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Adding and Subtracting Rational Expressions where One Denominator is the LCD

Combine fractions including variables, one-step LCM

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Adding and Subtracting Rational Expressions where One Denominator is the LCD

The length of a garden plot is \begin{align*}\frac{6x^2-5}{2x^2 + 4x - 6}\end{align*}. The width of the plot is \begin{align*}\frac{2x-7}{x+3}\end{align*}. How much longer is the garden plot than it is wide?

Adding and Subtracting Rational Expressions

Recall when two fractions do not have the same denominator. You have to multiply one or both fractions by a number to create equivalent fractions in order to combine them.

\begin{align*}\frac{1}{2} + \frac{3}{4}\end{align*}

Here, 2 goes into 4 twice. So, we will multiply the first fraction by \begin{align*}{\color{red}\frac{2}{2}}\end{align*} to get a denominator of 4. Then, the two fractions can be added.

\begin{align*}{\color{red}\frac{2}{2} \cdot} \frac{1}{2} + \frac{3}{4} = \frac{2}{4} + \frac{3}{4} = \frac{5}{4}\end{align*}

Once the denominators are the same, the fractions can be combined. We will apply this idea to rational expressions in order to add or subtract ones without like denominators.

Add or Subtract the Rational Expressions

Subtract \begin{align*}\frac{3x-5}{2x+8} - \frac{x^2-6}{x+4}\end{align*}.

Factoring the denominator of the first fraction, we have \begin{align*}2(x+4)\end{align*}. The second fraction needs to be multiplied by \begin{align*}{\color{red}\frac{2}{2}}\end{align*} in order to make the denominators the same.

\begin{align*}\frac{3x-5}{2x+8} - \frac{x^2-6}{x+4} &= \frac{3x-5}{2(x+4)} - \frac{x^2-6}{x+4} \cdot {\color{red}\frac{2}{2}} \\ &= \frac{3x-5}{2(x+4)} - \frac{2x^2-12}{2(x+4)}\end{align*}

Now that the denominators are the same, subtract the second rational expression just like in the previous concept.

\begin{align*}&= \frac{3x-5-(2x^2-12)}{2(x+4)} \\ &= \frac{3x-5-2x^2+12}{2(x+4)} \\ &= \frac{-2x^2+3x+7}{2(x+4)}\end{align*}

The numerator is not factorable, so we are done.

Add \begin{align*}\frac{2x-3}{x+5} + \frac{x^2+1}{x^2-2x-35}\end{align*}

Factoring the second denominator, we have \begin{align*}x^2-2x-35=(x+5)(x-7)\end{align*}. So, we need to multiply the first fraction by \begin{align*}{\color{red}\frac{x-7}{x-7}}\end{align*}.

\begin{align*}\overbrace{{\color{red}\frac{(x-7)}{(x-7)}} \cdot \frac{(2x-3)}{(x+5)}}^{FOIL} + \frac{x^2+1}{(x-7)(x+5)} &= \frac{2x^2-17x+21}{(x-7)(x+5)} + \frac{x^2+1}{(x-7)(x+5)} \\ &= \frac{3x^2-17x+22}{(x-7)(x+5)}\end{align*}

Subtract \begin{align*}\frac{7x+2}{2x^2+18x+40} - \frac{6}{x+5}\end{align*}.

Factoring the first denominator, we have \begin{align*}2x^2+18x+40=2(x^2+9x+20)=2(x+4)(x+5)\end{align*}. This is the Lowest Common Denominator, or LCD. The second fraction needs the 2 and the \begin{align*}(x+4)\end{align*}.

\begin{align*}\frac{7x+2}{2x^2+18x+40} - \frac{6-x}{x+5} &= \frac{7x+2}{2(x+5)(x+4)} - \frac{6-x}{x+5}{\color{red}\cdot \frac{2(x+4)}{2(x+4)}} \\ &= \frac{7x+2}{2(x+5)(x+4)} - \frac{2(6-x)(x+4)}{2(x+5)(x+4)} \\ &= \frac{7x+2}{2(x+5)(x+4)} - \frac{48+4x-2x^2}{2(x+5)(x+4)} \\ &= \frac{7x+2-(48+4x-2x^2)}{2(x+5)(x+4)} \\ &= \frac{7x+2-48-4x+2x^2}{2(x+5)(x+4)} \\ &= \frac{2x^2+3x-46}{2(x+5)(x+4)}\end{align*}

Examples

Example 1

Earlier, you were asked how much longer is the garden plot than it is wide. 

