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Intermediate Algebra for College Students

Allen R. Angel; Dennis C. Runde

Chapter 6

Rational Expressions and Equations - all with Video Answers

Educators

Section 1

The Domains of Rational Functions and Multiplication and Division of Rational Expressions

00:26

Problem 1

Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.
$\text { multiply }\quad \text { divide } \quad \text { add } \quad \text { function } \quad \text { domain }\quad \text { rational } \quad \text { range } \quad \text { simplify}$
A ______ expression is of the form $\frac{p}{q^{\prime}}$, where $p$ and $q$ are polynomials and $q \neq 0$.

AG
Ankit Gupta
Numerade Educator
00:26

Problem 2

Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.
$\text { multiply }\quad \text { divide } \quad \text { add } \quad \text { function } \quad \text { domain }\quad \text { rational } \quad \text { range } \quad \text { simplify}$
A rational ___________ is of the form $y=f(x)=\frac{p}{q}$, where $p$ and $q$ are polynomials and $q \neq 0$.

AG
Ankit Gupta
Numerade Educator
01:13

Problem 3

Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.
$\text { multiply }\quad \text { divide } \quad \text { add } \quad \text { function } \quad \text { domain }\quad \text { rational } \quad \text { range } \quad \text { simplify}$
The _______ of a rational function is the set of all real numbers for which the denominator is not equal to 0 .

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:34

Problem 4

Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.
$\text { multiply }\quad \text { divide } \quad \text { add } \quad \text { function } \quad \text { domain }\quad \text { rational } \quad \text { range } \quad \text { simplify}$
To __________ a rational expression, first factor the numerator and denominator and then divide out any common factors.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
00:26

Problem 5

Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.
$\text { multiply }\quad \text { divide } \quad \text { add } \quad \text { function } \quad \text { domain }\quad \text { rational } \quad \text { range } \quad \text { simplify}$
To ________ two rational expressions, first factor all numerators and denominators and divide out any common factors.

AG
Ankit Gupta
Numerade Educator
00:26

Problem 6

Fill in the blanks with the appropriate word, phrase, or symbol(s) from the following list.
$\text { multiply }\quad \text { divide } \quad \text { add } \quad \text { function } \quad \text { domain }\quad \text { rational } \quad \text { range } \quad \text { simplify}$
To ______ two rational expressions, multiply the first rational expression by the reciprocal of the second rational expression.

AG
Ankit Gupta
Numerade Educator
01:02

Problem 7

Determine the values that are excluded in the following expressions.
$$
\frac{x+3}{x-4}
$$

Aatish Gupta
Aatish Gupta
Numerade Educator
05:50

Problem 8

Determine the values that are excluded in the following expressions.
$$
\frac{t-2}{t+5}
$$

Aditya Modi
Aditya Modi
Monroe Township High School
03:37

Problem 9

Determine the values that are excluded in the following expressions.
$$
\frac{2 x+1}{3 x-4}
$$

Courtney Solomon
Courtney Solomon
Numerade Educator
00:12

Problem 10

Determine the values that are excluded in the following expressions.
$$
\frac{4 x-3}{7 x+5}
$$

Amy Jiang
Amy Jiang
Numerade Educator
01:43

Problem 11

Determine the values that are excluded in the following expressions.
$$
\frac{4}{2 x^{2}-15 x+25}
$$

Aditya Modi
Aditya Modi
Monroe Township High School
03:12

Problem 12

Determine the values that are excluded in the following expressions.
$$
\frac{3 x+5}{5 x^{2}-x-4}
$$

Ruby P
Ruby P
Numerade Educator
00:49

Problem 13

Determine the values that are excluded in the following expressions.
$$
\frac{x^{2}+81}{x^{2}-81}
$$

Aditya Modi
Aditya Modi
Monroe Township High School
01:34

Problem 14

Determine the values that are excluded in the following expressions.
$$
\frac{x+2}{x^{2}-64}
$$

Aditya Modi
Aditya Modi
Monroe Township High School
01:35

Problem 15

Determine the values that are excluded in the following expressions.
$$
\frac{x-3}{x^{2}+12}
$$

Aditya Modi
Aditya Modi
Monroe Township High School
01:16

Problem 16

Determine the values that are excluded in the following expressions.
$$
\frac{x^{2}-36}{x^{2}+36}
$$

Aditya Modi
Aditya Modi
Monroe Township High School
01:03

Problem 17

Determine the domain of each function.
$$
f(x)=\frac{x-4}{x-5}
$$

Alexandra Ali
Alexandra Ali
Numerade Educator
00:25

Problem 18

Determine the domain of each function.
$$
f(t)=\frac{t+7}{t+3}
$$

Michelle Nguyen
Michelle Nguyen
Numerade Educator
00:13

Problem 19

Determine the domain of each function.
$$
s(t)=\frac{t+5}{3 t^{2}-11 t-4}
$$

Michelle Nguyen
Michelle Nguyen
Numerade Educator
00:16

Problem 20

Determine the domain of each function.
$$
h(r)=\frac{2 r-7}{2 r^{2}+7 r-15}
$$

Michelle Nguyen
Michelle Nguyen
Numerade Educator
01:10

Problem 21

Determine the domain of each function.
$$
f(a)=\frac{a^{2}+3 a+2}{a^{2}+4 a+3}
$$

Kian Manafi
Kian Manafi
Numerade Educator
02:11

Problem 22

Determine the domain of each function.
$$
f(b)=\frac{b^{2}+8 b}{h^{2}-64}
$$

Jasmine Gordy
Jasmine Gordy
Numerade Educator
00:53

Problem 23

Determine the domain of each function.
$$
g(x)=\frac{x^{2}-x+8}{x^{2}+4}
$$

Erika Bustos
Erika Bustos
Numerade Educator
00:11

Problem 24

Determine the domain of each function.
$$
h(x)=\frac{x^{3}-64 x}{x^{2}+81}
$$

Michelle Nguyen
Michelle Nguyen
Numerade Educator
01:07

Problem 25

Determine the domain of each function.
$$
f(x)=\frac{x+7}{x^{3}+3 x^{2}-4 x-12}
$$

Linh Vu
Linh Vu
Numerade Educator
01:10

Problem 26

Determine the domain of each function.
$$
f(x)=\frac{x-3}{x^{3}-2 x^{2}-9 x+18}
$$

Kian Manafi
Kian Manafi
Numerade Educator
00:43

Problem 27

Simplify each rational expression.
$$
\frac{x^{2}+10 x}{2 x^{2}-5 x}
$$

Victor Salazar
Victor Salazar
Numerade Educator
00:25

Problem 28

Simplify each rational expression.
$$
\frac{3 y^{2}-4 y}{y^{2}+8 y}
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:30

Problem 29

Simplify each rational expression.
$$
\frac{15-6 x}{14 x-35}
$$

Erika Bustos
Erika Bustos
Numerade Educator
00:37

Problem 30

Simplify each rational expression.
$$
\frac{15 x-10}{8-12 x}
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:06

Problem 31

Simplify each rational expression.
$$
\frac{a^{2}-3 a-10}{a^{2}+5 a+6}
$$

AG
Ankit Gupta
Numerade Educator
01:25

Problem 32

Simplify each rational expression.
$$
\frac{b^{2}+5 b-14}{b^{2}+4 b-12}
$$

AG
Ankit Gupta
Numerade Educator
00:45

Problem 33

Simplify each rational expression.
$$
\frac{p^{2}-2 p-24}{6-p}
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:01

Problem 34

Simplify each rational expression.
$$
\frac{3 q^{2}+8 q-35}{7-3 q}
$$

Narayan Hari
Narayan Hari
Numerade Educator
00:50

Problem 35

Simplify each rational expression.
$$
\frac{16 x^{2}+34 x-15}{18 x^{2}+37 x-20}
$$

Ramzi Deek
Ramzi Deek
Numerade Educator
01:05

Problem 36

Simplify each rational expression.
$$
\frac{6 x^{2}+13 x-63}{12 x^{2}-13 x-35}
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:05

Problem 37

Simplify each rational expression.
$$
\frac{x^{2}+3 x y-18 y^{2}}{x^{2}+5 x y-6 y^{2}}
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:03

Problem 38

Simplify each rational expression.
$$
\frac{y^{2}-10 y z+24 z^{2}}{y^{2}-5 y z+4 z^{2}}
$$

Prashansha Kaushik
Prashansha Kaushik
Numerade Educator
00:43

Problem 39

Simplify each rational expression.
$$
\frac{9 x^{2}-49}{6 x^{2}-29 x+35}
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:28

Problem 40

Simplify each rational expression.
$$
\frac{4 x^{2}-9}{2 x^{2}-x-3}
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:20

Problem 41

Simplify each rational expression.
$$
\frac{-12 x^{2}-14 x+10}{-8 x^{2}-8 x+6}
$$

Alison Rodriguez
Alison Rodriguez
Numerade Educator
02:03

Problem 42

Simplify each rational expression.
$$
\frac{-18 y^{2}+39 y+15}{-36 y^{2}+120 y-75}
$$

AG
Ankit Gupta
Numerade Educator
01:29

Problem 43

Simplify each rational expression.
$$
\frac{6 y^{2}-17 y-14}{49-4 y^{2}}
$$

AG
Ankit Gupta
Numerade Educator
01:06

Problem 44

Simplify each rational expression.
$$
\frac{10 z^{2}+11 z-6}{4-25 z^{2}}
$$

Angela Guo
Angela Guo
Numerade Educator
00:18

Problem 45

Simplify each rational expression.
$$
\frac{a^{2}+7 a-a b-7 b}{a^{2}-a b+5 a-5 b}
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:06

Problem 46

Simplify each rational expression.
$$
\frac{x y-y w+x z-z w}{x y+y w+x z+z w}
$$

Ashley Volpe
Ashley Volpe
Numerade Educator
00:32

Problem 47

Simplify each rational expression.
$$
\frac{x^{2}-x-12}{x^{3}+27}
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:30

Problem 48

Simplify each rational expression.
$$
\frac{x^{2}-16}{x^{3}-64}
$$

Erika Bustos
Erika Bustos
Numerade Educator
00:45

Problem 49

Multiply or divide as indicated. Simplify all answers.
$$
\frac{3 x}{5 y^{2}} \cdot \frac{y}{9}
$$

James Kiss
James Kiss
Numerade Educator
00:52

Problem 50

Multiply or divide as indicated. Simplify all answers.
$$
\frac{5 a^{2}}{7 b^{2}} \cdot \frac{b}{15 a}
$$

Ashley High
Ashley High
Numerade Educator
01:33

Problem 51

Multiply or divide as indicated. Simplify all answers.
$$
\frac{18 x^{2} y^{3}}{25 p^{2}} \div \frac{27 x^{2} y}{15 p}
$$

James Kiss
James Kiss
Numerade Educator
01:38

Problem 52

Multiply or divide as indicated. Simplify all answers.
$$
\frac{10 m^{4}}{49 x^{5} y^{7}} \div \frac{25 m^{5}}{21 x^{12} y^{5}}
$$

Brianna Orr
Brianna Orr
Numerade Educator
01:02

Problem 53

Multiply or divide as indicated. Simplify all answers.
$$
\frac{2 x^{2}-2 x y}{3 x y+3 y^{2}} \cdot \frac{9 x y^{2}+9 y^{3}}{4 x y-4 x^{2}}
$$

James Kiss
James Kiss
Numerade Educator
06:08

Problem 54

Multiply or divide as indicated. Simplify all answers.
$$
\frac{5 v^{2}+5 v w}{7 v w-7 w^{2}} \cdot \frac{21 w^{2}-21 v w}{10 v^{3}+10 v^{2} w}
$$

Ruby P
Ruby P
Numerade Educator
03:19

Problem 55

Multiply or divide as indicated. Simplify all answers.
$$
\frac{x^{2}+3 x-10}{4 x} \cdot \frac{x^{2}-3 x}{x^{2}-5 x+6}
$$

Gladys Fowler
Gladys Fowler
Numerade Educator
01:03

Problem 56

Multiply or divide as indicated. Simplify all answers.
$$
\frac{9 y}{y^{2}+4 y-21} \cdot \frac{y^{2}+y-12}{3 y+12}
$$

Amy Jiang
Amy Jiang
Numerade Educator
02:23

Problem 57

Multiply or divide as indicated. Simplify all answers.
$$
\frac{r^{2}+10 r+21}{r+7} \div \frac{r^{2}-5 r-24}{r^{3}}
$$

Alison Rodriguez
Alison Rodriguez
Numerade Educator
01:41

Problem 58

Multiply or divide as indicated. Simplify all answers.
$$
\frac{s^{3}}{2 s^{2}-11 s-21} \div \frac{s^{2}-7 s}{2 s^{2}-3 s-9}
$$

