALT=\"y=-5-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-8 ALIGN=MIDDLE  ALT=\"y=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add<BR>\n" );
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document.write( "  So here's another point (-5,-8). Add this to our graph<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-16%2C5%2C+-19%2C+2%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++blue%28+circle%28+-5%2C-8%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-5%2C-8%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+-6%2C-9%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-6%2C-9%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now draw a line through these points<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-16%2C5%2C+-19%2C+2%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++line%28-16-10%2C1%2A%28-16-10%29%2B-3%2C5%2B10%2C1%2A%285%2B10%29%2B-3%29%2C%0D%0A++++blue%28+circle%28+-5%2C-8%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-5%2C-8%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+-6%2C-9%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-6%2C-9%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > So this is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2Ax-3 ALIGN=MIDDLE  ALT=\"y=1%2Ax-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points (-6,-9) and (-5,-8)<BR>\n" );
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document.write( "  So from the graph we can see that the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F1 ALIGN=MIDDLE  ALT=\"1%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which tells us that in order to go from point to point we have to start at one point and go  up 1 units and to the right 1 units to get to the next point) the y-intercept is (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >)and the x-intercept is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,0) . So all of this information verifies our graph.<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  We could graph this equation another way. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-3 ALIGN=MIDDLE  ALT=\"b=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > this tells us that the y-intercept (the point where the graph intersects with the y-axis) is (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >). <BR>\n" );
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document.write( "  So we have one point (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-10%2C11%2C+-13%2C+8%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++%0D%0A++++blue%28+circle%28+0%2C-3%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C-3%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  Now since the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F1 ALIGN=MIDDLE  ALT=\"1%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that in order to go from point to point we can use the slope to do so. So starting at (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >), we can go up 1 units <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28++600%2C+600%2C+-10%2C11%2C+-13%2C+8%2C%0D%0A++++grid%281%29%2C%0D%0A++++blue%28+circle%28+0%2C-3%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C-3%2C+.1%2C+1.5+%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+-3%2B%281%2F2%29%2C+2%2C+1%2C+90%2C+270+%29+%29%0D%0A++%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  and to the right 1 units to get to our next point<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28++600%2C+600%2C+-10%2C11%2C+-13%2C+8%2C%0D%0A++++grid%281%29%2C%0D%0A++++blue%28+circle%28+0%2C-3%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C-3%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+1%2C-2%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+1%2C-2%2C+.1%2C+1.5+%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+-3%2B%281%2F2%29%2C+2%2C+1%2C+90%2C+270+%29+%29%2C%0D%0A++blue%28+arc%28+%281%2F2%29%2C+-2%2C+1%2C+2%2C+180%2C+360+%29+%29%0D%0A++%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now draw a line through those points to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2Ax-3 ALIGN=MIDDLE  ALT=\"y=1%2Ax-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-10%2C11%2C+-13%2C+8%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++line%28-10-10%2C1%2A%28-10-10%29%2B-3%2C11%2B10%2C1%2A%2811%2B10%29%2B-3%29%2C%0D%0A++++blue%28+circle%28+1%2C-2%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+1%2C-2%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+0%2C-3%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C-3%2C+.1%2C+1.5+%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+-3%2B%281%2F2%29%2C+2%2C+1%2C+90%2C+270+%29+%29%2C%0D%0A++blue%28+arc%28+%281%2F2%29%2C+-2%2C+1%2C+2%2C+180%2C+360+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > So this is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=1%2Ax-3 ALIGN=MIDDLE  ALT=\"y=1%2Ax-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points (0,-3) and (1,-2)<BR>\n" );
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document.write( "Now lets graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y=6 ALIGN=MIDDLE  ALT=\"x%2B2y=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=graphing_linear_equations.solver>Graphing Linear Equations</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B2%2Ay=6 ALIGN=MIDDLE  ALT=\"1%2Ax%2B2%2Ay=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >Start with the given equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ay=6-1%2Ax ALIGN=MIDDLE  ALT=\"2%2Ay=6-1%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax ALIGN=MIDDLE  ALT=\"1%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29%286-1%2Ax%29 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29%286-1%2Ax%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F2%29%286%29-%281%2F2%29%281%29x%29 ALIGN=MIDDLE  ALT=\"y=%281%2F2%29%286%29-%281%2F2%29%281%29x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=6%2F2-%281%2F2%29x ALIGN=MIDDLE  ALT=\"y=6%2F2-%281%2F2%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29%2Ax%2B6%2F2 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29%2Ax%2B6%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29%2Ax%2B3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29%2Ax%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce any fractions<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So the equation is now in slope-intercept form (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx%2Bb ALIGN=MIDDLE  ALT=\"y=mx%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" >) where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-1%2F2 ALIGN=MIDDLE  ALT=\"m=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the slope) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=3 ALIGN=MIDDLE  ALT=\"b=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the y-intercept)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So to graph this equation lets plug in some points<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Plug in x=-8<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29%2A%28-8%29%2B3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29%2A%28-8%29%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=8%2F2%2B3 ALIGN=MIDDLE  ALT=\"y=8%2F2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=14%2F2 ALIGN=MIDDLE  ALT=\"y=14%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=7 ALIGN=MIDDLE  ALT=\"y=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So here's one point (-8,7)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-18%2C4%2C+-4%2C+17%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++blue%28+circle%28+-8%2C7%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-8%2C7%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now lets find another point <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Plug in x=-6<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29%2A%28-6%29%2B3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29%2A%28-6%29%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=6%2F2%2B3 ALIGN=MIDDLE  ALT=\"y=6%2F2%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=12%2F2 ALIGN=MIDDLE  ALT=\"y=12%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=6 ALIGN=MIDDLE  ALT=\"y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So here's another point (-6,6). Add this to our graph<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-18%2C4%2C+-4%2C+17%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++blue%28+circle%28+-6%2C6%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-6%2C6%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+-8%2C7%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-8%2C7%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now draw a line through these points<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-18%2C4%2C+-4%2C+17%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++line%28-18-10%2C-0.5%2A%28-18-10%29%2B3%2C4%2B10%2C-0.5%2A%284%2B10%29%2B3%29%2C%0D%0A++++blue%28+circle%28+-6%2C6%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-6%2C6%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+-8%2C7%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-8%2C7%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > So this is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29%2Ax%2B3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29%2Ax%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points (-8,7) and (-6,6)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So from the graph we can see that the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2 ALIGN=MIDDLE  ALT=\"-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which tells us that in order to go from point to point we have to start at one point and go  down -1 units and to the right 2 units to get to the next point), the y-intercept is (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >)and the x-intercept is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" >,0) . So all of this information verifies our graph.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  We could graph this equation another way. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=3 ALIGN=MIDDLE  ALT=\"b=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > this tells us that the y-intercept (the point where the graph intersects with the y-axis) is (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >). <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So we have one point (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-10%2C12%2C+-8%2C+13%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++%0D%0A++++blue%28+circle%28+0%2C3%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C3%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now since the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F2 ALIGN=MIDDLE  ALT=\"-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that in order to go from point to point we can use the slope to do so. So starting at (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >), we can go down 1 units <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28++600%2C+600%2C+-10%2C12%2C+-8%2C+13%2C%0D%0A++++grid%281%29%2C%0D%0A++++blue%28+circle%28+0%2C3%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C3%2C+.1%2C+1.5+%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+3%2B%28-1%2F2%29%2C+2%2C+-1%2C+90%2C+270+%29+%29%0D%0A++%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  and to the right 2 units to get to our next point<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28++600%2C+600%2C+-10%2C12%2C+-8%2C+13%2C%0D%0A++++grid%281%29%2C%0D%0A++++blue%28+circle%28+0%2C3%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C3%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+2%2C2%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+2%2C2%2C+.1%2C+1.5+%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+3%2B%28-1%2F2%29%2C+2%2C+-1%2C+90%2C+270+%29+%29%2C%0D%0A++blue%28+arc%28+%282%2F2%29%2C+2%2C+2%2C+2%2C+0%2C+180+%29+%29%0D%0A++%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now draw a line through those points to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29%2Ax%2B3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29%2Ax%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-10%2C12%2C+-8%2C+13%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++line%28-10-10%2C-0.5%2A%28-10-10%29%2B3%2C12%2B10%2C-0.5%2A%2812%2B10%29%2B3%29%2C%0D%0A++++blue%28+circle%28+2%2C2%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+2%2C2%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+0%2C3%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C3%2C+.1%2C+1.5+%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+3%2B%28-1%2F2%29%2C+2%2C+-1%2C+90%2C+270+%29+%29%2C%0D%0A++blue%28+arc%28+%282%2F2%29%2C+2%2C+2%2C+2%2C+0%2C+180+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > So this is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29%2Ax%2B3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29%2Ax%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points (0,3) and (2,2)<BR>\n" );
document.write( "  </TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So lets graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x-3 ALIGN=MIDDLE  ALT=\"y=x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29x%2B3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > together\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2C%0D%0A++grid%28+1+%29%2C%0D%0A++graph%28+300%2C+300%2C+-5%2C+5%2C+-5%2C+5%2C+x-3%2C+%28-1%2F2%29x%2B3%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now lets pick the test point (0,0) and evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y%3C3 ALIGN=MIDDLE  ALT=\"x-y%3C3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%29-%280%29%3C3 ALIGN=MIDDLE  ALT=\"%280%29-%280%29%3C3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C3 ALIGN=MIDDLE  ALT=\"0%3C3\"   DISABLED_style= \"max-width:90%; height:auto;\" > true. Shade the region that does contain (0,0) for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y=3 ALIGN=MIDDLE  ALT=\"x-y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this means you shade above the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x-3 ALIGN=MIDDLE  ALT=\"y=x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF=\"http://photobucket.com\" target=\"_blank\"><IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/ineq2-1.jpg\" border=\"0\" alt=\"Photo Sharing and Video Hosting at Photobucket\"></a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now lets use the same test point (0,0) and evaluate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y%3E6 ALIGN=MIDDLE  ALT=\"x%2B2y%3E6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%29%2B2%280%29%3E6 ALIGN=MIDDLE  ALT=\"%280%29%2B2%280%29%3E6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3E6 ALIGN=MIDDLE  ALT=\"0%3E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > false. Shade the region that doesn't contain (0,0) for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y=6 ALIGN=MIDDLE  ALT=\"x%2B2y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this means you shade above the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29x%2B3 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF=\"http://photobucket.com\" target=\"_blank\"><IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/ineq1-1.jpg\" border=\"0\" alt=\"Photo Sharing and Video Hosting at Photobucket\"></a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now the overlapping regions is the complete shading. In other words, every point that satisfies<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-y%3C3 ALIGN=MIDDLE  ALT=\"x-y%3C3\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y%3E6 ALIGN=MIDDLE  ALT=\"x%2B2y%3E6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "is in this region. So this is what the overlapped shaded region looks like:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<a  rel=nofollow HREF=\"http://photobucket.com\" target=\"_blank\"><IMG STYLE=\"max-width:80%;\" src=\"http://i150.photobucket.com/albums/s91/jim_thompson5910/ineq3-1.jpg\" border=\"0\" alt=\"Photo Sharing and Video Hosting at Photobucket\"></a>\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );