document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=825741>Question 825741</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve\r<BR>\n" );
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document.write( "r+(r^2-5)/(r^2-1)=(r^2+r+2)/(r+1)\r<BR>\n" );
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document.write( "Please I need help with this! Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #497580 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=497580\">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=r%2B%28r%5E2-5%29%2F%28r%5E2-1%29=%28r%5E2%2Br%2B2%29%2F%28r%2B1%29 ALIGN=MIDDLE  ALT=\"r%2B%28r%5E2-5%29%2F%28r%5E2-1%29=%28r%5E2%2Br%2B2%29%2F%28r%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Solving equations with fractions is harder than solving equations without fractions. So we will make the problem easier if we eliminate the fractions as soon as possible.<br><BR>\n" );
document.write( "Fractions in an equation can be eliminated by...<ol><li>Find the lowest common denominator (LCD) of all the fractions (on both sides of the equation).</li><li>Multiply both sides of the equation by the LCD.</li></ol>The first denominator will factor:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2-1+=+%28r%2B1%29%28r-1%29 ALIGN=MIDDLE  ALT=\"r%5E2-1+=+%28r%2B1%29%28r-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The second denominator, r+1, does not factor. Looking at the second denominator and the factored first denominator, we should be able to figure out that the LCD is: (r+1)(r-1). So we will multiply both sides by (r+1)(r-1):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%2B1%29%28r-1%29%28r%2B%28r%5E2-5%29%2F%28r%5E2-1%29%29=%28r%2B1%29%28r-1%29%28%28r%5E2%2Br%2B2%29%2F%28r%2B1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "First we must use the Distributive Property on the left side:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%2B1%29%28r-1%29%28r%29%2B%28r%2B1%29%28r-1%29%28%28r%5E2-5%29%2F%28r%5E2-1%29%29=%28r%2B1%29%28r-1%29%28%28r%5E2%2Br%2B2%29%2F%28r%2B1%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now as we multiply, each denominator will cancel with some part of (r+1)(r-1):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%2B1%29%28r-1%29%28r%29%2Bcross%28%28r%2B1%29%28r-1%29%29%28%28r%5E2-5%29%2Fcross%28%28r%5E2-1%29%29%29=cross%28%28r%2B1%29%29%28r-1%29%28%28r%5E2%2Br%2B2%29%2Fcross%28%28r%2B1%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Leaving:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%2B1%29%28r-1%29%28r%29%2B+1%2A%28r%5E2-5%29=%28r-1%29%28r%5E2%2Br%2B2%29 ALIGN=MIDDLE  ALT=\"%28r%2B1%29%28r-1%29%28r%29%2B+1%2A%28r%5E2-5%29=%28r-1%29%28r%5E2%2Br%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we solve. First we simplify:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%5E2-1%29%28r%29%2B+r%5E2-5=r%2Ar%5E2%2Br%2Ar%2Br%2A2%2B%28-1%29%2Ar%5E2%2B%28-1%29%2Ar%2B%28-1%29%2A2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E3-r%2B+r%5E2-5=r%5E3%2Br%5E2%2B2r%2B%28-r%5E2%29%2B%28-r%29%2B%28-2%29 ALIGN=MIDDLE  ALT=\"r%5E3-r%2B+r%5E2-5=r%5E3%2Br%5E2%2B2r%2B%28-r%5E2%29%2B%28-r%29%2B%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E3%2B+r%5E2-r-5=r%5E3%2Br%2B%28-2%29 ALIGN=MIDDLE  ALT=\"r%5E3%2B+r%5E2-r-5=r%5E3%2Br%2B%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we get a zero on one side. Subtracting the entire right side from both sides:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2-2r-3=0 ALIGN=MIDDLE  ALT=\"r%5E2-2r-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we factor:<BR>\n" );
document.write( "(r-3)(r+1) = 0<BR>\n" );
document.write( "From the Zero Product Property:<BR>\n" );
document.write( "r-3 = 0 or r+1 = 0<BR>\n" );
document.write( "Solving these we get:<BR>\n" );
document.write( "r = 3 or r = -1<br><BR>\n" );
document.write( "Last we check. This is <u>not</u> optional! When both sides of an equation are multiplied by something that might be zero, like (r+1)(r-1), then a check is required. One must make sure that the solution does not make a factor of (r+1)(r-1) equal to zero.<br><BR>\n" );
document.write( "A quick visual check should tell us that if r = 3 (one solution) then neither (r+1) nor (r-1) will be a zero. So this solution checks!<br><BR>\n" );
document.write( "A quick visual check should tell us that if r = -1 (the other solution) then (r+1) will be a zero! So this solution fails the check and must be rejected!<br><BR>\n" );
document.write( "So there is only one solution to this problem: r = 3<br><BR>\n" );
document.write( "P.S. This equation has rational expressions. It is not a rational function. It should be posted in the \"polynomials, rational expressions ...\" category. </SPAN>\n" );
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