Elementary and Intermediate Algebra

Educators

Problem 1

In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.
$$f(x)=\frac{2-x}{x-5}$$
a) $\{x | x \neq-5, x \neq 2\}$
b) $\{x | x \neq 3\}$
c) $\{x | x \neq-2\}$
d) $\{x | x \neq-3\}$
e) $\{x | x \neq 5\}$
f) $\{x | x \neq-2, x \neq 5\}$
g) $\{x | x \neq 2\}$
h) $\{x | x \neq-2, x \neq-5\}$
i) $\quad\{x | x \neq 2, x \neq 5\}$
j) $\{x | x \neq-5\}$

Check back soon!

Problem 2

In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.
$$f(x)=\frac{x+5}{x+2}$$
a) $\{x | x \neq-5, x \neq 2\}$
b) $\{x | x \neq 3\}$
c) $\{x | x \neq-2\}$
d) $\{x | x \neq-3\}$
e) $\{x | x \neq 5\}$
f) $\{x | x \neq-2, x \neq 5\}$
g) $\{x | x \neq 2\}$
h) $\{x | x \neq-2, x \neq-5\}$
i) $\quad\{x | x \neq 2, x \neq 5\}$
j) $\{x | x \neq-5\}$

Check back soon!

Problem 3

In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.
$$g(x)=\frac{x-3}{(x-2)(x-5)}$$
a) $\{x | x \neq-5, x \neq 2\}$
b) $\{x | x \neq 3\}$
c) $\{x | x \neq-2\}$
d) $\{x | x \neq-3\}$
e) $\{x | x \neq 5\}$
f) $\{x | x \neq-2, x \neq 5\}$
g) $\{x | x \neq 2\}$
h) $\{x | x \neq-2, x \neq-5\}$
i) $\quad\{x | x \neq 2, x \neq 5\}$
j) $\{x | x \neq-5\}$

Check back soon!

Problem 4

In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.
$$g(x)=\frac{x+3}{(x+2)(x-5)}$$
a) $\{x | x \neq-5, x \neq 2\}$
b) $\{x | x \neq 3\}$
c) $\{x | x \neq-2\}$
d) $\{x | x \neq-3\}$
e) $\{x | x \neq 5\}$
f) $\{x | x \neq-2, x \neq 5\}$
g) $\{x | x \neq 2\}$
h) $\{x | x \neq-2, x \neq-5\}$
i) $\quad\{x | x \neq 2, x \neq 5\}$
j) $\{x | x \neq-5\}$

Check back soon!

Problem 5

In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.
$$-h(x)=\frac{(x-2)(x-3)}{x+3}$$
a) $\{x | x \neq-5, x \neq 2\}$
b) $\{x | x \neq 3\}$
c) $\{x | x \neq-2\}$
d) $\{x | x \neq-3\}$
e) $\{x | x \neq 5\}$
f) $\{x | x \neq-2, x \neq 5\}$
g) $\{x | x \neq 2\}$
h) $\{x | x \neq-2, x \neq-5\}$
i) $\quad\{x | x \neq 2, x \neq 5\}$
j) $\{x | x \neq-5\}$

Check back soon!

Problem 6

In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.
$$f(x)=\frac{(x+2)(x+3)}{x-3}$$
a) $\{x | x \neq-5, x \neq 2\}$
b) $\{x | x \neq 3\}$
c) $\{x | x \neq-2\}$
d) $\{x | x \neq-3\}$
e) $\{x | x \neq 5\}$
f) $\{x | x \neq-2, x \neq 5\}$
g) $\{x | x \neq 2\}$
h) $\{x | x \neq-2, x \neq-5\}$
i) $\quad\{x | x \neq 2, x \neq 5\}$
j) $\{x | x \neq-5\}$

Check back soon!

Problem 7

Jasmine usually takes 3 hr more than Molly does to process a day’s orders at Books To Go. If Molly takes t hr to process a day’s orders, the function given by
$$H(t)=\frac{t^{2}+3 t}{2 t+3}$$
can be used to determine how long it would take if they worked together.
How long will it take them, working together, to complete a day's orders if Molly can process the orders alone in 5 hr?

Check back soon!

