Chapter 7, Problem 1RE

For the rational expression $\frac{t-2}{t+9}$

a. Evaluate the expression (if possible) for $t=0,1,2,-3,-9$

b. Identify the restricted values.

To determine

To calculate: The value of expression $\frac{t-2}{t+9}$ (if possible) for $t=0,1,2,-3,-9$.

Answer to Problem 1RE

Solution:

The value of expression $\frac{t-2}{t+9}$ for $t=0,1,2,-3,-9$ is $\underset{\_}{-\frac{2}{9},-\frac{1}{10},0,-\frac{5}{6}}$ and not defined

respectively.

Explanation of Solution

Given Information:

The provided expression is:

$\frac{t-2}{t+9}$

Calculation:

Consider the provided expression:

$\frac{t-2}{t+9}$

First, substitute $0$ for $t$ in the provided expression $\frac{t-2}{t+9}$:

$\begin{array}{ccc}\hfill \frac{t-2}{t+9}& =& \frac{0-2}{0+9}\hfill \\ \hfill & =& -\frac{2}{9}\hfill \end{array}$

Now, substitute $1$ for $t$ in the provided expression $\frac{t-2}{t+9}$:

$\begin{array}{ccc}\hfill \frac{t-2}{t+9}& =& \frac{1-2}{1+9}\hfill \\ \hfill & =& \frac{-1}{10}\hfill \end{array}$

Now, substitute $2$ for $t$ in the provided expression $\frac{t-2}{t+9}$:

$\begin{array}{ccc}\hfill \frac{t-2}{t+9}& =& \frac{2-2}{2+9}\hfill \\ \hfill & =& \frac{0}{10}\hfill \\ \hfill & =& 0\hfill \end{array}$

Now, substitute $-3$ for $t$ in the provided expression $\frac{t-2}{t+9}$:

$\begin{array}{ccc}\hfill \frac{t-2}{t+9}& =& \frac{-3-2}{-3+9}\hfill \\ \hfill & =& \frac{-5}{6}\hfill \end{array}$

Now, substitute $-9$ for $t$ in the provided expression $\frac{t-2}{t+9}$:

$\begin{array}{ccc}\hfill \frac{t-2}{t+9}& =& \frac{-9-2}{-9+9}\hfill \\ \hfill & =& \frac{-11}{0}\hfill \\ \hfill & =& \text{Undefined}\hfill \end{array}$

Hence, the value of expression $\frac{t-2}{t+9}$ for $t=0,1,2,-3,-9$ is $-\frac{2}{9},-\frac{1}{10},0,-\frac{5}{6}$ and not defined

respectively.

To determine

To calculate: The restricted value for the expression $\frac{t-2}{t+9}$.

Answer to Problem 1RE

Solution:

The restricted value for the expression $\frac{t-2}{t+9}$ is $\underset{\_}{-9}$.

Explanation of Solution

Given Information:

The provided expression is:

$\frac{t-2}{t+9}$

Formula used:

Restricted Values of a Rational Expression:

Restricted values of a rational expression are all the values that make the expression undefined. For a rational expression $\frac{ax+b}{cx+d}$, restricted values are obtained when denominator is equated to zero and solved for the variable:

$\begin{array}{ccc}\hfill cx+d& =& 0\hfill \\ \hfill x& =& -\frac{d}{c}\hfill \end{array}$

Rational value for the expression is $x=-\frac{d}{c}$.

Calculation:

Consider the provided expression:

$\frac{t-2}{t+9}$

Equate the denominator to zero in order to compute the restricted values:

$t+9=0$

Solve the above equation for $t$:

$\begin{array}{ccc}\hfill t& =& 0-9\hfill \\ \hfill & =& -9\hfill \end{array}$

Hence the restricted value for the expression $\frac{t-2}{t+9}$ is $t=-9$.