Chapter 4.6, Problem 27E

(a)

To determine

To find: The LCD of the rational expressions.

Answer to Problem 27E

  $\text{LCD: }a\left(a-6\right)$

Explanation of Solution

Given information: Consider the inequality $\frac{a-2}{a}<\frac{a-4}{a-6}$

Calculation:

  $\text{LCD: }a\left(a-6\right)$

(b)

To determine

To find: The zero(s) of the related function.

Answer to Problem 27E

  $a=3$

Explanation of Solution

Given information: Consider the inequality $\frac{a-2}{a}<\frac{a-4}{a-6}$

Calculation:

  $\begin{array}{l}\frac{a-2}{a}=\frac{a-4}{a-6}\\ \frac{a-2}{a}-\frac{a-4}{a-6}=0\\ \left(a-2\right)\left(a-6\right)-a\left(a-4\right)=0\\ a^2-8a+12-a^2+4a=0\\ -4a+12=0\\ a=3\end{array}$

(c)

To determine

To find: The excluded value(s) of the related function.

Answer to Problem 27E

  $\text{Exclude: }a=0,\text{ 6}$

Explanation of Solution

Given information: Consider the inequality $\frac{a-2}{a}<\frac{a-4}{a-6}$

Calculation:

The excluded values make the denominator = 0.

  $\text{Exclude: }a=0,\text{ 6}$

(d)

To determine

To solve: The inequality.

Answer to Problem 27E

  $\text{0}<a<3,a>6$

Explanation of Solution

Given information: Consider the inequality $\frac{a-2}{a}<\frac{a-4}{a-6}$

Calculation:

Test the values left and right of: 0, 3, 6.

  $\begin{array}{l}\text{Test}\left(-1\right)\to \frac{-1-2}{-1}<\frac{-1-4}{-1-6}\\ 3<\frac{5}{7}\text{ False}\end{array}$

  $\begin{array}{l}\text{Test}\left(1\right)\to \frac{1-2}{1}<\frac{1-4}{1-6}\\ -1<\frac{3}{5}\text{ True}\end{array}$

  $\begin{array}{l}\text{Test}\left(4\right)\to \frac{4-2}{4}<\frac{4-4}{4-6}\\ \frac{1}{2}<0\text{ False}\end{array}$

  $\begin{array}{l}\text{Test}\left(7\right)\to \frac{7-2}{7}<\frac{7-4}{7-6}\\ \frac{5}{7}\text{<3 True}\end{array}$

Solutions:

  $\begin{array}{l}a>0,a<3,a>6\\ \text{or}\\ 0<a<3,a>6.\end{array}$