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.


Example 3: $\displaystyle \frac{4x-5}{2x+3}=\displaystyle \frac{3x+8}{3x-5}+\displaystyle \frac{x+11}{x-7}$



Recall that you cannot divide by zero. Therefore, the first fraction is valid if , $\quad x\neq -\displaystyle \frac{3}{2},$ the second fraction is valid if $x\neq \displaystyle \frac{5}{3},\quad $ and the third fraction is valid is$ x\neq 7$. If $-\displaystyle \frac{3}{2},$
$\displaystyle \frac{5}{3},$ or 7 turn out to be the solutions, you must discard them as extraneous solutions.


Multiply both sides by the least common multiple $\left( x-7\right) \left( 2x+3\right) \left( 3x-5\right) \qquad $(the smallest number that all the denominators will divide into evenly). This step will eliminate all the denominators in the equation. The resulting equation may be equivalent (same solutions as the original equation) or it may not be equivalent (extraneous solutions),


\begin{eqnarray*}\displaystyle \frac{4x-5}{2x+3} &=&\displaystyle \frac{3x+8}{3x... ...laystyle \frac{3x+8}{3x-5}+\displaystyle \frac{x+11}{x-7}\right) \end{eqnarray*}


\begin{eqnarray*}\left( x-7\right) \left( 2x+3\right) \left( 3x-5\right) \left( ... ...\left( 3x-5\right) \left( \displaystyle \frac{x+11 }{x-7}\right) \end{eqnarray*}


which is equivalent to


\begin{eqnarray*}\frac{\left( x-7\right) \left( 2x+3\right) \left( 3x-5\right) }... ...t( 3x-5\right) }{1}\left( \displaystyle \frac{x+11}{x-7}\right) \end{eqnarray*}


which can be rewritten as


\begin{eqnarray*}\frac{\left( x-7\right) \left( 2x+3\right) \left( 3x-5\right) \... ...ght) \left( 3x-5\right) \left( x+11\right) }{\left( x-7\right) } \end{eqnarray*}


which can be rewritten again as


\begin{eqnarray*}\frac{\left( 2x+3\right) }{\left( 2x+3\right) }\cdot \frac{\lef... ...ac{\left( 2x+3\right) \left( 3x-5\right) \left( x+11\right) }{1} \end{eqnarray*}


which can be rewritten yet again as


\begin{eqnarray*}\cdot \frac{\left( x-7\right) \left( 3x-5\right) \left( 4x-5\ri... ...right) +\left( 2x+3\right) \left( 3x-5\right) \left( x+11\right) \end{eqnarray*}

\begin{eqnarray*}\left( x-7\right) \left[ \left( 3x-5\right) \left( 4x-5\right) ... ...2}+25x+24\right) +\left( 2x+3\right) \left( 3x^{2}+28x-55\right) \end{eqnarray*}

\begin{eqnarray*}\left( 12x^{3}-119x^{2}+270x-175\right) &=&\left( 6x^{3}-17x^{2... ...2x^{3}+48x^{2}-177x-333 \\ && \\ && \\ 0 &=&167x^{2}-447x-158 \end{eqnarray*}



Solve for x using the quadratic formula:


\begin{eqnarray*}167x^{2}-447x-158 &=&0 \\ && \\ && \\ x &=&\frac{-\left( -44... ...ight) } \\ && \\ && \\ x &=&\frac{447\pm \sqrt{305,353}}{334} \end{eqnarray*}


The exact answers are $x=\displaystyle \frac{447\pm \sqrt{305,353}}{334}$ and the approximate answers are x=2.992777 and x=-0.316130.



Check the answer in the original equation.


Check the solution $x=\displaystyle \frac{447+\sqrt{305,353}}{334}\smallskip $ by substituting x=2.992777 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.




It does not check exactly because we rounded the answer. However, it checks enough to tell us that the answer $x=\displaystyle \frac{447+\sqrt{305,353}}{334}\smallskip $ is a solution to the original problem.


Check the solution $x=\displaystyle \frac{447-+\sqrt{305,353}}{334}\smallskip $ by substituting x=-0.31613 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.



  • Left Side: $\qquad \displaystyle \frac{4x-5}{2x+3}$


    \begin{eqnarray*}&\approx &\displaystyle \frac{4\left( -0.31613\right) -5}{2\left( -0.31613\right) +3} \\ && \\ &\approx &-2.64578 \end{eqnarray*}



  • Right Side: $\qquad \displaystyle \frac{3x+8}{3x-5}+\displaystyle \frac{x+11}{x-7}$


    \begin{eqnarray*}&\approx &\displaystyle \frac{3\left( -0.31613\right) +8}{3\lef... ...3\right) +\left( -1.460317\right) \\ && \\ &\approx &-2.645782 \end{eqnarray*}


    It does not check exactly because we rounded the answer. However, it checks enough to tell us that the answer $x=\displaystyle \frac{447-\sqrt{305,353}}{334} \smallskip $ is a solution to the original problem.



    You can also check your answer by graphing $\quad f(x)=\displaystyle \frac{4x-5}{2x+3}- \displaystyle \frac{3x+8}{3x-5}-\displaystyle \frac{x+11}{x-7}\smallskip .\quad $(formed by subtracting the right side of the original equation from the left side). Look to see where the graph crosses the x-axis; the intercept(s) will be the real solution(s). Note that the graph crosses the x-axis in two places: -0.31313 and 2.992777.



    If you would like to work another example, click on Example


    If you would like to test yourself by working some problems similar to this example, click on Problem


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