# 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) $\{x | x \neq-5\}$

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### 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) $\{x | x \neq-5\}$

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### 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) $\{x | x \neq-5\}$

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### 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) $\{x | x \neq-5\}$

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### 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) $\{x | x \neq-5\}$

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### 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) $\{x | x \neq-5\}$

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### 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?

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### 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?

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### 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.

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### Problem 10

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

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### Problem 11

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

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### Problem 12

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

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### Problem 13

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

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### Problem 14

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

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### Problem 15

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

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### Problem 16

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

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### Problem 17

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

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### Problem 18

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

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### Problem 19

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

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### Problem 20

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

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### Problem 21

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

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### Problem 22

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

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### Problem 23

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

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### Problem 24

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

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### Problem 25

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

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### Problem 26

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

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### Problem 27

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

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### Problem 28

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

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### Problem 29

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

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### Problem 30

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

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### Problem 31

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

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### Problem 32

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

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### Problem 33

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

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### Problem 34

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

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### Problem 35

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

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### Problem 36

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

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### Problem 37

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

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### Problem 38

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

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### Problem 39

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

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### Problem 40

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

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### Problem 41

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

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### Problem 42

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

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### Problem 43

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

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### Problem 44

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

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### Problem 45

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

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### Problem 46

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

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### Problem 47

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

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### Problem 48

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

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### Problem 49

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

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### Problem 50

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

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### Problem 51

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

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### Problem 52

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

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### Problem 53

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

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### Problem 54

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

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### Problem 55

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

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### Problem 56

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

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### Problem 57

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

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### Problem 58

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

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### Problem 59

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

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### Problem 60

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

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### Problem 61

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

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### Problem 62

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

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### Problem 63

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

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### Problem 64

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

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### Problem 65

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

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### Problem 66

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

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### Problem 67

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

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### Problem 68

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

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### Problem 69

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

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### Problem 70

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

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### Problem 71

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

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### Problem 72

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

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### Problem 73

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

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### Problem 74

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

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### Problem 75

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

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### Problem 76

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

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### Problem 77

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

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### Problem 78

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

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### Problem 79

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

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### Problem 80

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

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### 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)

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### 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)

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### 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)

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### Problem 84

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

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### Problem 85

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

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### Problem 86

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

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### Problem 87

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

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### Problem 88

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

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### Problem 89

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

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### Problem 90

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

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### Problem 91

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

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### Problem 92

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

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### 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.

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### Problem 94

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

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### 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)

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### 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.

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### Problem 97

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

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### Problem 98

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

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### Problem 99

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

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### Problem 100

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

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### Problem 101

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

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### Problem 102

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

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### Problem 103

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

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### Problem 104

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

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### Problem 105

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

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### Problem 106

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

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### Problem 107

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

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### Problem 108

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

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### 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.

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