8.3 Add and subtract rational expressions with a common denominator

Solution

$\begin{array}{cccc}& & & \hfill \frac{5}{18}+\frac{7}{18}\hfill \\ \\ \\ \begin{array}{c}\text{The fractions have a common}\hfill \\ \text{denominator, so add the numerators and}\hfill \\ \text{place the sum over the common}\hfill \\ \text{denominator.}\hfill \end{array}\hfill & & & \hfill \frac{5+7}{18}\hfill \\ \\ \\ \text{Add in the numerator.}\hfill & & & \hfill \frac{12}{18}\hfill \\ \\ \\ \begin{array}{c}\text{Factor the numerator and denominator to}\hfill \\ \text{show the common factors.}\hfill \end{array}\hfill & & & \hfill \frac{6·2}{6·3}\hfill \\ \\ \\ \text{Remove common factors.}\hfill & & & \hfill \frac{\cancel{6}·2}{\cancel{6}·3}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \frac{2}{3}\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \hfill \frac{3y}{4y-3}+\frac{7}{4y-3}\hfill \\ \begin{array}{c}\text{The fractions have a common}\hfill \\ \text{denominator, so add the numerators and}\hfill \\ \text{place the sum over the common}\hfill \\ \text{denominator.}\hfill \end{array}\hfill & & & \hfill \frac{3y+7}{4y-3}\hfill \end{array}$

The numerator and denominator cannot be factored. The fraction is simplified.

Solution

$\begin{array}{cccc}& & & \hfill \frac{7x+12}{x+3}+\frac{x^2}{x+3}\hfill \\ \\ \\ \begin{array}{c}\text{The fractions have a common}\hfill \\ \text{denominator, so add the numerators and}\hfill \\ \text{place the sum over the common}\hfill \\ \text{denominator.}\hfill \end{array}\hfill & & & \hfill \frac{7x+12+x^2}{x+3}\hfill \\ \\ \\ \text{Write the degrees in descending order.}\hfill & & & \hfill \frac{x^2+7x+12}{x+3}\hfill \\ \\ \\ \text{Factor the numerator.}\hfill & & & \hfill \frac{\left(x+3\right)\left(x+4\right)}{x+3}\hfill \\ \\ \\ \text{Simplify by removing common factors.}\hfill & & & \hfill \frac{\cancel{\left(x+3\right)}\left(x+4\right)}{\cancel{x+3}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill x+4\hfill \end{array}$

Solution

$\begin{array}{cccc}& & & \hfill \frac{n^2}{n-10}-\frac{100}{n-10}\hfill \\ \\ \\ \begin{array}{c}\text{The fractions have a common}\hfill \\ \text{denominator, so subtract the numerators}\hfill \\ \text{and place the difference over the}\hfill \\ \text{common denominator.}\hfill \end{array}\hfill & & & \hfill \frac{n^2-100}{n-10}\hfill \\ \\ \\ \text{Factor the numerator.}\hfill & & & \hfill \frac{\left(n-10\right)\left(n+10\right)}{n-10}\hfill \\ \\ \\ \text{Simplify by removing common factors.}\hfill & & & \hfill \frac{\cancel{\left(n-10\right)}\left(n+10\right)}{\cancel{n-10}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill n+10\hfill \end{array}$