document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=76649>Question 76649</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need to solve for x. I believe the answer if 4 but I have no idea how to start.<BR>\n" );
document.write( "(2x-3/x-1)-(x-1/x+2)=(2x-5/x+2)+(2-x/1-x) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #54971 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=54971\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Sort of a long problem.  Given:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%282x-3%29%2F%28x-1%29%29-%28%28x-1%29%2F%28x%2B2%29%29=%28%282x-5%29%2F%28x%2B2%29%29%2B%28%282-x%29%2F%281-x%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Notice that the denominators are (x - 1), (x + 2), and (1 - x).  Sure would be nice if<BR>\n" );
document.write( "that denominator (1 - x) was (x - 1) so it was like the other one.  So let's apply a little<BR>\n" );
document.write( "math trick to the last term on the right side.  Let's multiply the numerator and the denominator<BR>\n" );
document.write( "inside the parentheses by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29%2F%28-1%29 ALIGN=MIDDLE  ALT=\"%28-1%29%2F%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Since the numerator and the denominator of that multiplier<BR>\n" );
document.write( "are equal, it is the same as multiplying the fraction inside the parentheses by 1, so it does<BR>\n" );
document.write( "not change anything ... except it reverses the signs of the terms in the parentheses.<BR>\n" );
document.write( "When you do that manipulation, the last term on the right side becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x-2%29%2F%28x-1%29%29 ALIGN=MIDDLE  ALT=\"%28%28x-2%29%2F%28x-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and the equation becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%282x-3%29%2F%28x-1%29%29-%28%28x-1%29%2F%28x%2B2%29%29=%28%282x-5%29%2F%28x%2B2%29%29%2B%28%28x-2%29%2F%28x-1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now we're ready to rumble!  Let's multiply every term on both sides by the product of<BR>\n" );
document.write( "the common terms in the denominator.  In other words, let's multiply everything by the<BR>\n" );
document.write( "product (x-1)*(x+2).  When we do, the problem becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28x-1%29%2A%28x%2B2%29%2A%282x-3%29%2F%28x-1%29%29-%28%28x-1%29%2A%28x%2B2%29%2A%28x-1%29%2F%28x%2B2%29%29=%28%28x-1%29%2A%28x%2B2%29%2A%282x-5%29%2F%28x%2B2%29%29%2B%28%28x-1%29%2A%28x%2B2%29%2A%28x-2%29%2F%28x-1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now we can cancel any terms in the denominator that have the same term in the numerator as<BR>\n" );
document.write( "shown:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cross%28x-1%29%2A%28x%2B2%29%2A%282x-3%29%2Fcross%28x-1%29%29-%28%28x-1%29%2Across%28x%2B2%29%2A%28x-1%29%2Fcross%28x%2B2%29%29=%28%28x-1%29%2Across%28x%2B2%29%2A%282x-5%29%2Fcross%28x%2B2%29%29%2B%28cross%28x-1%29%2A%28x%2B2%29%2A%28x-2%29%2Fcross%28x-1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "And 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%2B2%29%2A%282x-3%29-%28x-1%29%2A%28x-1%29+=+%28x-1%29%2A%282x-5%29%2B%28x%2B2%29%2A%28x-2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is now just an \"ordinary\" algebraic equation.  Let's multiply out the all the terms to<BR>\n" );
document.write( "get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x%5E2%2Bx-6%29-%28x%5E2+-2x+%2B+1%29+=+%282x%5E2+-7x+%2B+5%29+%2B%28x%5E2+-+4%29 ALIGN=MIDDLE  ALT=\"%282x%5E2%2Bx-6%29-%28x%5E2+-2x+%2B+1%29+=+%282x%5E2+-7x+%2B+5%29+%2B%28x%5E2+-+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Remove the parentheses.  If they are preceded by a + sign, just erase them. If they are<BR>\n" );
document.write( "preceded by a minus sign, when you remove them, change the sign of all the terms inside.  When<BR>\n" );
document.write( "you remove the parentheses you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2%2Bx-6-x%5E2+%2B2x-1+=+2x%5E2-7x%2B5%2Bx%5E2+-4 ALIGN=MIDDLE  ALT=\"2x%5E2%2Bx-6-x%5E2+%2B2x-1+=+2x%5E2-7x%2B5%2Bx%5E2+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Grouping like terms on the left and right side results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+-+x%5E2+%2Bx+%2B+2x+-6+-1+=+2x%5E2+%2B+x%5E2+-+7x+%2B5+-4 ALIGN=MIDDLE  ALT=\"2x%5E2+-+x%5E2+%2Bx+%2B+2x+-6+-1+=+2x%5E2+%2B+x%5E2+-+7x+%2B5+-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Combining like terms gives you:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+3x+-+7+=+3x%5E2+-7x+%2B1 ALIGN=MIDDLE  ALT=\"x%5E2+%2B+3x+-+7+=+3x%5E2+-7x+%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Since this is going to be a quadratic equation, let's get it into standard form by moving<BR>\n" );
document.write( "everything to the left side and just have a zero on the right side.  Do this by subtracting<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2+-+7x+%2B+1 ALIGN=MIDDLE  ALT=\"3x%5E2+-+7x+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.  After grouping like terms on the left side, this<BR>\n" );
document.write( "subtraction results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+3x%5E2+%2B+3x+%2B+7x+-7+-+1+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+3x%5E2+%2B+3x+%2B+7x+-7+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Combine like terms to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%5E2+%2B+10x+-+8+=+0 ALIGN=MIDDLE  ALT=\"-2x%5E2+%2B+10x+-+8+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Simplify by dividing both sides by -2:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+5x+%2B4+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+5x+%2B4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This quadratic factors to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x+-+4%29%2A%28x+-+1%29+=+0 ALIGN=MIDDLE  ALT=\"%28x+-+4%29%2A%28x+-+1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and this equation will be true if either factor on the left side is zero.  So one at a <BR>\n" );
document.write( "time set the factors equal to zero and solve for x.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+4+=+0 ALIGN=MIDDLE  ALT=\"x+-+4+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve by adding 4 to both sides to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4 ALIGN=MIDDLE  ALT=\"x+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Next:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+-+1+=+0 ALIGN=MIDDLE  ALT=\"x+-+1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve by adding 1 to both sides to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+1 ALIGN=MIDDLE  ALT=\"x+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "But this last solution has to be ignored.  Why??? Go way back to the original problem and<BR>\n" );
document.write( "note what happens if x = 1. If it does = 1 then a couple of the denominators become<BR>\n" );
document.write( "zero, and division by zero is not allowed.  Therefore, x = 1 is not a valid answer.  The<BR>\n" );
document.write( "only acceptable answer is x = 4.<BR>\n" );
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document.write( "You can check that by returning to the original problem and substituting 4 for x to see if<BR>\n" );
document.write( "the left side of the equation still equals the right side. You should find that it does.<BR>\n" );
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document.write( "Hope this helps you to see your way through the problem.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );