Simplify Rational Expressions: example 8.1

The expression will be undefined when the denominator is zero.

1. ⓐ
   $\begin{array}{cccc}& & & \hfill \frac{9y}{x}\hfill \\ \begin{array}{c}\text{Set the denominator equal to zero. Solve}\hfill \\ \text{for the variable.}\hfill \end{array}\hfill & & & \hfill x=0\hfill \\ & & & \hfill \frac{9y}{x}\text{is undefined for}x=0.\hfill \end{array}$
   
   
2. ⓑ
   $\begin{array}{cccc}& & & \hfill \frac{4b-3}{2b+5}\hfill \\ \begin{array}{c}\text{Set the denominator equal to zero. Solve}\hfill \\ \text{for the variable.}\hfill \end{array}\hfill & & & \hfill \begin{array}{ccc}\hfill 2b+5& =\hfill & 0\hfill \\ \hfill 2b& =\hfill & \mathrm{-5}\hfill \\ \hfill b& =\hfill & -\frac{5}{2}\hfill \end{array}\hfill \\ & & & \hfill \frac{4b-3}{2b+5}\text{is undefined for}b=-\frac{5}{2}.\hfill \end{array}$
   
   
3. ⓒ
   $\begin{array}{cccc}& & & \hfill \frac{x+4}{x^2+5x+6}\hfill \\ \begin{array}{c}\text{Set the denominator equal to zero. Solve}\hfill \\ \text{for the variable.}\hfill \end{array}\hfill & & & \hfill \begin{array}{ccc}\hfill x^2+5x+6& =\hfill & 0\hfill \\ \hfill \left(x+2\right)\left(x+3\right)& =\hfill & 0\hfill \\ \hfill x+2=0\text{or}x+3& =\hfill & 0\hfill \\ \hfill x=\mathrm{-2}\text{or}x& =\hfill & \mathrm{-3}\hfill \end{array}\hfill \\ & & & \hfill \frac{x+4}{x^2+5x+6}\text{is undefined for}x=\mathrm{-2}\text{or}x=\mathrm{-3}.\hfill \end{array}$

Saying that the rational expression $\frac{x+4}{x^2+5x+6}$ is undefined for $x=\mathrm{-2}\text{or}x=\mathrm{-3}$ is similar to writing the phrase “void where prohibited” in contest rules.