EQUATIONS INVOLVING FRACTIONS (RATIONAL EQUATIONS)


Note:


If you would like an in-depth review of fractions, click on Fractions.



Solve for x in the following equation.


Problem 5.3c: $\displaystyle \frac{10x+3}{5x-1}=\displaystyle \frac{3x+7}{3x-8}+\displaystyle \frac{ x+2}{x+8}$



Answer: $x=\displaystyle \frac{763\pm \sqrt{600,409}}{60}\approx 25.631011, $-0.197677


Solution:


Recall that you cannot divide by zero. Therefore, the first fraction is valid if , $\quad x\neq \displaystyle \frac{1}{5},$ $x\neq \displaystyle \frac{8}{3},\quad $ and the third fraction is valid is $ x\neq -8$ If $\quad \displaystyle \frac{1}{5},$ $\displaystyle \frac{8}{3}$ or $-8\quad $ turn out to be the solutions, you must discard them as extraneous solutions.




Multiply both sides by the least common multiple $\left( 5x-1\right) \left( 3x-8\right) \left( x+8\right) \qquad $(the smallest number that all the denominators will divide into evenly). This step will eliminate all the denominators.




\begin{eqnarray*}\displaystyle \frac{10x+3}{5x-1} &=&\displaystyle \frac{3x+7}{3... ...playstyle \frac{3x+7}{3x-8}+\displaystyle \frac{x+2}{x+8}\right] \end{eqnarray*}



\begin{eqnarray*}\left( 5x-1\right) \left( 3x-8\right) \left( x+8\right) \left[ ... ...) \left( x+8\right) \left[ \displaystyle \frac{x+2 }{x+8}\right] \end{eqnarray*}


which is equivalent to

\begin{eqnarray*}\frac{\left( 5x-1\right) \left( 3x-8\right) \left( x+8\right) }... ...eft( x+8\right) }{1}\left[ \displaystyle \frac{x+2}{x+8}\right] \end{eqnarray*}


which can be rewritten as

\begin{eqnarray*}\frac{\left( 5x-1\right) \left( 3x-8\right) \left( x+8\right) \... ...right) \left( x+8\right) \left( x+2\right) }{\left( x+8\right) } \end{eqnarray*}


which can be rewritten as

\begin{eqnarray*}\frac{\left( 5x-1\right) }{\left( 5x-1\right) }\cdot \frac{\lef... ...rac{\left( 5x-1\right) \left( 3x-8\right) \left( x+2\right) }{1} \end{eqnarray*}


which can be simplified to

\begin{eqnarray*}1\cdot \frac{\left( 3x-8\right) \left( x+8\right) \left( 10x+3\... ...\right) +\left( 5x-1\right) \left( 3x-8\right) \left( x+2\right) \end{eqnarray*}



\begin{eqnarray*}\left( 3x-8\right) \left[ \left( x+8\right) \left( 10x+3\right)... ...{2}+31x+56\right] +\left( 5x-1\right) \left[ 3x^{2}-2x-16\right] \end{eqnarray*}



\begin{eqnarray*}3x\left[ 10x^{2}+83x+24\right] -8\left[ 10x^{2}+83x+24\right] &... ...^{2}+72x-80x^{2}-664x-192 &=&15x^{3}+155x^{2}+280x-3x^{2}-31x-56 \end{eqnarray*}



\begin{eqnarray*}&&+15x^{3}-10x^{2}-80x-3x^{2}+2x+16 \\ && \\ && \\ 30x^{3}+1... ...30x^{3}+139x^{2}+171x-40 \\ && \\ && \\ 30x^{2}-763x-152 &=&0 \end{eqnarray*}


All the work up to this point is translating the original equation, $\displaystyle \frac{10x+3}{5x-1}=\displaystyle \frac{3x+7}{3x-8}+\displaystyle \frac{ x+2}{x+8}$, into an equation that is easier to solve. The translated equation, 30x2-763x-152=0, can be solved using the quadratic formula.




Solve for x using the quadratic formula $x=\displaystyle \frac{-b\pm \sqrt{b^{2}-4ac}}{2a} $ $a=30,\quad b=-763,\quad c=-152$

\begin{eqnarray*}30x^{2}-763x-152 &=&0 \\ && \\ x &=&\displaystyle \frac{-\lef... ...& \\ && \\ x &=&\displaystyle \frac{763\pm \sqrt{600,409}}{60} \end{eqnarray*}



\begin{eqnarray*}x &=&\displaystyle \frac{763+\sqrt{600,409}}{60}\approx 25.6310... ... &=&\displaystyle \frac{763-\sqrt{600,409}}{60}\approx -0.197677 \end{eqnarray*}



The exacts answers are $x=\displaystyle \frac{763\pm \sqrt{600,409}}{60}$ and the approximate answers are 25.631011 and $-0.197677.\medskip\bigskip \bigskip\medskip $

Check the solution $x=\displaystyle \frac{763+\sqrt{600,409}}{60}\smallskip $ by substituting 25.631011 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.


Since the left side of the equation equals the right side of the equation after the substitution, we have verified that $x=\displaystyle \frac{763+\sqrt{600,409}}{60} \smallskip\approx $25.631011 is a solution.




Check the solution $x=\displaystyle \frac{763-\sqrt{600,409}}{60}$ by substituting -0.197677 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.