Chapter 8.6, Problem 55PPSE

Solution

a.

To identify: The one solution rational equation.

The rational equation has one solutions.

Required example $\frac{2}{x}=\frac{1}{2}$

Explanation:

One way to have a rational equation that have a one solution is that in any case, the variable must be in linear meaning that the exponenet of the variable is one..

Example of such rational equation where the variable is in linear is:

  $\frac{2}{x}=\frac{1}{2}$

Note that any numarical constants in this format will do as long as the variable is linear.

Solve for $x$

  $\begin{array}{l}\frac{2}{x}=\frac{1}{2}\\ x=4\end{array}$

Thus the rational equation has one solutions.

Required example $\frac{2}{x}=\frac{1}{2}$

b.

To identify:The two solution rational equation.

.The rational equation has two solutions

Required example is $\frac{x}{2}=\frac{2}{x}$

Explanation:

Write a rational equation that has two solutions.

Consider the below equation.

  $\frac{x}{2}=\frac{2}{x}$

Multiply on the both sides by $2x$ .

  $\begin{array}{l}\frac{x}{2}\cdot 2x=\frac{2}{x}\cdot 2x\\ x^2=4\\ x=\pm 2\end{array}$

Thus the rational equation has two solutions

Required example $\frac{x}{2}=\frac{2}{x}$

c.

To identify:The no real solution rational equation.

The rational equation has no real solutions

Required example is $\frac{x}{2}=\frac{-2}{x}$

Explanation:

Write a rational equation that has two solutions.

Consider the below equation. $\frac{x}{2}=\frac{-2}{x}$

Multiply on the both sides by $2x$ .

  $\begin{array}{l}\frac{x}{2}\cdot 2x=-\frac{2}{x}\cdot 2x\\ x^2=-4\end{array}$

Thus the rational equation has no real solutions

Required example is $\frac{x}{2}=\frac{-2}{x}$