Problem 8

Jasmine usually takes 3 hr more than Molly does to process a day’s orders at Books To Go. If Molly takes t hr to process a day’s orders, the function given by
$$H(t)=\frac{t^{2}+3 t}{2 t+3}$$
can be used to determine how long it would take if they worked together.
How long will it take them, working together, to complete a day's orders if Molly can process the srders alone in 7 hr?

Check back soon!

Problem 9

Jasmine usually takes 3 hr more than Molly does to process a day’s orders at Books To Go. If Molly takes t hr to process a day’s orders, the function given by
$$H(t)=\frac{t^{2}+3 t}{2 t+3}$$
can be used to determine how long it would take if they worked together.

Check back soon!

Problem 10

For each rational function, find the function values indicated, provided the value exists.
$$v(t)=\frac{4 t^{2}-5 t+2}{t+3} ; v(0), v(-2), v(7)$$

Check back soon!

Problem 11

For each rational function, find the function values indicated, provided the value exists.
$$g(x)=\frac{2 x^{3}-9}{x^{2}-4 x+4} ; g(0), g(2), g(-1)$$

Check back soon!

Problem 12

For each rational function, find the function values indicated, provided the value exists.
$$r(t)=\frac{t^{2}-5 t+4}{t^{2}-9} ; r(1), r(2), r(-3)$$

Check back soon!

Problem 13

List all numbers for which each rational expression is undefined.
$$\frac{25}{-7 x}$$

Check back soon!

Problem 14

List all numbers for which each rational expression is undefined.
$$\frac{14}{-5 y}$$

Check back soon!

Problem 15

List all numbers for which each rational expression is undefined.
$$\frac{t-3}{t+8}$$

Check back soon!

Problem 16

List all numbers for which each rational expression is undefined.
$$\frac{a-8}{a+7}$$

Check back soon!

Problem 17

List all numbers for which each rational expression is undefined.
$$\frac{a}{3 a-12}$$

Check back soon!

Problem 18

List all numbers for which each rational expression is undefined.
$$\frac{x^{2}}{4 x-12}$$

Check back soon!

Problem 19

List all numbers for which each rational expression is undefined.
$$\frac{x^{2}-16}{x^{2}-3 x-28}$$

Check back soon!

Problem 20

List all numbers for which each rational expression is undefined.
$$\frac{p^{2}-9}{p^{2}-7 p+10}$$

Check back soon!

Problem 21

List all numbers for which each rational expression is undefined.
$$\frac{m^{3}-2 m}{m^{2}-25}$$

Check back soon!

Problem 22

List all numbers for which each rational expression is undefined.
$$\frac{7-3 x+x^{2}}{49-x^{2}}$$

Check back soon!

Problem 23

Simplify by removing a factor equal to 1.
$$\frac{15 x}{5 x^{2}}$$

Check back soon!

Problem 24

Simplify by removing a factor equal to 1.
$$\frac{7 a^{3}}{21 a}$$

Check back soon!

Problem 25

Simplify by removing a factor equal to 1.
$$\frac{18 t^{3} w^{2}}{27 t^{7} w}$$

Check back soon!

Problem 26

Simplify by removing a factor equal to 1.
$$\frac{8 y^{5} z}{4 y^{9} z^{3}}$$

Check back soon!

Problem 27

Simplify by removing a factor equal to 1.
$$\frac{2 a-10}{2}$$

Check back soon!

Problem 28

Simplify by removing a factor equal to 1.
$$\frac{3 a+12}{3}$$

Check back soon!

Problem 29

Simplify by removing a factor equal to 1.
$$\frac{3 x^{2}-12 x}{3 x^{2}+15 x}$$

Check back soon!

Problem 30

Simplify by removing a factor equal to 1.
$$\frac{4 y^{2}-20 y}{4 y^{2}+12 y}$$

Check back soon!

Problem 31

Simplify by removing a factor equal to 1.
$$\frac{6 a^{2}-3 a}{7 a^{2}-7 a}$$

Check back soon!

Problem 32

Simplify by removing a factor equal to 1.
$$\frac{3 m^{2}+3 m}{6 m^{2}+9 m}$$

Check back soon!

Problem 33

Simplify by removing a factor equal to 1.
$$\frac{3 a-1}{2-6 a}$$

Check back soon!

Problem 34

Simplify by removing a factor equal to 1.
$$\frac{6-5 a}{10 a-12}$$

Check back soon!

Problem 35

Simplify by removing a factor equal to 1.
$$\frac{3 a^{2}+9 a-12}{6 a^{2}-30 a+24}$$

Check back soon!

Problem 36

Simplify by removing a factor equal to 1.
$$\frac{2 t^{2}-6 t+4}{4 t^{2}+12 t-16}$$

Check back soon!

Problem 37

Simplify by removing a factor equal to 1.
$$\frac{x^{2}+8 x+16}{x^{2}-16}$$

Check back soon!

Problem 38

Simplify by removing a factor equal to 1.
$$\frac{x^{2}-25}{x^{2}-10 x+25}$$

Check back soon!

Problem 39

Simplify by removing a factor equal to 1.
$$\frac{t^{2}-1}{t+1}$$

Check back soon!

Problem 40

Simplify by removing a factor equal to 1.
$$\frac{a^{2}-1}{a-1}$$

Check back soon!

Problem 41

Simplify by removing a factor equal to 1.
$$\frac{y^{2}+4}{y+2}$$

Check back soon!

Problem 42

Simplify by removing a factor equal to 1.
$$\frac{x^{2}+1}{x+1}$$

Check back soon!

Problem 43

Simplify by removing a factor equal to 1.
$$\frac{5 x^{2}-20}{10 x^{2}-40}$$

Check back soon!

Problem 44

Simplify by removing a factor equal to 1.
$$\frac{6 x^{2}-54}{4 x^{2}-36}$$

Check back soon!

Problem 45

Simplify by removing a factor equal to 1.
$$\frac{x-8}{8-x}$$

Check back soon!

Problem 46

Simplify by removing a factor equal to 1.
$$\frac{6-x}{x-6}$$

Check back soon!

Problem 47

Simplify by removing a factor equal to 1.
$$\frac{2 t-1}{1-4 t^{2}}$$

Check back soon!

Problem 48

Simplify by removing a factor equal to 1.
$$\frac{3 a-2}{4-9 a^{2}}$$

Check back soon!

Problem 49

Simplify by removing a factor equal to 1.
$$\frac{a^{2}-25}{a^{2}+10 a+25}$$

Check back soon!

Problem 50

Simplify by removing a factor equal to 1.
$$\frac{a^{2}-16}{a^{2}-8 a+16}$$

Check back soon!

Problem 51

Simplify by removing a factor equal to 1.
$$\frac{7 s^{2}-28 t^{2}}{28 t^{2}-7 s^{2}}$$

Check back soon!

Problem 52

Simplify by removing a factor equal to 1.
$$\frac{9 m^{2}-4 n^{2}}{4 n^{2}-9 m^{2}}$$

Check back soon!

Problem 53

Simplify by removing a factor equal to 1.
$$\frac{x^{3}-1}{x^{2}-1}$$

Check back soon!

Problem 54

Simplify by removing a factor equal to 1.
$$\frac{a^{3}+8}{a^{2}-4}$$

Check back soon!

Problem 55

Simplify by removing a factor equal to 1.
$$\frac{3 y^{3}+24}{y^{2}-2 y+4}$$

Check back soon!

Problem 56

Simplify by removing a factor equal to 1.
$$\frac{x^{3}-27}{5 x^{2}+15 x+45}$$

Check back soon!

Problem 57

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$f(x)=\frac{3 x+21}{x^{2}+7 x}$$

Check back soon!

Problem 58

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$f(x)=\frac{5 x+20}{x^{2}+4 x}$$

Check back soon!

Problem 59

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$g(x)=\frac{x^{2}-9}{5 x+15}$$

Check back soon!

Problem 60

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$g(x)=\frac{8 x-16}{x^{2}-4}$$

Check back soon!

Problem 61

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$h(x)=\frac{4-x}{5 x-20}$$

Check back soon!

Problem 62

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$h(x)=\frac{7-x}{3 x-21}$$

Check back soon!

Problem 63

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$f(t)=\frac{t^{2}-9}{t^{2}+4 t+3}$$

Check back soon!

Problem 64

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$f(t)=\frac{t^{2}-25}{t^{2}-6 t+5}$$

Check back soon!

Problem 65

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$g(t)=\frac{21-7 t}{3 t-9}$$

Check back soon!

Problem 66

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$g(t)=\frac{12-6 t}{5 t-10}$$

Check back soon!

Problem 67

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$h(t)=\frac{t^{2}+5 t+4}{t^{2}-8 t-9}$$

Check back soon!

Problem 68

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$h(t)=\frac{t^{2}-3 t-4}{t^{2}+9 t+8}$$

Check back soon!

Problem 69

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$f(x)=\frac{9 x^{2}-4}{3 x-2}$$

Check back soon!

Problem 70

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$f(x)=\frac{4 x^{2}-1}{2 x-1}$$

Check back soon!

Problem 71

Determine the vertical asymptotes of the graph of each function.
$$f(x)=\frac{3 x-12}{3 x+15}$$

Check back soon!

Problem 72

Determine the vertical asymptotes of the graph of each function.
$$f(x)=\frac{4 x-20}{4 x+12}$$

Check back soon!

Problem 73

Determine the vertical asymptotes of the graph of each function.
$$g(x)=\frac{12-6 x}{5 x-10}$$

Check back soon!

Problem 74

Determine the vertical asymptotes of the graph of each function.
$$r(x)=\frac{21-7 x}{3 x-9}$$

Check back soon!

Problem 75

Determine the vertical asymptotes of the graph of each function.
$$t(x)=\frac{x^{3}+3 x^{2}}{x^{2}+6 x+9}$$

Check back soon!

Problem 76

Determine the vertical asymptotes of the graph of each function.
$$g(x)=\frac{x^{2}-4}{2 x^{2}-5 x+2}$$

Check back soon!

Problem 77

Determine the vertical asymptotes of the graph of each function.
$$f(x)=\frac{x^{2}-x-6}{x^{2}-6 x+8}$$

Check back soon!

Problem 78

Determine the vertical asymptotes of the graph of each function.
$$f(x)=\frac{x^{2}+2 x+1}{x^{2}-2 x+1}$$

Check back soon!

Problem 79

In Exercises 79–84, match each function with one of the following graphs.
$$h(x)=\frac{1}{x}$$
(GRAPH CANT COPY)

Check back soon!

Problem 80

In Exercises 79–84, match each function with one of the following graphs.
$$q(x)=-\frac{1}{x}$$
(GRAPH CANT COPY)

Check back soon!

Problem 81

In Exercises 79–84, match each function with one of the following graphs.
In Exercises 79–84, match each function with one of the following graphs.
$$h(x)=\frac{1}{x}$$
(GRAPH CANT COPY)
(GRAPH CANT COPY)

Check back soon!

Problem 82

In Exercises 79–84, match each function with one of the following graphs.
$$g(x)=\frac{x-3}{x+2}$$
(GRAPH CANT COPY)

Check back soon!

Problem 83

In Exercises 79–84, match each function with one of the following graphs.
$$r(x)=\frac{4 x-2}{x^{2}-2 x+1}$$
(GRAPH CANT COPY)

Check back soon!

Problem 84

Explain why the graphs of $f(x)=5 x$ and $g(x)=\frac{5 x^{2}}{x}$ differ

Check back soon!

Problem 85

Explain why the graphs of $f(x)=5 x$ and $g(x)=\frac{5 x^{2}}{x}$ differ.

Check back soon!

Problem 86

If a rational expression is undefined for $x=5$ and $x=-3,$ what is the degree of the denominator? Why?

Check back soon!

Problem 87

To prepare for Section 7.2, review multiplication and division using fraction notation.
Simplify.
$$-\frac{2}{15} \cdot \frac{10}{7}[1.7]$$

Check back soon!

Problem 88

To prepare for Section 7.2, review multiplication and division using fraction notation.
Simplify.
$$\left(\frac{3}{4}\right)\left(\frac{-20}{9}\right)[1.7]$$

Check back soon!

Problem 89

To prepare for Section 7.2, review multiplication and division using fraction notation.
Simplify.
$$\frac{7}{10} \div\left(-\frac{8}{15}\right)[1.7]$$

Check back soon!

Problem 90

To prepare for Section 7.2, review multiplication and division using fraction notation.
Simplify.
$$\frac{7}{10} \div\left(-\frac{8}{15}\right)[1.7]$$

Check back soon!

Problem 91

To prepare for Section 7.2, review multiplication and division using fraction notation.
Simplify.
$$\frac{7}{9}-\frac{2}{3} \cdot \frac{6}{7}[1.8]$$

Check back soon!

Problem 92

To prepare for Section 7.2, review multiplication and division using fraction notation.
Simplify.
$$\frac{2}{3}-\left(\frac{3}{4}\right)^{2}$$

Check back soon!

Problem 93

Keith incorrectly simplifies
$$\frac{x^{2}+x-2}{x^{2}+3 x+2} \text { as } \frac{x-1}{x+2}$$
He then checks his simplification by evaluating both expressions for $x=1 .$ Use this situation to explain why evaluating is not a foolproof check.

Check back soon!

Problem 94

How could you convince someone that $a-b$ and $b-a$ are opposites of each other?

Check back soon!

Problem 95

Calculate the slope of the line passing through $(a, f(a))$ and $(a+h, f(a+h))$ for the function $f$ given by $f(x)=x^{2}+5 .$ Be sure your answer is simplified.
(GRAPH CANT COPY)

Check back soon!

Problem 96

Calculate the slope of the line passing through the points $(a, f(a))$ and $(a+h, f(a+h))$ for the function $f$ given by $f(x)=3 x^{2} .$ Be sure your answer is simplified.

Check back soon!

Problem 97

Simplify.
$$\frac{x^{4}-y^{4}}{(y-x)^{4}}$$

Check back soon!

Problem 98

Simplify.
$$\frac{16 y^{4}-x^{4}}{\left(x^{2}+4 y^{2}\right)(x-2 y)}$$

Check back soon!

Problem 99

Simplify.
$$\frac{(x-1)\left(x^{4}-1\right)\left(x^{2}-1\right)}{\left(x^{2}+1\right)(x-1)^{2}\left(x^{4}-2 x^{2}+1\right)}$$

Check back soon!

Problem 100

Simplify.
$$\frac{\left(t^{4}-1\right)\left(t^{2}-9\right)(t-9)^{2}}{\left(t^{4}-81\right)\left(t^{2}+1\right)(t+1)^{2}}$$

Check back soon!

Problem 101

Simplify.
$$\frac{a^{3}-2 a^{2}+2 a-4}{a^{3}-2 a^{2}-3 a+6}$$

Check back soon!

Problem 102

Simplify.
$$\frac{x^{3}+x^{2}-y^{3}-y^{2}}{x^{2}-2 x y+y^{2}}$$

Check back soon!

Problem 103

Simplify.
$$\frac{(t+2)^{3}\left(t^{2}+2 t+1\right)(t+1)}{(t+1)^{3}\left(t^{2}+4 t+4\right)(t+2)}$$

Check back soon!

Problem 104

Simplify.
$$\frac{\left(x^{2}-y^{2}\right)\left(x^{2}-2 x y+y^{2}\right)}{(x+y)^{2}\left(x^{2}-4 x y-5 y^{2}\right)}$$

Check back soon!

Problem 105

Determine the domain and the range of each function from its graph.
(GRAPH CANT COPY)

Check back soon!

Problem 106

Determine the domain and the range of each function from its graph.
(GRAPH CANT COPY)

Check back soon!

Problem 107

Determine the domain and the range of each function from its graph.
(GRAPH CANT COPY)

Check back soon!

Problem 108

Graph the function given by
$$f(x)=\frac{x^{2}-9}{x-3}$$
(Hint. Determine the domain of $f$ and simplify.)

Check back soon!

Problem 109

Select any number $x$, multiply by $2,$ add $5,$ multiply by $5,$ subtract $25,$ and divide by $10 .$ What do you get? Explain how this procedure can be used for a number trick.

Check back soon!