document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=692374>Question 692374</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Graph the following system of linear inequalities  {5x-4y&#8805;0<BR>\n" );
document.write( "{5x+7y&#8805;55<BR>\n" );
document.write( "{5x-3y&#8804;70    <BR>\n" );
document.write( " x,y&#8805;0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Thanks for any help, I do not understand this at all.   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #427329 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=427329\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You have 5 inequalities.<BR>\n" );
document.write( "Each has a boundary (such as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x-4y=0 ALIGN=MIDDLE  ALT=\"5x-4y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >) that is included, and can be graphed as a line.<BR>\n" );
document.write( "The other (x,y) pairs that satisfy the inequality are represented by all the points to one side of that graphed line.<BR>\n" );
document.write( "You can use a convenient test point that is not on the line to see which side is in the solution.<BR>\n" );
document.write( "You could graph all those inequalities separately, and find what parts of the x-y plane are common to all those graphs.<BR>\n" );
document.write( "However, the points that satisfy all the inequalities may be part of a polygon limited by all those boundary lines.<BR>\n" );
document.write( "The points where the boundary lines intersect could be the vertices of such a polygon and finding those points may be a better strategy.<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE EASY CONSTRAINTS:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=0 ALIGN=MIDDLE  ALT=\"x%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is graphed as the y-axis (the line with <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 all the points to the right of the y-axis (all have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E0 ALIGN=MIDDLE  ALT=\"x%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E=0 ALIGN=MIDDLE  ALT=\"y%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is graphed as the x-axis (the line with <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;\" >) and all the points above the x-axis (all have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3E0 ALIGN=MIDDLE  ALT=\"y%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >).<BR>\n" );
document.write( "Those 2 constraints tell us we are limited to the first quadrant.<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE INTERSECTION POINTS:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%285x-4y=0%2C5x-3y=70%29 ALIGN=MIDDLE  ALT=\"system%285x-4y=0%2C5x-3y=70%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%285x=4y%2C4y-3y=70%29 ALIGN=MIDDLE  ALT=\"system%285x=4y%2C4y-3y=70%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%285x=280%2Cy=70%29 ALIGN=MIDDLE  ALT=\"system%285x=280%2Cy=70%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=56%2Cy=70%29 ALIGN=MIDDLE  ALT=\"system%28x=56%2Cy=70%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives point A(56,70)<BR>\n" );
document.write( "AND<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%285x-4y=0%2C5x%2B7y=55%29 ALIGN=MIDDLE  ALT=\"system%285x-4y=0%2C5x%2B7y=55%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%285x=4y%2C4y%2B7y=55%29 ALIGN=MIDDLE  ALT=\"system%285x=4y%2C4y%2B7y=55%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%285x=4y%2C11y=55%29 ALIGN=MIDDLE  ALT=\"system%285x=4y%2C11y=55%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%285x=20%2Cy=5%29 ALIGN=MIDDLE  ALT=\"system%285x=20%2Cy=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=4%2Cy=5%29 ALIGN=MIDDLE  ALT=\"system%28x=4%2Cy=5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives point B(4,5)<BR>\n" );
document.write( "Those 2 points can be used to graph boundary line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x-4y=0 ALIGN=MIDDLE  ALT=\"5x-4y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (line AB)<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%285x%2B7y=55%2Cy=0%29 ALIGN=MIDDLE  ALT=\"system%285x%2B7y=55%2Cy=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%285x=55%2Cy=0%29 ALIGN=MIDDLE  ALT=\"system%285x=55%2Cy=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=11%2Cy=0%29 ALIGN=MIDDLE  ALT=\"system%28x=11%2Cy=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives point C(11,0)<BR>\n" );
document.write( "Point (11,0) plus previously found point (4,5) can be used to graph boundary line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%2B7y=55 ALIGN=MIDDLE  ALT=\"5x%2B7y=55\"   DISABLED_style= \"max-width:90%; height:auto;\" > (line BC)<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%285x-3y=70%2Cy=0%29 ALIGN=MIDDLE  ALT=\"system%285x-3y=70%2Cy=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%285x=70%2Cy=0%29 ALIGN=MIDDLE  ALT=\"system%285x=70%2Cy=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=14%2Cy=0%29 ALIGN=MIDDLE  ALT=\"system%28x=14%2Cy=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives point D(14,0)<BR>\n" );
document.write( "Point (14,0) plus previously found point (56,70) can be used to graph boundary line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x-3y=70 ALIGN=MIDDLE  ALT=\"5x-3y=70\"   DISABLED_style= \"max-width:90%; height:auto;\" > (line DA)<BR>\n" );
document.write( " <BR>\n" );
document.write( "All of the above, along with test point P(10,10), can be graphed as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-20%2C80%2C-20%2C80%2C%0D%0Agrid%280%29%2C%0D%0Ablue%28circle%284%2C5%2C2%29%29%2Cblue%28circle%2856%2C70%2C2%29%29%2C%0D%0Ablue%28circle%2811%2C0%2C2%29%29%2Cblue%28circle%2814%2C0%2C2%29%29%2C%0D%0Ablue%28line%2864%2C80%2C-16%2C-20%29%29%2Cred%28line%282%2C-20%2C62%2C80%29%29%2C%0D%0Agreen%28line%28-17%2C20%2C39%2C-20%29%29%2Cred%28circle%2810%2C10%2C2%29%29%2Clocate%2813%2C13%2CP%29%2C%0D%0Alocate%2860%2C75%2CA%29%2Clocate%282%2C12%2CB%29%2C%0D%0Alocate%286%2C0%2CC%29%2Clocate%2817%2C6%2CD%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Substituted, P's coordinates satisfy all the inequalities.<BR>\n" );
document.write( "All the points on the same side of all boundary lines as point P are solutions.<BR>\n" );
document.write( "The solution to the system is the quadrilateral ABCD, including its boundaries. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );