## Section13.6Rational Functions and Equations Chapter Review

### Subsection13.6.1Introduction to Rational Functions

In Section 13.1 we learned about rational functions and explored them with tables and graphs.

###### Example13.6.1Graphs of Rational Functions

In an apocalypse, a zombie infestation begins with $1$ zombie and spreads rapidly. The population of zombies can be modeled by $Z(x)=\frac{200000x+100}{5x+100}\text{,}$ where $x$ is the number of days after the apocalypse began. Use technology to graph the function and answer these questions:

1. How many zombies are there $2$ days after the apocalypse began?

2. After how many days will the zombie population be $20{,}000\text{?}$

3. As time goes on, the population will level off at about how many zombies?

Explanation

We will graph the function with technology. After adjusting window settings, we have:

1. To find the number of zombies after $2$ days, we locate the point $(2,3637.27)\text{.}$ Since we can only have a whole number of zombies, we round to $3{,}637$ zombies.

2. To find the number of days it will take for the zombie population reach $20{,}000\text{,}$ we locate the point $(19.999,20000)$ so it will take about 20 days.

3. When we look far to the right on the graph using technology we can see that the population will level off at about $40{,}000$ zombies.

### Subsection13.6.2Multiplication and Division of Rational Expressions

In Section 13.2 we covered how to simplify rational expressions. It is very important to list any domain restrictions from factors that are canceled. We also multiplied and divided rational expressions.

###### Example13.6.3Simplifying Rational Expressions

Simplify the expression $\frac{8t+4t^2-12t^3}{1-t}\text{.}$

Explanation

To begin simplifying this expression, we will rewrite each polynomial in descending order. Then we'll factor out the GCF, including the constant $-1$ from both the numerator and denominator because their leading terms are negative.

\begin{align*} \frac{8t+4t^2-12t^3}{1-t}\amp=\frac{-12t^3+4t^2+8t}{-t+1}\\ \amp=\frac{-4t(3t^2-t-2)}{-(t-1)}\\ \amp=\frac{-4t(3t+2)(t-1)}{-(t-1)}\\ \amp=\frac{-4t(3t+2)\cancelhighlight{(t-1)}}{-\cancelhighlight{(t-1)}}\\ \amp=\frac{-4t(3t+2)}{-1}, \text{ for } t\neq 1\\ \amp=4t(3t+2), \text{ for } t\neq 1 \end{align*}
###### Example13.6.4Multiplication of Rational Functions and Expressions

Multiply the rational expressions: $\frac{r^3s}{4t}\cdot\frac{2t^2}{r^2s^3}\text{.}$

Explanation

Note that we won't need to factor anything in this problem, and can simply multiply across and then simplify. With multivariable expressions, this textbook ignores domain restrictions.

\begin{align*} \frac{r^3s}{4t}\cdot\frac{2t^2}{r^2s^3}\amp=\frac{r^3s\cdot2t^2}{4t\cdot r^2s^3}\\ \amp=\frac{2r^3st^2}{4r^2s^3t}\\ \amp=\frac{rt}{2s^2} \end{align*}
###### Example13.6.5Division of Rational Functions and Expressions

Divide the rational expressions: $\frac{2x^2+8xy}{x^2-4x+3}\div\frac{x^3+4x^2y}{x^2+4x-5}\text{.}$

Explanation

To divide rational expressions, we multiply by the reciprocal of the second fraction. Then we will factor and cancel any common factors. With multivariable expressions, this textbook ignores domain restrictions.

\begin{align*} \frac{2x^2+8xy}{x^2-4x+3}\div\frac{x^3+4x^2y}{x^2+4x-5}\amp=\frac{2x^2+8xy}{x^2-4x+3}\cdot\frac{x^2+4x-5}{x^3+4x^2y}\\ \amp=\frac{2x\cancelhighlight{(x+4y)}}{\cancelhighlight{(x-1)}(x-3)}\cdot\frac{\cancelhighlight{(x-1)}(x+5)}{x^2\cancelhighlight{(x+4y)}}\\ \amp=\frac{2x}{x-3}\cdot\frac{x+5}{x^2}\\ \amp=\frac{2(x+5)}{x(x-3)} \end{align*}

### Subsection13.6.3Addition and Subtraction of Rational Expressions

In Section 13.3 we covered how to add and subtract rational expressions.

###### Example13.6.6Addition and Subtraction of Rational Expressions with the Same Denominator

Add the rational expressions: $\dfrac{5x}{x+5}+\dfrac{25}{x+5}\text{.}$

Explanation

These expressions already have a common denominator:

\begin{align*} \frac{5x}{x+5}+\frac{25}{x+5}\amp=\frac{5x+25}{x+5}\\ \amp=\frac{5\cancelhighlight{(x+5)}}{\cancelhighlight{x+5}}\\ \amp=\frac{5}{1}, \text{ for } x\neq -5\\ \amp=5, \text{ for } x\neq -5 \end{align*}

Note that we didn't stop at $\frac{5x+25}{x+5}\text{.}$ If possible, we must simplify the numerator and denominator.

###### Example13.6.7Addition and Subtraction of Rational Expressions with Different Denominators

Add and subtract the rational expressions: $\dfrac{6y}{y+2}+\dfrac{y}{y-2}-7$

Explanation

The denominators can't be factored, so we'll find the least common denominator and build each expression to that denominator. Then we will be able to combine the numerators and simplify the expression.

\begin{align*} \frac{6y}{y+2}+\frac{y}{y-2}-7\amp=\frac{6y}{y+2}\multiplyright{\frac{y-2}{y-2}}+\frac{y}{y-2}\multiplyright{\frac{y+2}{y+2}}-7\multiplyright{\frac{(y-2)(y+2)}{(y-2)(y+2)}}\\ \amp=\frac{6y(y-2)}{(y-2)(y+2)}+\frac{y(y+2)}{(y-2)(y+2)}-\frac{7(y-2)(y+2)}{(y-2)(y+2)}\\ \amp=\frac{6y^2-12y+y^2+2y-\highlight{\attention{(}}7(y^2-4)\highlight{\attention{)}}}{(y-2)(y+2)}\\ \amp=\frac{6y^2-12y+y^2+2y-7y^2+28}{(y-2)(y+2)}\\ \amp=\frac{-10y+28}{(y-2)(y+2)}\\ \amp=\frac{-2(5y-14)}{(y-2)(y+2)} \end{align*}

### Subsection13.6.4Complex Fractions

In Section 13.4 we covered how to simplify a rational expression that has fractions in the numerator and/or denominator.

###### Example13.6.8Simplifying Complex Fractions

Simplify the complex fraction $\dfrac{\frac{2t}{t^2-9}+3}{\frac{6}{t+3}+\frac{1}{t-3}}\text{.}$

Explanation

First, we check all quadratic polynomials to see if they can be factored and factor them:

\begin{equation*} \frac{\frac{2t}{t^2-9}+3}{\frac{6}{t+3}+\frac{1}{t-3}}=\frac{\frac{2t}{(t-3)(t+3)}+3}{\frac{6}{t+3}+\frac{1}{t-3}} \end{equation*}

Next, we identify the common denominator of the three fractions, which is $(t+3)(t-3)\text{.}$ We then multiply the main numerator and denominator by that expression:

\begin{align*} \frac{\frac{2t}{(t-3)(t+3)}+3}{\frac{6}{t+3}+\frac{1}{t-3}}\amp=\frac{\frac{2t}{(t-3)(t+3)}+3}{\frac{6}{t+3}+\frac{1}{t-3}}\multiplyright{\frac{(t-3)(t+3)}{(t-3)(t+3)}}\\ \amp=\frac{\frac{2t}{\highlight{\xcancel{(t-3)(t+3)}}}\highlight{\xcancel{(t-3)(t+3)}}+3(t-3)(t+3)}{\frac{6}{\highlight{\cancel{t+3}}}(t-3)\highlight{\cancel{(t+3)}}+\frac{1}{\lighthigh{\bcancel{t-3}}}\lighthigh{\bcancel{(t-3)}(t+3)}}\\ \amp=\frac{2t+3(t-3)(t+3)}{6(t-3)+1(t+3)} \text{ for }t\neq -3, t\neq 3\\ \amp=\frac{2t+3(t^2-9)}{6t-18+t+3} \text{ for }t\neq -3, t\neq 3\\ \amp=\frac{2t+3t^2-27}{7t-15} \text{ for }t\neq -3, t\neq 3\\ \amp=\frac{3t^2+2t-27}{7t-15} \text{ for }t\neq -3, t\neq 3 \end{align*}

Both the numerator and denominator are prime polynomials so this expression can neither factor nor simplify any further.

### Subsection13.6.5Solving Rational Equations

In Section 13.5 we covered how to solve rational equations. We looked at rate problems, solved for a specified variable and used technology to solve rational equations.

###### Example13.6.9Solving Rational Equations

Two pipes are being used to fill a large tank. Pipe B can fill the tank twice as fast as Pipe A can. When both pipes are turned on, it takes 12 hours to fill the tank. Write and solve a rational equation to answer the following questions:

1. If only Pipe A is turned on, how many hours would it take to fill the tank?

2. If only Pipe B is turned on, how many hours would it take to fill the tank?

Explanation

Since both pipes can fill the tank in $12$ hours, they fill $\frac{1}{12}$ of the tank together each hour. We will let $a$ represent the number of hours it takes pipe A to fill the tank alone, so pipe A will fill $\frac{1}{a}$ of the tank each hour. Pipe B can fill the tank twice as fast so it fills $2\cdot \frac{1}{a}$ of the tank each hour or $\frac{2}{a}\text{.}$ When they are both turned on, they fill $\frac{1}{a}+\frac{2}{a}$ of the tank each hour.

Now we can write this equation:

\begin{equation*} \frac{1}{a}+\frac{2}{a}=\frac{1}{12} \end{equation*}

To clear away denominators, we multiply both sides of the equation by the common denominator of $12$ and $a\text{,}$ which is $12a\text{:}$

\begin{align*} \frac{1}{a}+\frac{2}{a}\amp=\frac{1}{12}\\ \multiplyleft{12a}\left(\frac{1}{a}+\frac{2}{a}\right)\amp=\multiplyleft{12a}\frac{1}{12}\\ 12a\cdot\frac{1}{a}+12a\cdot\frac{2}{a}\amp=12a\cdot\frac{1}{12}\\ 12+24\amp=a\\ 36\amp=a\\ a\amp=36 \end{align*}

The possible solution $a=36$ should be checked

\begin{align*} \frac{1}{\substitute{36}}+\frac{2}{\substitute{36}}\amp\stackrel{?}{=}\frac{1}{12}\\ \frac{3}{36}\amp\stackrel{\checkmark}{=}\frac{1}{12} \end{align*}

So it is a solution.

1. If only Pipe A is turned on, it would take $36$ hours to fill the tank.

2. Since Pipe B can fill the tank twice as fast, it would take half the time, or $18$ hours to fill the tank.

###### Example13.6.10Solving Rational Equations for a Specific Variable

Solve the rational equation $y=\frac{2x+5}{3x-1}$ for $x\text{.}$

Explanation

To get the $x$ out of the denominator, our first step will be to multiply each side by the LCD, which is $3x-1\text{.}$ Then we'll isolate all terms containing $x\text{,}$ factor out $x\text{,}$ and then finish solving for that variable.

\begin{align*} y\amp=\frac{2x+5}{3x-1}\\ y\multiplyright{(3x-1)}\amp=\frac{2x+5}{\highlight{\cancel{3x-1}}}\multiplyright{\highlight{\cancel{(3x-1)}}}\\ 3xy-y\amp=2x+5\\ 3xy\amp=2x+5+y\\ 3xy-2x\amp=y+5\\ x(3y-2)\amp=y+5\\ \divideunder{x(3y-2)}{3y-2}\amp=\divideunder{y+5}{3y-2}\\ x\amp=\frac{y+5}{3y-2} \end{align*}
###### Example13.6.11Solving Rational Equations Using Technology

Solve the equation $\frac{1}{x+2}+1=\frac{10x}{x^2+5}$ using graphing technology.

Explanation

We will define $f(x)=\frac{1}{x+2}+1$ and $g(x)=\frac{10x}{x^2+5}\text{,}$ and then find a window where we can see all of the points of intersection.

Since the two functions intersect at approximately $(-2.309,-2.235)\text{,}$ $(0.76,1.362)$ and $(8.549,1.095)\text{,}$ the solutions to $\frac{1}{x+2}+1=\frac{10x}{x^2+5}$ are approximately $-2.309\text{,}$ $0.76$ and $8.549\text{.}$ The solution set is approximately $\{-2.309\ldots, 0.76\ldots, 8.549\ldots\}\text{.}$

### Subsection13.6.6Exercises

###### 1

A function is graphed.

This function has domain and range .

###### 2

A function is graphed.

This function has domain and range .

###### 3

The population of deer in a forest can be modeled by

\begin{equation*} P(x) = {\frac{780x+2660}{3x+7}} \end{equation*}

where $x$ is the number of years in the future. Answer the following questions.

1. How many deer live in this forest this year?

2. How many deer will live in this forest $24$ years later? Round your answer to an integer.

3. After how many years, the deer population will be $276\text{?}$ Round your answer to an integer.

4. Use a calculator to answer this question: As time goes on, the population levels off at about how many deer?

###### 4

The population of deer in a forest can be modeled by

\begin{equation*} P(x) = {\frac{400x+1950}{4x+5}} \end{equation*}

where $x$ is the number of years in the future. Answer the following questions.

1. How many deer live in this forest this year?

2. How many deer will live in this forest $27$ years later? Round your answer to an integer.

3. After how many years, the deer population will be $143\text{?}$ Round your answer to an integer.

4. Use a calculator to answer this question: As time goes on, the population levels off at about how many deer?

###### 5

In a certain store, cashiers can serve $55$ customers per hour on average. If $x$ customers arrive at the store in a given hour, then the average number of customers $C$ waiting in line can be modeled by the function

\begin{equation*} C(x) = {\frac{x^{2}}{3025-55x}} \end{equation*}

where $x\lt 55\text{.}$

1. If $42$ customers arrived in the store in the past hour, there are approximately customers waiting in line.

2. If there are $7$ customers waiting in line, approximately customers arrived in the past hour.

###### 6

In a certain store, cashiers can serve $60$ customers per hour on average. If $x$ customers arrive at the store in a given hour, then the average number of customers $C$ waiting in line can be modeled by the function

\begin{equation*} C(x) = {\frac{x^{2}}{3600-60x}} \end{equation*}

where $x\lt 60\text{.}$

1. If $51$ customers arrived in the store in the past hour, there are approximately customers waiting in line.

2. If there are $2$ customers waiting in line, approximately customers arrived in the past hour.

###### 7

The concentration of a drug in a patient’s blood stream, in milligrams per liter, can be modeled by the function $C(t)={\frac{6t}{t^{2}+4}}\text{,}$ where $t$ is the number of hours since the drug is injected. Answer the following question with technology. Round your answer to two decimal places if needed.

hours since injection, the drug’s concentration is at the maximum value of milligrams per liter.

###### 8

The concentration of a drug in a patient’s blood stream, in milligrams per liter, can be modeled by the function $C(t)={\frac{7t}{t^{2}+8}}\text{,}$ where $t$ is the number of hours since the drug is injected. Answer the following question with technology. Round your answer to two decimal places if needed.

hours since injection, the drug’s concentration is at the maximum value of milligrams per liter.

###### 9

Simplify this expression.

$\displaystyle{{\frac{-t^{2}+6ty-5y^{2}}{t^{2}-25y^{2}}}=}$

###### 10

Simplify this expression.

$\displaystyle{{\frac{-t^{2}-8tx-15x^{2}}{t^{2}-25x^{2}}}=}$

###### 11

Simplify the function formula, and if applicable, write the restricted domain.

$\displaystyle{ G(x) = {\frac{x^{4}+8x^{3}+16x^{2}}{3x^{4}+13x^{3}+4x^{2}}} }$

Reduced $G(x)=$

###### 12

Simplify the function formula, and if applicable, write the restricted domain.

$\displaystyle{ h(x) = {\frac{x^{4}-4x^{3}+4x^{2}}{2x^{4}-3x^{3}-2x^{2}}} }$

Reduced $h(x)=$

###### 13

Simplify this expression, and if applicable, write the restricted domain.

$\displaystyle{{\frac{y^{2}-16y}{y^{2}-16}} \cdot {\frac{y^{2}-4y}{y^{2}-19y+48}} =}$

###### 14

Simplify this expression, and if applicable, write the restricted domain.

$\displaystyle{{\frac{y^{2}-4y}{y^{2}-4}} \cdot {\frac{y^{2}-2y}{y^{2}-3y-4}} =}$

###### 15

Simplify this expression, and if applicable, write the restricted domain.

$\displaystyle{{\frac{9r^{2}-25}{3r^{2}+8r+5}} \div ({5-3r}) =}$

###### 16

Simplify this expression, and if applicable, write the restricted domain.

$\displaystyle{{\frac{25r^{2}-9}{5r^{2}+8r+3}} \div ({3-5r}) =}$

###### 17

Simplify this expression.

$\displaystyle{{\frac{r^{3}}{r^{2}y+4r}} \div {\frac{1}{r^{2}y^{2}+5ry+4}} =}$

###### 18

Simplify this expression.

$\displaystyle{{\frac{t^{4}}{t^{2}y+4t}} \div {\frac{1}{t^{2}y^{2}+5ty+4}} =}$

###### 19

Add or subtract the rational expressions to a single rational expression and then simplify. If applicable, state the restricted domain.

$\displaystyle{{\frac{1}{t-4}-\frac{8}{t^{2}-16}} =}$

###### 20

Add or subtract the rational expressions to a single rational expression and then simplify. If applicable, state the restricted domain.

$\displaystyle{{\frac{1}{x+1}+\frac{2}{x^{2}-1}} =}$

###### 21

Add or subtract the rational expressions to a single rational expression and then simplify. If applicable, state the restricted domain.

$\displaystyle{{-\frac{5x}{x^{2}-3x+2}-\frac{5x}{x-1}} =}$

###### 22

Add or subtract the rational expressions to a single rational expression and then simplify. If applicable, state the restricted domain.

$\displaystyle{{\frac{8y}{y^{2}+2y-3}+\frac{2y}{y+3}} =}$

###### 23

Add or subtract the rational expressions to a single rational expression and then simplify. If applicable, state the restricted domain.

$\displaystyle{{\frac{y^{2}-20}{y^{2}+5y}-\frac{y-4}{y}} =}$

###### 24

Add or subtract the rational expressions to a single rational expression and then simplify. If applicable, state the restricted domain.

$\displaystyle{{\frac{y^{2}-10}{y^{2}-2y}-\frac{y+5}{y}} =}$

###### 25

Add or subtract the rational expressions to a single rational expression and then simplify.

$\displaystyle{{-\frac{4r}{3t^{5}}-\frac{6}{5rt}} =}$

###### 26

Add or subtract the rational expressions to a single rational expression and then simplify.

$\displaystyle{{-\frac{6r}{5y^{3}}+\frac{3}{2ry}} =}$

###### 27

Add or subtract the rational expressions to a single rational expression and then simplify.

$\displaystyle{{\frac{6tx}{t^{2}-4tx-5x^{2}}-\frac{t}{t-5x}} =}$

###### 28

Add or subtract the rational expressions to a single rational expression and then simplify.

$\displaystyle{{-\frac{6ty}{t^{2}+14ty+40y^{2}}-\frac{t}{t+10y}} =}$

###### 29

1. $\displaystyle{\frac{ \frac{3}{7} }{ \frac{3}{8} } =}$

2. $\displaystyle{\frac{ \frac{x}{y} }{ \frac{r}{t} } =}$

###### 30

1. $\displaystyle{\frac{ \frac{3}{8} }{ \frac{3}{5} } =}$

2. $\displaystyle{\frac{ \frac{x}{t} }{ \frac{y}{r} } =}$

###### 31

Simplify this expression, and if applicable, write the restricted domain.

$\displaystyle{\frac{{\frac{8}{b-1}-4}}{{\frac{1}{b-1}+\frac{1}{b-10}}}=}$

###### 32

Simplify this expression, and if applicable, write the restricted domain.

$\displaystyle{\frac{{\frac{4}{b-1}-7}}{{\frac{1}{b-1}+\frac{1}{b-8}}}=}$

###### 33

Simplify this expression, and if applicable, write the restricted domain.

$\displaystyle{{\frac{\frac{2y}{y^{2}-9}-1}{\frac{3}{y+3}+\frac{2}{y-3}}}=}$

###### 34

Simplify this expression, and if applicable, write the restricted domain.

$\displaystyle{{\frac{\frac{2r}{r^{2}-9}+3}{\frac{3}{r+3}+\frac{1}{r-3}}}=}$

###### 35

Simplify this expression.

$\displaystyle{\frac{{\frac{p}{q}}}{{\frac{4p}{3q^{2}}}}=}$

###### 36

Simplify this expression.

$\displaystyle{\frac{{\frac{m}{n}}}{{\frac{6m}{5n^{2}}}}=}$

###### 37

Simplify this expression.

$\displaystyle{{\frac{\frac{3}{t}}{3+\frac{3r}{2}}} =}$

###### 38

Simplify this expression.

$\displaystyle{{\frac{\frac{2}{x}}{2-\frac{2y}{3}}} =}$

###### 39

Solve the equation.

$\displaystyle{ {\frac{2}{x-1}+\frac{4}{x+7}}={-\frac{2}{x^{2}+6x-7}} }$

###### 40

Solve the equation.

$\displaystyle{ {\frac{3}{y+7}-\frac{9}{y-4}}={-\frac{9}{y^{2}+3y-28}} }$

###### 41

Solve the equation.

$\displaystyle{ {\frac{1}{y-5}-\frac{5}{y^{2}-5y}} = \frac{1}{4} }$

###### 42

Solve the equation.

$\displaystyle{ {\frac{1}{y+8}+\frac{8}{y^{2}+8y}} = -\frac{1}{4} }$

###### 43

Solve the equation.

$\displaystyle{ {-\frac{6}{r+9}+\frac{2r}{r-3}}={\frac{8}{r^{2}+6r-27}} }$

###### 44

Solve the equation.

$\displaystyle{ {\frac{3}{r-1}+\frac{6r}{r-7}}={\frac{9}{r^{2}-8r+7}} }$

###### 45

Solve the equation.

$\displaystyle{ {\frac{t-3}{t+1}-\frac{6}{t-4}} = 6 }$

###### 46

Solve the equation.

$\displaystyle{ {\frac{t+7}{t-9}+\frac{6}{t+6}} = -1 }$

###### 47

Solve this equation for $r\text{:}$

$\displaystyle{ \frac{1}{x} = \frac{3}{r+4} }$

###### 48

Solve this equation for $n\text{:}$

$\displaystyle{ \frac{1}{r} = \frac{3}{n+8} }$

###### 49

Use technology to solve the equation

\begin{equation*} 2x-\frac{1}{x+4}=\frac{3}{x+6}\text{.} \end{equation*}
###### 50

Use technology to solve the equation

\begin{equation*} \frac{1}{x^2-1}-\frac{2}{x-4}=\frac{3}{x-2}\text{.} \end{equation*}
###### 51

Two pipes are being used to fill a tank. Pipe A can fill the tank $4.5$ times as fast as Pipe B does. When both pipes are turned on, it takes $18$ hours to fill the tank. Answer the following questions:

If only Pipe A is turned on, it would take hours to fill the tank.

If only Pipe B is turned on, it would take hours to fill the tank.

###### 52

Two pipes are being used to fill a tank. Pipe A can fill the tank $5.5$ times as fast as Pipe B does. When both pipes are turned on, it takes $11$ hours to fill the tank. Answer the following questions:

If only Pipe A is turned on, it would take hours to fill the tank.

If only Pipe B is turned on, it would take hours to fill the tank.

###### 53

Town A and Town B are $360$ miles apart. A boat traveled from Town A to Town B, and then back to Town A. Since the river flows from Town B to Town A, the boat’s speed was $15$ miles per hour faster when it traveled from Town B to Town A. The whole trip took $20$ hours. Answer the following questions:

The boat traveled from Town A to Town B at the speed of miles per hour.

The boat traveled from Town B back to Town A at the speed of miles per hour.

###### 54

Town A and Town B are $560$ miles apart. A boat traveled from Town A to Town B, and then back to Town A. Since the river flows from Town B to Town A, the boat’s speed was $30$ miles per hour faster when it traveled from Town B to Town A. The whole trip took $28$ hours. Answer the following questions:

The boat traveled from Town A to Town B at the speed of miles per hour.

The boat traveled from Town B back to Town A at the speed of miles per hour.