document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=114567>Question 114567</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0321334337/algebrahomeworkh>Mathematics with Applications</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> 2x is greater or equal to y. I'm not sure how to graph it. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #83375 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20855)</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=83375\">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=2x+%3E=+y ALIGN=MIDDLE  ALT=\"2x+%3E=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=2x ALIGN=MIDDLE  ALT=\"y%3C=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "	<BR>\n" );
document.write( "Think about how you've done inequalities on the number line. For instance, they'd ask you to graph something like x > 2 or y<2\r<BR>\n" );
document.write( "\n" );
document.write( "greater than\" meant \"everything off to the right\" on the number line<BR>\n" );
document.write( "	<BR>\n" );
document.write( "less than\" meant \"everything off to the left\" the number line\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "now do your inequality:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C=2x ALIGN=MIDDLE  ALT=\"y%3C=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "the first step is to find the \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=equals ALIGN=MIDDLE  ALT=\"equals\"   DISABLED_style= \"max-width:90%; height:auto;\" >\" part, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x ALIGN=MIDDLE  ALT=\"y=2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "then graph it\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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>In order to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%2Ax%2B0 ALIGN=MIDDLE  ALT=\"y=2%2Ax%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we only need to plug in two points to draw the line<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So lets plug in some points<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Plug in x=-4<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%2A%28-4%29%2B0 ALIGN=MIDDLE  ALT=\"y=2%2A%28-4%29%2B0\"   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%2B0 ALIGN=MIDDLE  ALT=\"y=-8%2B0\"   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" );
document.write( "  <BR>\n" );
document.write( "  So here's one point (-4,-8)<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+-14%2C7%2C+-18%2C+4%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++blue%28+circle%28+-4%2C-8%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-4%2C-8%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=-3<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%2A%28-3%29%2B0 ALIGN=MIDDLE  ALT=\"y=2%2A%28-3%29%2B0\"   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%2B0 ALIGN=MIDDLE  ALT=\"y=-6%2B0\"   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=-6 ALIGN=MIDDLE  ALT=\"y=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So here's another point (-3,-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+-14%2C7%2C+-18%2C+4%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++blue%28+circle%28+-3%2C-6%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-3%2C-6%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+-4%2C-8%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-4%2C-8%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+-14%2C7%2C+-18%2C+4%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++line%28-14-10%2C2%2A%28-14-10%29%2B0%2C7%2B10%2C2%2A%287%2B10%29%2B0%29%2C%0D%0A++++blue%28+circle%28+-3%2C-6%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-3%2C-6%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+-4%2C-8%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-4%2C-8%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=2%2Ax%2B0 ALIGN=MIDDLE  ALT=\"y=2%2Ax%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points (-4,-8) and (-3,-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=2%2F1 ALIGN=MIDDLE  ALT=\"2%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 2 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=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >)and the x-intercept is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >,0) <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=0 ALIGN=MIDDLE  ALT=\"b=0\"   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=0 ALIGN=MIDDLE  ALT=\"0\"   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=0 ALIGN=MIDDLE  ALT=\"0\"   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%2C11%2C+-10%2C+12%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++%0D%0A++++blue%28+circle%28+0%2C0%2C+0.14%29%29%2C%0D%0A++++blue%28+circle%28+0%2C0%2C+0.14-0.05%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 since the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F1 ALIGN=MIDDLE  ALT=\"2%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=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" >), we can go up 2 units <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%2C11%2C+-10%2C+12%2C%0D%0A++++grid%281%29%2C%0D%0A++++blue%28+circle%28+0%2C0%2C+0.14%29%29%2C%0D%0A++++blue%28+circle%28+0%2C0%2C+0.14-0.05%29+%29%0D%0A++blue%28+arc%28+0%2C+0%2B%282%2F2%29%2C+2%2C+2%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" );
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+-10%2C+12%2C%0D%0A++++grid%281%29%2C%0D%0A++++blue%28+circle%28+1%2C2%2C+0.14%29%29%2C%0D%0A++++blue%28+circle%28+1%2C2%2C+0.14-0.05+%29+%29%2C%0D%0A++++blue%28+circle%28+0%2C0%2C+0.14%29%29%2C%0D%0A++++blue%28+circle%28+0%2C0%2C+0.14-0.05%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+0%2B%282%2F2%29%2C+2%2C+2%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( "  Now draw a line through those points to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%2Ax%2B0 ALIGN=MIDDLE  ALT=\"y=2%2Ax%2B0\"   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%2C11%2C+-10%2C+12%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++line%28-10-10%2C2%2A%28-10-10%29%2B0%2C11%2B10%2C2%2A%2811%2B10%29%2B0%29%2C%0D%0A++++blue%28+circle%28+1%2C2%2C+0.14%29%29%2C%0D%0A++++blue%28+circle%28+1%2C2%2C+0.14-0.05+%29+%29%2C%0D%0A++++blue%28+circle%28+0%2C0%2C+0.14%29%29%2C%0D%0A++++blue%28+circle%28+0%2C0%2C+0.14-0.05%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+0%2B%282%2F2%29%2C+2%2C+2%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=2%2Ax%2B0 ALIGN=MIDDLE  ALT=\"y=2%2Ax%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points (0,0) and (1,2)<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( "You have already graphed the \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=or+ ALIGN=MIDDLE  ALT=\"or+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=equal ALIGN=MIDDLE  ALT=\"equal\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+to ALIGN=MIDDLE  ALT=\"+to\"   DISABLED_style= \"max-width:90%; height:auto;\" >\" part (it is just the line);<BR>\n" );
document.write( " <BR>\n" );
document.write( "now you are ready to do the \"<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y_+less_+than ALIGN=MIDDLE  ALT=\"y_+less_+than\"   DISABLED_style= \"max-width:90%; height:auto;\" >\" part; it means, this is where you need to <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=+one ALIGN=MIDDLE  ALT=\"+one\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+side ALIGN=MIDDLE  ALT=\"+side\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=or ALIGN=MIDDLE  ALT=\"or\"   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=other ALIGN=MIDDLE  ALT=\"other\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
document.write( "\n" );
document.write( "All you need to be sure is do you need <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+ ALIGN=MIDDLE  ALT=\"y+\"   DISABLED_style= \"max-width:90%; height:auto;\" >LESS THAN the line, do you need ABOVE the line, or BELOW the line? \r<BR>\n" );
document.write( "\n" );
document.write( "Naturally, you want below the line. So shade it in: the LINE and BELOW the line.\r<BR>\n" );
document.write( "\n" );
document.write( "the side you shaded is the \"solution region\"<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );