Intermediate Algebra

Problem 5

Find the domain of each rational function. See Example 1
$f(x)=\frac{3 x}{7-x}$

Amy Jiang

Numerade Educator

00:26

Problem 6

Find the domain of each rational function. See Example 1
$f(x)=\frac{-4 x}{-2+x}$

Amy Jiang

Numerade Educator

00:27

Problem 7

Find the domain of each rational function. See Example 1
$f(x)=\frac{x}{3 x-1}$

Amy Jiang

Numerade Educator

00:28

Problem 8

Find the domain of each rational function. See Example 1
$g(x)=\frac{-2}{2 x+5}$

Amy Jiang

Numerade Educator

01:00

Problem 9

Find the domain of each rational function. See Example 1
$R(x)=\frac{3+2 x}{x^{3}+x^{2}-2 x}$

Amy Jiang

Numerade Educator

00:52

Problem 10

Find the domain of each rational function. See Example 1
$C(x)=\frac{x+3}{x^{2}-4}$

Amy Jiang

Numerade Educator

Problem 12

Find the domain of each rational function. See Example 1
$R(x)=\frac{5}{x^{2}-7 x}$

Amy Jiang

Numerade Educator

00:24

Problem 13

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{8 x-16 x^{2}}{8 x}
$$

Amy Jiang

Numerade Educator

00:17

Problem 14

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{3 x-6 x^{2}}{3 x}
$$

Amy Jiang

Numerade Educator

00:20

Problem 15

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{x^{2}-9}{3+x}
$$

Amy Jiang

Numerade Educator

00:18

Problem 16

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{x^{2}-25}{5+x}
$$

Amy Jiang

Numerade Educator

00:20

Problem 17

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{9 y-18}{7 y-14}
$$

Amy Jiang

Numerade Educator

00:32

Problem 18

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{6 y-18}{2 y-6}
$$

Amy Jiang

Numerade Educator

00:21

Problem 19

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{x^{2}+6 x-40}{x+10}
$$

Amy Jiang

Numerade Educator

00:18

Problem 20

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{x^{2}-8 x+16}{x-4}
$$

Amy Jiang

Numerade Educator

Problem 21

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{x-9}{9-x}
$$

Amy Jiang

Numerade Educator

00:19

Problem 22

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{x-4}{4-x}
$$

Amy Jiang

Numerade Educator

00:24

Problem 23

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{x^{2}-49}{7-x}
$$

Amy Jiang

Numerade Educator

00:26

Problem 24

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{x^{2}-y^{2}}{y-x}
$$

Amy Jiang

Numerade Educator

00:33

Problem 25

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{2 x^{2}-7 x-4}{x^{2}-5 x+4}
$$

Amy Jiang

Numerade Educator

00:33

Problem 26

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{3 x^{2}-11 x+10}{x^{2}-7 x+10}
$$

Amy Jiang

Numerade Educator

Problem 27

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{x^{3}-125}{2 x-10}
$$

Amy Jiang

Numerade Educator

00:34

Problem 28

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{4 x+4}{x^{3}+1}
$$

Amy Jiang

Numerade Educator

00:45

Problem 29

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{3 x^{2}-5 x-2}{6 x^{3}+2 x^{2}+3 x+1}
$$

Amy Jiang

Numerade Educator

00:49

Problem 30

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{2 x^{2}-x-3}{2 x^{3}-3 x^{2}+2 x-3}
$$

Amy Jiang

Numerade Educator

00:53

Problem 31

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{9 x^{2}-15 x+25}{27 x^{3}+125}
$$

Amy Jiang

Numerade Educator

00:36

Problem 32

Simplify each rational expression. See Examples 2 through $5 .$
$$
\frac{8 x^{3}-27}{4 x^{2}+6 x+9}
$$

Amy Jiang

Numerade Educator

00:34

Problem 33

Multiply and simplify. See Example 6
$$
\frac{2 x-4}{15} \cdot \frac{6}{2-x}
$$

Amy Jiang

Numerade Educator

Problem 34

Multiply and simplify. See Example 6
$$
\frac{10-2 x}{7} \cdot \frac{14}{5 x-25}
$$

Amy Jiang

Numerade Educator

01:36

Problem 35

Multiply and simplify. See Example 6
$$
\frac{18 a-12 a^{2}}{4 a^{2}+4 a+1} \cdot \frac{4 a^{2}+8 a+3}{4 a^{2}-9}
$$

Amy Jiang

Numerade Educator

00:54

Problem 36

Multiply and simplify. See Example 6
$$
\frac{a-5 b}{a^{2}+a b} \cdot \frac{b^{2}-a^{2}}{10 b-2 a}
$$

Amy Jiang

Numerade Educator

00:50

Problem 37

Multiply and simplify. See Example 6
$$
\frac{9 x+9}{4 x+8} \cdot \frac{2 x+4}{3 x^{2}-3}
$$

Amy Jiang

Numerade Educator

00:59

Problem 38

Multiply and simplify. See Example 6
$$
\frac{2 x^{2}-2}{10 x+30} \cdot \frac{12 x+36}{3 x-3}
$$

Amy Jiang

Numerade Educator

01:31

Problem 39

Multiply and simplify. See Example 6
$$
\frac{2 x^{3}-16}{6 x^{2}+6 x-36} \cdot \frac{9 x+18}{3 x^{2}+6 x+12}
$$

Amy Jiang

Numerade Educator

Problem 40

Multiply and simplify. See Example 6
$$
\frac{x^{2}-3 x+9}{5 x^{2}-20 x-105} \cdot \frac{x^{2}-49}{x^{3}+27}
$$

Amy Jiang

Numerade Educator

01:09

Problem 41

Multiply and simplify. See Example 6
$$
\frac{a^{3}+a^{2} b+a+b}{5 a^{3}+5 a} \cdot \frac{6 a^{2}}{2 a^{2}-2 b^{2}}
$$

Amy Jiang

Numerade Educator

01:02

Problem 42

Multiply and simplify. See Example 6
$$
\frac{4 a^{2}-8 a}{a b-2 b+3 a-6} \cdot \frac{8 b+24}{3 a+6}
$$

Amy Jiang

Numerade Educator

01:06

Problem 43

Multiply and simplify. See Example 6
$$
\frac{x^{2}-6 x-16}{2 x^{2}-128} \cdot \frac{x^{2}+16 x+64}{3 x^{2}+30 x+48}
$$

Amy Jiang

Numerade Educator

01:38

Problem 44

Multiply and simplify. See Example 6
$$
\frac{2 x^{2}+12 x-32}{x^{2}+16 x+64} \cdot \frac{x^{2}+10 x+16}{x^{2}-3 x-10}
$$

Amy Jiang

Numerade Educator

00:33

Problem 45

Divide and simplify. See Example 7
$$
\frac{2 x}{5} \div \frac{6 x+12}{5 x+10}
$$

Amy Jiang

Numerade Educator

00:36

Problem 46

Divide and simplify. See Example 7
$$
\frac{7}{3 x} \div \frac{14-7 x}{18-9 x}
$$

Amy Jiang

Numerade Educator

Problem 47

Divide and simplify. See Example 7
$$
\frac{a+b}{a b} \div \frac{a^{2}-b^{2}}{4 a^{3} b}
$$

Amy Jiang

Numerade Educator

00:39

Problem 48

Divide and simplify. See Example 7
$$
\frac{6 a^{2} b^{2}}{a^{2}-4} \div \frac{3 a b^{2}}{a-2}
$$

Amy Jiang

Numerade Educator

00:40

Problem 49

Divide and simplify. See Example 7
$$
\frac{x^{2}-6 x+9}{x^{2}-x-6} \div \frac{x^{2}-9}{4}
$$

Amy Jiang

Numerade Educator

01:03

Problem 50

Divide and simplify. See Example 7
$$
\frac{x^{2}-4}{3 x+6} \div \frac{2 x^{2}-8 x+8}{x^{2}+4 x+4}
$$

Amy Jiang

Numerade Educator

01:02

Problem 51

Divide and simplify. See Example 7
$$
\frac{x^{2}-6 x-16}{2 x^{2}-128} \div \frac{x^{2}+10 x+16}{x^{2}+16 x+64}
$$

Amy Jiang

Numerade Educator

01:04

Problem 52

Divide and simplify. See Example 7
$$
\frac{a^{2}-a-6}{a^{2}-81} \div \frac{a^{2}-7 a-18}{4 a+36}
$$

Amy Jiang

Numerade Educator

00:54

Problem 53

Divide and simplify. See Example 7
$$
\frac{3 x-x^{2}}{x^{3}-27} \div \frac{x}{x^{2}+3 x+9}
$$

Amy Jiang

Numerade Educator

00:56

Problem 54

Divide and simplify. See Example 7
$$
\frac{x^{2}-3 x}{x^{3}-27} \div \frac{2 x}{2 x^{2}+6 x+18}
$$

Amy Jiang

Numerade Educator

01:20

Problem 55

Divide and simplify. See Example 7
$$
\frac{8 b+24}{3 a+6} \div \frac{a b-2 b+3 a-6}{a^{2}-4 a+4}
$$

Amy Jiang

Numerade Educator

01:07

Problem 56

Divide and simplify. See Example 7
$$
\frac{2 a^{2}-2 b^{2}}{a^{3}+a^{2} b+a+b} \div \frac{6 a^{2}}{a^{3}+a}
$$

Amy Jiang

Numerade Educator

00:44

Problem 57

Perform each indicated operation. See Examples 2 through $8 .$
$$
\frac{x^{2}-9}{4} \cdot \frac{x^{2}-x-6}{x^{2}-6 x+9}
$$

Amy Jiang

Numerade Educator

00:44

Problem 58

Perform each indicated operation. See Examples 2 through $8 .$
$$
\frac{x^{2}-4}{9} \cdot \frac{x^{2}-6 x+9}{x^{2}-5 x+6}
$$

Amy Jiang

Numerade Educator

02:03

Problem 59

Perform each indicated operation. See Examples 2 through $8 .$
$$
\frac{2 x^{2}-4 x-30}{5 x^{2}-40 x-75} \div \frac{x^{2}-8 x+15}{x^{2}-6 x+9}
$$

Amy Jiang

Numerade Educator

01:16

Problem 60

Perform each indicated operation. See Examples 2 through $8 .$
$$
\frac{4 a+36}{a^{2}-7 a-18} \div \frac{a^{2}-a-6}{a^{2}-81}
$$

Amy Jiang

Numerade Educator

00:25

Problem 61

Perform each indicated operation. See Examples 2 through $8 .$
$$
\text { Simplify: } \frac{r^{3}+s^{3}}{r+s}
$$

Amy Jiang

Numerade Educator

Problem 62

Perform each indicated operation. See Examples 2 through $8 .$
$$
\text { Simplify: } \frac{m^{3}-n^{3}}{m-n}
$$

Amy Jiang

Numerade Educator

Problem 63

Perform each indicated operation. See Examples 2 through $8 .$
$$
\frac{4}{x} \div \frac{3 x y}{x^{2}} \cdot \frac{6 x^{2}}{x^{4}}
$$

Amy Jiang

Numerade Educator

Problem 64

Perform each indicated operation. See Examples 2 through $8 .$
$$
\frac{4}{x} \cdot \frac{3 x y}{x^{2}} \div \frac{6 x^{2}}{x^{4}}
$$

Amy Jiang

Numerade Educator

Problem 65

Perform each indicated operation. See Examples 2 through $8 .$
$$
\frac{3 x^{2}-5 x-2}{y^{2}+y-2} \cdot \frac{y^{2}+4 y-5}{12 x^{2}+7 x+1} \div \frac{5 x^{2}-9 x-2}{8 x^{2}-2 x-1}
$$

Amy Jiang

Numerade Educator

Problem 66

Perform each indicated operation. See Examples 2 through $8 .$
$$
\frac{x^{2}+x-2}{3 y^{2}-5 y-2} \cdot \frac{12 y^{2}+y-1}{x^{2}+4 x-5} \div \frac{8 y^{2}-6 y+1}{5 y^{2}-9 y-2}
$$

Amy Jiang

Numerade Educator

01:16

Problem 67

Perform each indicated operation. See Examples 2 through $8 .$
$$
\frac{5 a^{2}-20}{3 a^{2}-12 a} \div \frac{a^{3}+2 a^{2}}{2 a^{2}-8 a} \cdot \frac{9 a^{3}+6 a^{2}}{2 a^{2}-4 a}
$$

Amy Jiang

Numerade Educator

01:16

Problem 68

Perform each indicated operation. See Examples 2 through $8 .$
$$
\frac{5 a^{2}-20}{3 a^{2}-12 a} \div\left(\frac{a^{3}+2 a^{2}}{2 a^{2}-8 a} \cdot \frac{9 a^{3}+6 a^{2}}{2 a^{2}-4 a}\right)
$$

Amy Jiang

Numerade Educator

06:41

Problem 69

Perform each indicated operation. See Examples 2 through $8 .$
$$
\frac{5 x^{4}+3 x^{2}-2}{x-1} \cdot \frac{x+1}{x^{4}-1}
$$

Nick Johnson

Numerade Educator

02:03

Problem 70

Perform each indicated operation. See Examples 2 through $8 .$
$$
\frac{3 x^{4}-10 x^{2}-8}{x-2} \cdot \frac{3 x+6}{15 x^{2}+10}
$$

Amy Jiang

Numerade Educator

Problem 71

Find each function value. See Example $9 .$
If $f(x)=\frac{x+8}{2 x-1},$ find $f(2), f(0),$ and $f(-1)$

Amy Jiang

Numerade Educator

00:42

Problem 72

Find each function value. See Example $9 .$
If $f(x)=\frac{x-2}{-5+x},$ find $f(-5), f(0),$ and $f(10)$

Amy Jiang

Numerade Educator

01:08

Problem 73

Find each function value. See Example $9 .$
If $g(x)=\frac{x^{2}+8}{x^{3}-25 x},$ find $g(3), g(-2),$ and $g(1)$

Amy Jiang

Numerade Educator

00:59

Problem 74

Find each function value. See Example $9 .$
If $s(t)=\frac{t^{3}+1}{t^{2}+1},$ find $s(-1), s(1),$ and $s(2)$

Amy Jiang

Numerade Educator

02:46

Find each function value. See Example $9 .$
The total revenue from the sale of a popular book is approximated by the rational function $R(x)=\frac{1000 x^{2}}{x^{2}+4},$ where $x$ is the number of years since publication and $R(x)$ is the total revenue in millions of dollars.
A. Find the total revenue at the end of the first year.
B. Find the total revenue at the end of the second year.
C. Find the revenue during the second year only.
D. Find the domain of function $R$

Heather Zimmers

Numerade Educator

04:36

Problem 76

Find each function value. See Example $9 .$
The function $f(x)=\frac{100,000 x}{100-x}$ models the cost in dollars for removing $x$ percent of the pollutants from a bayou in which
a nearby company dumped creosol.
a. Find the cost of removing $20 \%$ of the pollutants from the bayou. [Hint: Find $f(20) .]$
b. Find the cost of removing $60 \%$ of the pollutants and then $80 \%$ of the pollutants.
c. Find $f(90),$ then $f(95),$ and then $f(99) .$ What happens to the cost as $x$ approaches $100 \% ?$
d. Find the domain of function $f$

Heather Zimmers

Numerade Educator

00:11

Problem 77

Perform each indicated operation. See Section 1.3
$$
\frac{4}{5}+\frac{3}{5}
$$

Amy Jiang

Numerade Educator

00:11

Problem 78

Perform each indicated operation. See Section 1.3
$$
\frac{4}{10}-\frac{7}{10}
$$

