document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=86197>Question 86197</A>: <I><SPAN STYLE=\"word-break: break-all;\"> (3 over x-5)+1 divided by  1- (4 over x-5)\r<BR>\n" );
document.write( "\n" );
document.write( "I hope this makes sense.  Thanks for any help you can offer.  I'm stuck on this one.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #62316 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=62316\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are many ways to solve this problem, here are two routes you could take:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Route 1:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29 ALIGN=MIDDLE  ALT=\"%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=x-5 ALIGN=MIDDLE  ALT=\"y=x-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2Fy%2B1%29%2F%281-4%2Fy%29 ALIGN=MIDDLE  ALT=\"%283%2Fy%2B1%29%2F%281-4%2Fy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiply the expression by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2Fy ALIGN=MIDDLE  ALT=\"y%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is a form of 1)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%283%2Fy%2B1%29%2F%281-4%2Fy%29%29%28y%2Fy%29 ALIGN=MIDDLE  ALT=\"%28%283%2Fy%2B1%29%2F%281-4%2Fy%29%29%28y%2Fy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%283%2Fy%2B1%29%2Fy%281-4%2Fy%29 ALIGN=MIDDLE  ALT=\"y%283%2Fy%2B1%29%2Fy%281-4%2Fy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y%2A%283%2Fy%29%2By%2A1%29%2F%28y%2A1-y%284%2Fy%29%29 ALIGN=MIDDLE  ALT=\"%28y%2A%283%2Fy%29%2By%2A1%29%2F%28y%2A1-y%284%2Fy%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2By%29%2F%28y-4%29 ALIGN=MIDDLE  ALT=\"%283%2By%29%2F%28y-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply and simplify\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2B%28x-5%29%29%2F%28%28x-5%29-4%29 ALIGN=MIDDLE  ALT=\"%283%2B%28x-5%29%29%2F%28%28x-5%29-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace y with <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" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%2F%28x-9%29 ALIGN=MIDDLE  ALT=\"%28x-2%29%2F%28x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------------------------\r<BR>\n" );
document.write( "\n" );
document.write( "Route 2:\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29 ALIGN=MIDDLE  ALT=\"%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given expression\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F%28x-5%29%2B%28x-5%29%2F%28x-5%29%29%2F%28%28x-5%29%2F%28x-5%29-4%2F%28x-5%29%29 ALIGN=MIDDLE  ALT=\"%283%2F%28x-5%29%2B%28x-5%29%2F%28x-5%29%29%2F%28%28x-5%29%2F%28x-5%29-4%2F%28x-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Convert each \"1\" into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-5%29%2F%28x-5%29 ALIGN=MIDDLE  ALT=\"%28x-5%29%2F%28x-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%283%2Bx-5%29%2F%28x-5%29%29%2F%28%28x-5-4%29%2F%28x-5%29%29 ALIGN=MIDDLE  ALT=\"%28%283%2Bx-5%29%2F%28x-5%29%29%2F%28%28x-5-4%29%2F%28x-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the fractions\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%283%2Bx-5%29%2F%28x-5%29%29%2A%28%28x-5%29%2F%28x-5-4%29%29 ALIGN=MIDDLE  ALT=\"%28%283%2Bx-5%29%2F%28x-5%29%29%2A%28%28x-5%29%2F%28x-5-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the fractions by flipping the 2nd fraction\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%283%2Bx-5%29%2Fcross%28%28x-5%29%29%29%2A%28cross%28%28x-5%29%29%2F%28x-5-4%29%29 ALIGN=MIDDLE  ALT=\"%28%283%2Bx-5%29%2Fcross%28%28x-5%29%29%29%2A%28cross%28%28x-5%29%29%2F%28x-5-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Notice these terms cancel\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2Bx-5%29%2F%28x-5-4%29 ALIGN=MIDDLE  ALT=\"%283%2Bx-5%29%2F%28x-5-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now combine like terms\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%2F%28x-9%29 ALIGN=MIDDLE  ALT=\"%28x-2%29%2F%28x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which looks just like the original answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So no matter what route you take, the expression\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29 ALIGN=MIDDLE  ALT=\"%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "simplifies to \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%2F%28x-9%29 ALIGN=MIDDLE  ALT=\"%28x-2%29%2F%28x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check:\r<BR>\n" );
document.write( "\n" );
document.write( "One way we could verify this is to graph both expressions as equations like this\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Graph the first equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29 ALIGN=MIDDLE  ALT=\"y=%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just set the expression equal to y)\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%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=y=%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29 ALIGN=MIDDLE  ALT=\"y=%283%2F%28x-5%29%2B1%29%2F%281-4%2F%28x-5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now graph the 2nd equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x-2%29%2F%28x-9%29 ALIGN=MIDDLE  ALT=\"y=%28x-2%29%2F%28x-9%29\"   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=+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+%28x-2%29%2F%28x-9%29%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+%28x-2%29%2F%28x-9%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28x-2%29%2F%28x-9%29 ALIGN=MIDDLE  ALT=\"y=%28x-2%29%2F%28x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice how they trace out the same path. If they were plotted on the same graph, one would overlap another perfectly. This means they are equivalent (minus the discontinuous hole) and this verifies our answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );