document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209876>Question 1209876</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the largest value of x where the plots of<BR>\n" );
document.write( "f(x) = - \frac{2x + 5}{x + 3} and g(x) = \frac{12}{x - 1}<BR>\n" );
document.write( "intersect.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850964 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850964\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To find the intersection points of $f(x)$ and $g(x)$, we set $f(x) = g(x)$:\r<BR>\n" );
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document.write( "$$ - \frac{2x + 5}{x + 3} = \frac{12}{x - 1} $$\r<BR>\n" );
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document.write( "Multiply both sides by $(x + 3)(x - 1)$ to eliminate the denominators:\r<BR>\n" );
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document.write( "$$ -(2x + 5)(x - 1) = 12(x + 3) $$<BR>\n" );
document.write( "$$ -(2x^2 - 2x + 5x - 5) = 12x + 36 $$<BR>\n" );
document.write( "$$ -(2x^2 + 3x - 5) = 12x + 36 $$<BR>\n" );
document.write( "$$ -2x^2 - 3x + 5 = 12x + 36 $$<BR>\n" );
document.write( "$$ 0 = 2x^2 + 15x + 31 $$\r<BR>\n" );
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document.write( "Now, we use the quadratic formula to solve for $x$:\r<BR>\n" );
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document.write( "$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$\r<BR>\n" );
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document.write( "In this case, $a = 2$, $b = 15$, and $c = 31$.\r<BR>\n" );
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document.write( "$$ x = \frac{-15 \pm \sqrt{15^2 - 4(2)(31)}}{2(2)} $$<BR>\n" );
document.write( "$$ x = \frac{-15 \pm \sqrt{255 - 248}}{4} $$<BR>\n" );
document.write( "$$ x = \frac{-15 \pm \sqrt{7}}{4} $$\r<BR>\n" );
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document.write( "We have two solutions:\r<BR>\n" );
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document.write( "$$ x_1 = \frac{-15 + \sqrt{7}}{4} $$<BR>\n" );
document.write( "$$ x_2 = \frac{-15 - \sqrt{7}}{4} $$\r<BR>\n" );
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document.write( "We want to find the largest value of $x$, so we choose the solution with the plus sign:\r<BR>\n" );
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document.write( "$$ x_1 = \frac{-15 + \sqrt{7}}{4} $$\r<BR>\n" );
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document.write( "We approximate the value of $\sqrt{7} \approx 2.646$.\r<BR>\n" );
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document.write( "$$ x_1 \approx \frac{-15 + 2.646}{4} \approx \frac{-12.354}{4} \approx -3.0885 $$<BR>\n" );
document.write( "$$ x_2 \approx \frac{-15 - 2.646}{4} \approx \frac{-17.646}{4} \approx -4.4115 $$\r<BR>\n" );
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document.write( "Therefore, the largest value of $x$ is:\r<BR>\n" );
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document.write( "$$ x = \frac{-15 + \sqrt{7}}{4} $$\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{\frac{-15 + \sqrt{7}}{4}}$<BR>\n" );
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