### Rational Equations Worksheets

Equations involving rational expressions are solved in the same way as solving equations with whole numbers. Simply apply the different properties of equality, whichever is most appropriate. It is advised to undo the denominators first then follow the steps in solving equations with whole numbers.

Examples

Solve for the unknown variable.

EASY

\small \begin{align*}\displaystyle &1.\;\;\;\frac{3x}{2}-\frac{4x}{3}=8\\ &2.\;\;\; \frac{5}{x}+\frac{7}{2x}=8\\&3.\;\;\;\frac{2x+6}{x-2}=9 \end{align*}

HARD

\small \begin{align*}\displaystyle &4.\;\;\;\frac{6}{x+2}+\frac{4}{x-2}=\frac{2-8x}{4-x^2}\\ &5.\;\;\; \frac{7}{x-4}-\frac{3}{x+4}=\frac{x+25}{x^2-16} \end{align*}

SOLUTIONS:

\small \begin{align*}\displaystyle &1.\;\;\;\frac{3x}{2}-\frac{4x}{3}=8 \\& \qquad 6\left [ \frac{3x}{2}-\frac{4x}{3} = 8 \right ]&&\textup{Multiply both sides by the LCM of 2 and 3} \\& \qquad \quad \;\; 9x-8x=48 && \textup{Distribute 6 to each term}\\ & \qquad \qquad \quad \therefore x=48 &&\textup{Combine like terms} \end{align*}

\small \begin{align*}\displaystyle &2.\;\;\;\frac{5}{x}+\frac{7}{2x}=8 \\& \qquad 2x\left [ \frac{5}{x}+\frac{7}{2x} = 8 \right ]&&\textup{Multiply both sides by the LCM of x and 2x} \\& \qquad \quad \quad \; 10+7=16x && \textup{Distribute 2x to each term}\\ & \qquad \qquad \quad \; \therefore x=\frac{17}{16} &&\textup{Divide both sides by 16} \end{align*}

\small \begin{align*}\displaystyle &3.\;\;\;\frac{2x+6}{x-2}=9 \\& \qquad \left [ \frac{2x+6}{x-2}= 9 \right ]\left ( x-2 \right )&&\textup{Multiply both sides by}\; x-2 \\& \qquad \quad \; 2x+6=9x-18 && \textup{Distribute}\; x-2\; \textup{to each term}\\& \qquad \quad 2x-9x=-18-6 && \textup{Combine like terms}\\ &\qquad \quad \;\;\; -7x=-24 &&\textup{Divide both sides by -7}\\ & \qquad \qquad \;\; \therefore x=\frac{24}{7} \end{align*}

\small \begin{align*}\displaystyle 4.\;\;\;\frac{6}{x+2}+\frac{4}{x-2}=\frac{2-8x}{4-x^2} \end{align*}

\small \begin{align*}\displaystyle \frac{6}{x+2}+\frac{4}{x-2}&=\frac{2-8x}{4-x^2} \\ \frac{6}{x+2}+\frac{4}{x-2}&=\frac{2-8x}{4-x^2}\left ( \frac{-1}{-1}\right )&& \textup{Multiply}\;\frac{-1}{-1}\; \textup{to right side of the eq.}\\ \frac{6}{x+2}+\frac{4}{x-2}&=\frac{8x-2}{x^2-4}\left ( x+2 \right )\left ( x-2 \right ) &&\textup{Multiply both sides by}\;\left ( x+2 \right )\left ( x-2 \right ) \\6x-12+4x+8&=8x-2&&\textup{Distribute}\;\left ( x+2 \right )\left ( x-2 \right ) \textup{to each term}\\10x-4&=8x-2&&\textup{Combine like terms}\\2x&=2&& \textup{Divide both sides by 2}\\ \therefore x&=1 \end{align*}

\small \begin{align*}5.\;\;\; \displaystyle \frac{7}{x-4}-\frac{3}{x+4}=\frac{x+25}{x^2-16} \end{align*}

\small \begin{align*} \displaystyle &\;\;\;\;\; \frac{7}{x-4}-\frac{3}{x+4}=\frac{x+25}{x^2-16} \\ &\;\;\;\left [\frac{7}{x-4}-\frac{3}{x+4}=\frac{x+25}{x^2-16} \right ]\left ( x-4 \right ) \left ( x+4 \right )\\ &7x+28-3x+12=x+25 \\ &\qquad\qquad \;\; 4x+40=x+25\\&\qquad \qquad \qquad \;\;\; 3x=-15 \\ &\qquad \qquad \qquad \; \therefore x=-5 \end{align*}

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