Amy Jiang

Numerade Educator

01:00

Problem 79

Perform each indicated operation. See Section 1.3
$$
\frac{5}{28}-\frac{2}{21}
$$

Amy Jiang

Numerade Educator

Problem 80

Perform each indicated operation. See Section 1.3
$$
\frac{5}{13}+\frac{2}{7}
$$

Amy Jiang

Numerade Educator

00:22

Problem 81

Perform each indicated operation. See Section 1.3
$$
\frac{3}{8}+\frac{1}{2}-\frac{3}{16}
$$

Amy Jiang

Numerade Educator

00:39

Problem 82

Perform each indicated operation. See Section 1.3
$$
\frac{2}{9}-\frac{1}{6}+\frac{2}{3}
$$

Amy Jiang

Numerade Educator

01:10

Problem 83

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
Which of the expressions are equivalent to $\frac{x}{5-x} ?$
a. $\frac{-x}{5-x}$
b. $\frac{-x}{-5+x}$
c. $\frac{x}{x-5}$
d. $\frac{-x}{x-5}$

Amy Jiang

Numerade Educator

Problem 84

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
Which of the expressions are equivalent to $\frac{-2+x}{x} ?$
a. $\frac{2-x}{-x}$
b. $-\frac{2-x}{x}$
c. $\frac{x-2}{x}$
d. $\frac{x-2}{-x}$

Amy Jiang

Numerade Educator

Problem 85

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
Does $\frac{x}{x+5}$ simplify to $\frac{1}{5} ?$ Why or why not?

Amy Jiang

Numerade Educator

Problem 86

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
Does $\frac{x+7}{x}$ simplify to $7 ?$ Why or why not?

Amy Jiang

Numerade Educator

Problem 87

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
Find the area of the rectangle.
(IMAGE CANNOT COPY)

Amy Jiang

Numerade Educator

00:45

Problem 88

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
Find the area of the triangle.
(IMAGE CANNOT COPY)

Mrinal Rana

Numerade Educator

Problem 89

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
A parallelogram has an area of $\frac{x^{2}+x-2}{x^{3}}$ square feet and a height of $\frac{x^{2}}{x-1}$ feet. Express the length of its base as a rational expression in $x .$ (Hint: since $A=b \cdot h, b=\frac{A}{h}$ or $b=A \div h .)$
(IMAGE CANNOT COPY)

Amy Jiang

Numerade Educator

Problem 90

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
A lottery prize of $\frac{15 x^{3}}{y^{2}}$ dollars is to be divided among $5 x$ people. Express the amount of money each person is to receive as a rational expression in $x$ and $y .$

Amy Jiang

Numerade Educator

Problem 91

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
In your own words, explain how to simplify a rational expression.

Amy Jiang

Numerade Educator

Problem 92

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
In your own words, explain the difference between multiplying rational expressions and dividing rational expressions.

Amy Jiang

Numerade Educator

01:10

Problem 93

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
Decide whether each rational expression equals $1,-1,$ or neither.
a. $\frac{x+5}{5+x}$
b. $\frac{x-5}{5-x}$
c. $\frac{x+5}{x-5}$
d. $\frac{-x-5}{x+5}$
e. $\frac{x-5}{-x+5}$
f. $\frac{-5+x}{x-5}$

Amy Jiang

Numerade Educator

Problem 94

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
Find the polynomial in the second numerator such that the following statement is true.
$$
\frac{x^{2}-4}{x^{2}-7 x+10} \cdot \frac{?}{2 x^{2}+11 x+14}=1
$$

Amy Jiang

Numerade Educator

Problem 95

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
In our definition of division for
$$
\frac{P}{Q} \div \frac{R}{S}
$$
we stated that $Q \neq 0, S \neq 0,$ and $R \neq 0 .$ Explain why $R$ cannot equal 0.

Amy Jiang

Numerade Educator

View

Problem 96

Solve. For Exercises 83 and $84,$ see the second Concept Check in this section, for Exercises 85 and $86,$ see the third Concept Check.
In your own words, explain how to find the domain of a rational function.

Cara Casner

Numerade Educator