We need to subtract the width from the length.

\begin{align*}\frac{6x^2-5}{2x^2 + 4x - 6} - \frac{2x-7}{x+3}\end{align*}

Factoring the first denominator, we have \begin{align*}2x^2+4x-6=(2x-2)(x+3)\end{align*}. So, we need to multiply the second fraction by \begin{align*}{\color{red}\frac{2x-2}{2x-2}}\end{align*}.

\begin{align*}\frac{6x^2-5}{2x^2 + 4x - 6} -\overbrace{{\color{red}\frac{(2x-2)}{(2x-2)}} \cdot \frac{(2x-7)}{(x+3)}}^{FOIL} &= \frac{4x^2-18x+14}{2x^2+4x-6} \\ \frac{6x^2-5}{2x^2 + 4x - 6} - \frac{4x^2-18x+14}{2x^2+4x-6}\\ \frac{6x^2-5-(4x^2-18x+14)}{2x^2+4x-6}\\ \frac{6x^2-5-4x^2+18x-14}{2x^2+4x-6}\\ \frac{2x^2+18x-19}{2x^2+4x-6}\end{align*}

Therefore, the garden plot is \begin{align*}\frac{2x^2+18x-19}{2x^2+4x-6}\end{align*} longer than it is wide

Perform the indicated operation.

Example 2

\begin{align*}\frac{2}{x+1} - \frac{x}{3x+3}\end{align*}

The LCD is \begin{align*}3x+3\end{align*} or \begin{align*}3(x+1)\end{align*}. Multiply the first fraction by \begin{align*}\frac{3}{3}\end{align*}.

\begin{align*}\frac{2}{x+1} - \frac{x}{3x+3} &= \frac{3}{3} \cdot \frac{2}{x+1} - \frac{x}{3(x+1)} \\ &= \frac{6}{3(x+1)} - \frac{x}{3(x+1)} \\ &= \frac{6-x}{3(x+1)}\end{align*}

Example 3

\begin{align*}\frac{x-10}{x^2+4x-24} + \frac{x+3}{x+6}\end{align*}

Here, the LCD \begin{align*}x^2+4x-24\end{align*} or \begin{align*}(x+6)(x-4)\end{align*}. Multiply the second fraction by \begin{align*}\frac{x-4}{x-4}\end{align*}.

\begin{align*}\frac{x-10}{x^2+4x-24} + \frac{x+3}{x+6} &= \frac{x-10}{(x+6)(x-4)} + \frac{x+3}{x+6} \cdot \frac{x-4}{x-4} \\ &= \frac{x-10}{(x+6)(x-4)} + \frac{x^2-x-12}{(x+6)(x-4)} \\ &= \frac{x-10+x^2-x-12}{(x+6)(x-4)} \\ &= \frac{x^2-22}{(x+6)(x-4)}\end{align*}

Example 4

\begin{align*}\frac{3x^2-5}{3x^2-12} + \frac{x+8}{3x+6}\end{align*}

The LCD is \begin{align*}3x^2-12=3(x-2)(x+2)\end{align*}. The second fraction’s denominator factors to be \begin{align*}3x+6=3(x+2)\end{align*}, so it needs to be multiplied by \begin{align*}\frac{x-2}{x-2}\end{align*}.

\begin{align*}\frac{3x^2-5}{3x^2-12} + \frac{x+8}{3x+6} &= \frac{3x^2-5}{3(x-2)(x+2)} + \frac{x+8}{3(x+2)} \cdot \frac{x-2}{x-2} \\ &= \frac{3x^2-5}{3(x-2)(x+2)} + \frac{x^2+6x-16}{3(x-2)(x+2)} \\ &= \frac{3x^2-5+x^2+6x-16}{3(x-2)(x+2)} \\ &= \frac{4x^2+6x-21}{3(x-2)(x+2)}\end{align*}

Review

Find the LCD.

  1. \begin{align*}x, \ 6x\end{align*}
  2. \begin{align*}x, \ x+1\end{align*}
  3. \begin{align*}x+2, \ x-4\end{align*}
  4. \begin{align*}x, \ x-1, \ x^2 - 1\end{align*}

Perform the indicated operations.

  1. \begin{align*}\frac{3}{x} - \frac{5}{4x}\end{align*}
  2. \begin{align*}\frac{x+2}{x+3} + \frac{x-1}{x^2+3x}\end{align*}
  3. \begin{align*}\frac{x}{x-7} - \frac{2x+7}{3x-21}\end{align*}
  4. \begin{align*}\frac{x^2+3x-10}{x^2-4} - \frac{x}{x+2}\end{align*}
  5. \begin{align*}\frac{5x+14}{2x^2-7x-15} - \frac{3}{x-5}\end{align*}
  6. \begin{align*}\frac{x-3}{3x^2+x-10} + \frac{3}{x+2}\end{align*}
  7. \begin{align*}\frac{x+1}{6x+2} + \frac{x^2-7x}{12x^2-14x-6}\end{align*}
  8. \begin{align*}\frac{-3x^2-10x+15}{10x^2-x-3} + \frac{x+4}{2x+1}\end{align*}
  9. \begin{align*}\frac{8}{2x-5} - \frac{x+5}{2x^2+x-15}\end{align*}
  10. \begin{align*}\frac{2}{x+2} + \frac{3x+16}{x^2-x-6} - \frac{2}{x-3}\end{align*}
  11. \begin{align*}\frac{6x^2+4x+8}{x^3+3x^2-x-3} + \frac{x-4}{x^2-1} - \frac{3x}{x^2+2x-3}\end{align*}

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 9.11. 

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