AG
Ankit Gupta
Numerade Educator
02:22

Problem 59

Multiply or divide as indicated. Simplify all answers.
$$
\frac{x^{2}+12 x+35}{x^{2}+4 x-5} \div \frac{x^{2}+3 x-28}{7 x-7}
$$

AG
Ankit Gupta
Numerade Educator
03:21

Problem 60

Multiply or divide as indicated. Simplify all answers.
$$
\frac{x+1}{x^{2}-17 x+30} \div \frac{8 x+8}{x^{2}+7 x-18}
$$

James Kiss
James Kiss
Numerade Educator
01:38

Problem 61

Multiply or divide as indicated. Simplify all answers.
$$
\frac{a-b}{9 a+9 b} \div \frac{a^{2}-b^{2}}{a^{2}+2 a+1}
$$

AG
Ankit Gupta
Numerade Educator
00:39

Problem 62

Multiply or divide as indicated. Simplify all answers.
$$
\frac{p^{2}-q^{2}}{p^{2}+2 p q+q^{2}} \div \frac{5 p-5 q}{p^{2}-2 p q+q^{2}}
$$

Erika Bustos
Erika Bustos
Numerade Educator
03:37

Problem 63

Multiply or divide as indicated. Simplify all answers.
$$
\frac{3 x^{2}-x-4}{4 x^{2}+5 x+1} \cdot \frac{2 x^{2}-5 x-12}{6 x^{2}+x-12}
$$

Gladys Fowler
Gladys Fowler
Numerade Educator
02:40

Problem 64

Multiply or divide as indicated. Simplify all answers.
$$
\frac{6 x^{3}-x^{2}-x}{2 x^{2}+x-1} \cdot \frac{x^{2}-1}{x^{3}-2 x^{2}+x}
$$

Taylor Shimono
Taylor Shimono
Numerade Educator
00:37

Problem 65

Multiply or divide as indicated. Simplify all answers.
$$
\frac{x+2}{x^{3}-8} \cdot \frac{(x-2)^{2}}{x^{2}+4}
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:18

Problem 66

Multiply or divide as indicated. Simplify all answers.
$$
\frac{(a-b)^{3}}{a^{3}-b^{3}} \cdot \frac{a^{2}-b^{2}}{(a-b)^{2}}
$$

James Kiss
James Kiss
Numerade Educator
01:42

Problem 67

Multiply or divide as indicated. Simplify all answers.
$$
\frac{x^{2}-y^{2}}{x^{2}-2 x y+y^{2}} \div \frac{(x+y)^{2}}{(x-y)^{2}}
$$

AG
Ankit Gupta
Numerade Educator
01:26

Problem 68

Multiply or divide as indicated. Simplify all answers.
$$
\frac{\left(x^{2}-y^{2}\right)^{2}}{\left(x^{2}-y^{2}\right)^{3}} \div \frac{x^{2}+y^{2}}{x^{4}-y^{4}}
$$

Erika Bustos
Erika Bustos
Numerade Educator
03:11

Problem 69

Multiply or divide as indicated. Simplify all answers.
$$
\frac{2 x^{4}+4 x^{2}}{6 x^{2}+14 x+4} \div \frac{x^{2}+2}{3 x^{2}+x}
$$

Gladys Fowler
Gladys Fowler
Numerade Educator
00:32

Problem 70

Multiply or divide as indicated. Simplify all answers.
$$
\frac{2 x^{3}-7 x^{2}+3 x}{x^{2}+2 x-3} \cdot \frac{x^{2}+3 x}{(x-3)^{2}}
$$

AG
Ankit Gupta
Numerade Educator
05:25

Problem 71

Multiply or divide as indicated. Simplify all answers.
$$
\frac{8 a^{3}-1}{4 a^{2}+2 a+1} \div \frac{a^{2}-2 a+1}{(a-1)^{2}}
$$

James Kiss
James Kiss
Numerade Educator
01:34

Problem 72

Multiply or divide as indicated. Simplify all answers.
$$
\frac{r^{2}-16}{r^{3}-64} \div \frac{r^{2}+8 r+16}{r^{2}+4 r+16}
$$

Daniel Nolan
Daniel Nolan
Numerade Educator
01:42

Problem 73

Multiply or divide as indicated. Simplify all answers.
$$
\frac{2 a^{2}-17 a+21}{2 a^{2}-13 a+15} \div \frac{3 a-21}{35-7 a}
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:54

Problem 74

Multiply or divide as indicated. Simplify all answers.
$$
\frac{12 b^{2}+13 b-35}{6 b^{2}-13 b-63} \div \frac{30-24 b}{10 b-45}
$$

Ernest Castorena
Ernest Castorena
Numerade Educator
03:21

Problem 75

Multiply or divide as indicated. Simplify all answers.
$$
\frac{-8 x^{2}+18}{-36 x^{2}+100} \cdot \frac{3 x^{2}+17 x+20}{2 x^{2}+5 x-12}
$$

James Kiss
James Kiss
Numerade Educator
01:05

Problem 76

Multiply or divide as indicated. Simplify all answers.
$$
\frac{-48 x^{2}-114 x+15}{32 x^{2}-52 x+6} \cdot \frac{4 x^{2}-9}{4 x^{2}-25}
$$

AG
Ankit Gupta
Numerade Educator
01:20

Problem 77

Multiply or divide as indicated. Simplify all answers.
$$
\frac{a c-a d+b c-b d}{a c+a d+b c+b d} \cdot \frac{p c+p d-q c-q d}{p c-p d+q c-q d}
$$

Erika Bustos
Erika Bustos
Numerade Educator
00:39

Problem 78

Multiply or divide as indicated. Simplify all answers.
$$
\frac{2 p^{2}+2 p q-p q^{2}-q^{3}}{p^{3}+p^{2}+p q^{2}+q^{2}} \div \frac{p^{3}+p+p^{2} q+q}{p^{3}+p+p^{2}+1}
$$

Erika Bustos
Erika Bustos
Numerade Educator
00:32

Problem 79

Determine the polynomial to be placed in the shaded area to give a true statement.
$$
\frac{}{x^{2}+2 x-15}=\frac{1}{x-3}
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:16

Problem 80

Determine the polynomial to be placed in the shaded area to give a true statement.
$$
\overline{3 x+2}=x-3
$$

Robert Daugherty
Robert Daugherty
Numerade Educator
02:02

Problem 81

Determine the polynomial to be placed in the shaded area to give a true statement.
$$
\underline{y^{2}-y-20}=\frac{y+4}{y+1}
$$

Bailey Brooks
Bailey Brooks
Numerade Educator
01:19

Problem 82

Determine the polynomial to be placed in the shaded area to give a true statement.
$$
\frac{2 p-1}{6 p^{2}+p-15}=\frac{2 p}{2 p-3}
$$

Erika Bustos
Erika Bustos
Numerade Educator
03:14

Problem 83

Determine the polynomial to be placed in the shaded area to give a true statement.
$$
\frac{x^{2}-x-12}{x^{2}+2 x-3} \cdot \frac{}{x^{2}-2 x-8}=1
$$

Vikash Ranjan
Vikash Ranjan
Numerade Educator
03:14

Problem 84

Determine the polynomial to be placed in the shaded area to give a true statement.
$$
\frac{x^{2}-4}{(x+2)^{2}} \cdot \frac{2 x^{2}+x-6} ? =\frac{x-2}{2 x+5}
$$

Vikash Ranjan
Vikash Ranjan
Numerade Educator
01:11

Problem 85

Determine the polynomial to be placed in the shaded area to give a true statement.
$$
\frac{x^{2}-9}{2 x^{2}+3 x-2} \div \frac{2 x^{2}-9 x+9}?=\frac{x+3}{2 x-1}
$$

Adrian Co
Adrian Co
Numerade Educator
01:19

Problem 86

Determine the polynomial to be placed in the shaded area to give a true statement.
$$
\frac{4 r^{2}-r-18}? \div \frac{4 r^{3}-9 r^{2}}{6 r^{2}-9 r+3}=\frac{3(r-1)}{r^{2}}
$$

Erika Bustos
Erika Bustos
Numerade Educator
00:35

Problem 87

Consider the rectangle below. Its area is $3 a^{2}+7 a b+2 b^{2}$ and its length is $2 a+4 b$. Determine its width, $w$, in terms of $a$ and $b$, by dividing its area by its length.

AG
Ankit Gupta
Numerade Educator
01:12

Problem 88

Consider the rectangle below. Its area is $a^{2}+2 a b+b^{2}$ and its length is $3 a+3 b$. Determine its width, $w$, in terms of $a$ and $b$, by dividing its area by its length.

Julie Silva
Julie Silva
Numerade Educator
01:54

Problem 89

Consider the triangle below. If its area is $a^{2}+4 a b+3 b^{2}$ and its base is $a+3 b$, determine its height $h$. Use area $=\frac{1}{2}$ (base) (height).

Narayan Hari
Narayan Hari
Numerade Educator
06:42

Problem 90

Consider the trapezoid below. If its area is $a^{2}+2 a b+b^{2}$, determine its height, $h$. Use area $=\frac{1}{2} h(a+b)$.

Denae Mcneily
Denae Mcneily
Numerade Educator
01:04

Problem 91

Perform each indicated operation.
$$
\left(\frac{2 x^{2}-3 x-14}{2 x^{2}-9 x+7} \div \frac{6 x^{2}+x-15}{3 x^{2}+2 x-5}\right) \cdot \frac{6 x^{2}-7 x-3}{2 x^{2}-x-3}
$$

Brandon Fox
Brandon Fox
Numerade Educator
02:16

Problem 92

Perform each indicated operation.
$$
\left(\frac{a^{2}-b^{2}}{2 a^{2}-3 a b+b^{2}} \cdot \frac{2 a^{2}-7 a b+3 b^{2}}{a^{2}+a b}\right) \div \frac{a b-3 b^{2}}{a^{2}+2 a b+b^{2}}
$$

AG
Ankit Gupta
Numerade Educator
05:55

Problem 93

Perform each indicated operation.
$$
\frac{5 x^{2}(x-1)-3 x(x-1)-2(x-1)}{10 x^{2}(x-1)+9 x(x-1)+2(x-1)} \cdot \frac{2 x+1}{x+3}
$$

Aryan Malesha
Aryan Malesha
Numerade Educator
04:04

Problem 94

Perform each indicated operation.
$$
\frac{x^{2}(3 x-y)-5 x(3 x-y)-24(3 x-y)}{x^{2}(3 x-y)-9 x(3 x-y)+8(3 x-y)} \cdot \frac{x-1}{x+3}
$$

Anthony Agnessanto
Anthony Agnessanto
Numerade Educator
00:29

Problem 95

Perform each indicated operation.
$$
\frac{(x-p)^{n}}{x^{-2}} \div \frac{(x-p)^{2 n}}{x^{-6}}
$$

Alison Rodriguez
Alison Rodriguez
Numerade Educator
00:53

Problem 96

Perform each indicated operation.
$$
\frac{x^{-3}}{(a-b)^{r}} \div \frac{x^{-5}}{(a-b)^{r+2}}
$$

Alison Rodriguez
Alison Rodriguez
Numerade Educator
01:20

Problem 97

Simplify.
$$
\frac{x^{5 y}+3 x^{4 y}}{3 x^{3 y}+x^{4 y}}
$$

Sneha Ravi
Sneha Ravi
Numerade Educator
00:30

Problem 98

Simplify.
$$
\frac{m^{2 x}-m^{x}-2}{m^{2 x}-4}
$$

Amy Jiang
Amy Jiang
Numerade Educator
00:34

Problem 99

For Exercises 99-102,
a) Determine the domain of the function.
b) Graph the function in connected mode.
$$
f(x)=\frac{1}{x-2}
$$

Linh Vu
Linh Vu
Numerade Educator
00:27

Problem 100

For Exercises 99-102,
a) Determine the domain of the function.
b) Graph the function in connected mode.
$$
f(x)=\frac{x}{x-2}
$$

Linh Vu
Linh Vu
Numerade Educator
00:27

Problem 101

For Exercises 99-102,
a) Determine the domain of the function.
b) Graph the function in connected mode.
$$
f(x)=\frac{x^{2}}{x-2}
$$

Linh Vu
Linh Vu
Numerade Educator
00:27

Problem 102

For Exercises 99-102,
a) Determine the domain of the function.
b) Graph the function in connected mode.
$$
f(x)=\frac{x-2}{x-2}
$$

Linh Vu
Linh Vu
Numerade Educator
03:12

Problem 103

Consider the rational function $f(x)=\frac{1}{x}$.
a) Determine the domain of the function.
b) Complete the following table for the function.
$$
\begin{array}{l|l|l|l|l|l|l|l|l|l|l}
\hline x & -10 & -1 & -0.5 & -0.1 & -0.01 & 0.01 & 0.1 & 0.5 & 1 & 10 \\
\hline y & & & & & & & & & & \\
\hline
\end{array}
$$
c) Draw the graph of $f(x)=\frac{1}{x}$. Consider what happens to the function as $x$ gets closer and closer to 0 , approaching 0 from both the left and right sides.
d) Can this fraction ever have a value of 0? Explain your answer.

Monique Rousselle Maynard
Monique Rousselle Maynard
Numerade Educator
00:19

Problem 104

Make up a rational expression that is undefined at $x=4$ and $x=-5$. Explain how you determined your answer.

Amy Jiang
Amy Jiang
Numerade Educator
00:27

Problem 105

Make up a rational expression that is undefined at $x=2$ and $x=-3$. Explain how you determined your answer.

Erika Bustos
Erika Bustos
Numerade Educator
00:43

Problem 106

Consider the rational function $g(x)=\frac{2}{x+3}$. Explain why this function can never equal 0 .

Erika Bustos
Erika Bustos
Numerade Educator
00:46

Problem 107

Consider the rational function $f(x)=\frac{1}{x}$. Explain why this function can never equal 0 .

Palitha Reddy
Palitha Reddy
Numerade Educator
01:54

Problem 108

Consider the function $f(x)=\frac{x-2}{x^{2}-81}$. For what value of $x$, if any, will this function a) equal 0 ? b) be undefined? Explain.

Yifan Xu
Yifan Xu
Numerade Educator
02:56

Problem 109

Consider the rational function $f(x)=\frac{x-4}{x^{2}-36}$. For what value of $x$, if any, will this function
a) equal 0 ?
b) be undefined? Explain.

Suzanne Harwood
Suzanne Harwood
Numerade Educator
02:00

Problem 110

Give a function that is undefined at $x=-4$ and $x=-2$ and has a value of 0 at $x=5$. Explain how you determined your answer.

Dale Sanford
Dale Sanford
Numerade Educator
00:25

Problem 111

Give a function that is undefined at $x=3$ and $x=-1$ and has a value of 0 at $x=2$. Explain how you determined your answer.

Brandon Fox
Brandon Fox
Numerade Educator
02:56

Problem 112

Consider the rational function $f(x)=\frac{x^{2}-4}{x-2}$.
a) As a group, determine the domain of this function.
b) Have each member of the group individually complete the following table for the function.
$$
\begin{array}{l|l|l|l|l|l|l|l|l|l|l|l|l}
\hline x & -2 & -1 & 0 & 1 & 1.9 & 1.99 & 2.01 & 2.1 & 3 & 4 & 5 & 6 \\
\hline y & & & & & & & & & & & & \\
\hline
\end{array}
$$
c) Compare your answers to part b) and agree on the correct table values.
d) As a group draw the graph of $f(x)=\frac{x^{2}-4}{x-2}$. Is the function defined when $x=2 ?$
e) Can this function ever have a value of 0 ? If so, for what value(s) of $a$ is $f(a)=0$ ?

Suzanne Harwood
Suzanne Harwood
Numerade Educator
00:54

Problem 113

Solve $6(x-2)+6 y=12 x$ for $y$

Holly Miner
Holly Miner
Numerade Educator
01:22

Problem 114

Solve $4+\frac{4 x}{3}<6$ and give the answer in interval notation.

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:22

Problem 114

Solve $4+\frac{4 x}{3}<6$ and give the answer in interval notation.

Jonathon Brumley
Jonathon Brumley
Numerade Educator
01:01

Problem 115

Solve
$\left|\frac{2 x-4}{12}\right|=5$

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:01

Problem 115

Solve $\left|\frac{2 x-4}{12}\right|=5$.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
00:26

Problem 116

Let $f(x)=|6-3 x|-2$. Determine $f(1.3)$.

Amy Jiang
Amy Jiang
Numerade Educator
01:21

Problem 117

Solve the system of equations.
$$
\begin{aligned}
&3 x+4 y=2 \\
&2 x+5 y=-1
\end{aligned}
$$

Christopher Stanley
Christopher Stanley
Numerade Educator
00:51

Problem 118

Factor $9 x^{2}+6 x y+y^{2}-4$

Matt Gibson
Matt Gibson
Numerade Educator