document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=73730>Question 73730</A>: <I><SPAN STYLE=\"word-break: break-all;\"> equations involving rational expressions.<BR>\n" );
document.write( "solve and check each of the following equations for x.\r<BR>\n" );
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document.write( "x / x - 2  -  x + 1 / x  =  8 / x ^ 2 - 2 x\r<BR>\n" );
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document.write( "so far i have...\r<BR>\n" );
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document.write( "x / x - 2  -  x + 1 / x  =  8 / x ( x - 2 )<BR>\n" );
document.write( "x ( x - 2 )* x / x - 2  -  x ( x - 1 ) * x + 1 / x  =  x ( x - 2 ) * 8 / x ( x - 2 )   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #52955 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=52955\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Your problem statement was a little confusing.  It would have helped if you had used some<BR>\n" );
document.write( "more parentheses to separate the quantities.  But because you provided your work I think I<BR>\n" );
document.write( "was able to sort things out after seeing what you were doing.<BR>\n" );
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document.write( "I think that the problem you are working on is to find a solution to this equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F%28x-2%29+-+%28x%2B1%29%2Fx+=+8%2F%28x%5E2+-+2x%29 ALIGN=MIDDLE  ALT=\"x%2F%28x-2%29+-+%28x%2B1%29%2Fx+=+8%2F%28x%5E2+-+2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You saw that the critical beginning was to factor the denominator on the right side.  Good<BR>\n" );
document.write( "for you. Factoring that denominator results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F%28x-2%29+-+%28x%2B1%29%2Fx+=+8%2F%28x%2A%28x-2%29%29 ALIGN=MIDDLE  ALT=\"x%2F%28x-2%29+-+%28x%2B1%29%2Fx+=+8%2F%28x%2A%28x-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "After this point, I had a little difficulty seeing what you had done, so let me just proceed<BR>\n" );
document.write( "from this point without commenting on your work.  The next step is to put all the terms in<BR>\n" );
document.write( "this equation over a common denominator so you can combine them.  As it turns out, the<BR>\n" );
document.write( "common denominator is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28x-2%29 ALIGN=MIDDLE  ALT=\"x%2A%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the denominator of the right side. <BR>\n" );
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document.write( "To put the first term on the left side over the common denominator you need to multiply it <BR>\n" );
document.write( "by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fx ALIGN=MIDDLE  ALT=\"x%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" > and to put the second term on the left side over the common denominator <BR>\n" );
document.write( "you need to multiply it by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%2F%28x-2%29 ALIGN=MIDDLE  ALT=\"%28x-2%29%2F%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. When you do those multiplications you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Ax%29%2F%28x%2A%28x-2%29%29+-+%28%28x-2%29%2A%28x%2B1%29%29%2F%28x%2A%28x-2%29%29+=+8%2F%28x%2A%28x-2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now if you multiply all three terms in this equation by the common denominator <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2A%28x-2%29 ALIGN=MIDDLE  ALT=\"x%2A%28x-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "then that multiplier appears in the numerator of all three terms and it cancels with the<BR>\n" );
document.write( "common denominator.<BR>\n" );
document.write( "The result is that the problem is reduced to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2Ax%29+-+%28x-2%29%2A%28x%2B1%29+=+8 ALIGN=MIDDLE  ALT=\"%28x%2Ax%29+-+%28x-2%29%2A%28x%2B1%29+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "[Another way of looking at this is that since all the fractions in the equation had the same<BR>\n" );
document.write( "denominator, then the numerators must combine to be equal.]<BR>\n" );
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document.write( "Anyhow, the first term on the left side is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the second term multiplies out<BR>\n" );
document.write( "(using the FOIL method, or whatever method you are familiar with) to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-x-2 ALIGN=MIDDLE  ALT=\"x%5E2-x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+%28x%5E2-x-2%29+=+8 ALIGN=MIDDLE  ALT=\"x%5E2+-+%28x%5E2-x-2%29+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Remove the parentheses by changing the signs of all the terms within them to make the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+x%5E2+%2B+x+%2B+2+=+8 ALIGN=MIDDLE  ALT=\"x%5E2+-+x%5E2+%2B+x+%2B+2+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The first two terms <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2 ALIGN=MIDDLE  ALT=\"-x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > cancel each other out and you are left with:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x+%2B+2+=+8 ALIGN=MIDDLE  ALT=\"+x+%2B+2+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Eliminate the +2 from the left side by adding -2 to both sides and you finally end up with:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+6 ALIGN=MIDDLE  ALT=\"x+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "That's the answer. I hope you could follow the above and that it helps you to find whether<BR>\n" );
document.write( "you were on the right track to get a solution too.<BR>\n" );
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