Mathematics » Rational Expressions and Equations » Simplify Rational Expressions

# Determining the Values For Which a Rational Expression is Undefined

## Determining the Values For Which a Rational Expression is Undefined

When we work with a numerical fraction, it is easy to avoid dividing by zero, because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero.

If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator.

So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.

### Determine the Values for Which a Rational Expression is Undefined.

1. Set the denominator equal to zero.
2. Solve the equation in the set of reals, if possible.

## Example

Determine the values for which the rational expression is undefined:

1. $$\frac{9y}{x}$$
2. $$\frac{4b-3}{2b+5}$$
3. $$\frac{x+4}{{x}^{2}+5x+6}$$

### Solution

The expression will be undefined when the denominator is zero.

1.

$$\begin{array}{cccc}& & & \hfill \phantom{\rule{8em}{0ex}}\frac{9y}{x}\hfill \\ \begin{array}{c}\text{Set the denominator equal to zero. Solve}\hfill \\ \text{for the variable.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{8em}{0ex}}x=0\hfill \\ & & & \hfill \phantom{\rule{8em}{0ex}}\frac{9y}{x}\phantom{\rule{0.2em}{0ex}}\text{is undefined for}\phantom{\rule{0.2em}{0ex}}x=0.\hfill \end{array}$$

2.

$$\begin{array}{cccc}& & & \hfill \phantom{\rule{6em}{0ex}}\frac{4b-3}{2b+5}\hfill \\ \begin{array}{c}\text{Set the denominator equal to zero. Solve}\hfill \\ \text{for the variable.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{6em}{0ex}}\begin{array}{ccc}\hfill 2b+5& =\hfill & 0\hfill \\ \hfill 2b& =\hfill & -5\hfill \\ \hfill b& =\hfill & -\frac{5}{2}\hfill \end{array}\hfill \\ & & & \hfill \phantom{\rule{6em}{0ex}}\frac{4b-3}{2b+5}\phantom{\rule{0.2em}{0ex}}\text{is undefined for}\phantom{\rule{0.2em}{0ex}}b=-\frac{5}{2}.\hfill \end{array}$$

3.

$$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{x+4}{{x}^{2}+5x+6}\hfill \\ \begin{array}{c}\text{Set the denominator equal to zero. Solve}\hfill \\ \text{for the variable.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\begin{array}{ccc}\hfill {x}^{2}+5x+6& =\hfill & 0\hfill \\ \hfill \left(x+2\right)\left(x+3\right)& =\hfill & 0\hfill \\ \hfill x+2=0\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}x+3& =\hfill & 0\hfill \\ \hfill x=-2\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}x& =\hfill & -3\hfill \end{array}\hfill \\ & & & \hfill \phantom{\rule{4em}{0ex}}\frac{x+4}{{x}^{2}+5x+6}\phantom{\rule{0.2em}{0ex}}\text{is undefined for}\phantom{\rule{0.2em}{0ex}}x=-2\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}x=-3.\hfill \end{array}$$

Saying that the rational expression $$\frac{x+4}{{x}^{2}+5x+6}$$ is undefined for $$x=-2\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}x=-3$$ is similar to writing the phrase “void where prohibited” in contest rules.