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College Algebra 11th

Michael Sullivan

Chapter 1

Equations and Inequalities

Educators


Problem 1

The fact that $2(x+3)=2 x+6$ is attributable to the ___________ Property.

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Problem 2

The fact that $3 x=0$ implies that $x=0$ is a result of the _____________ Property.

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Problem 3

The domain of the variable in the expression $\frac{x}{x-4}$ is __________.

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Problem 4

True or False
Multiplying both sides of an equation by any number results in an equivalent equation.

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Problem 5

An equation that is satisfied for every value of the variable for which both sides are defined is called a(n) ________.

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Problem 6

An equation of the form $a x+b=0$ is called a(n) _______ equation or a(n) ___________ equation

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Problem 7

True or False
The solution of the equation $3 x-8=0$ is $\frac{3}{8}$

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Problem 8

True or False
Some equations have no solution.

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Problem 9

Multiple Choice
An admissible value for the variable that makes the equation a true statement is called a(n) ______ of the equation.
(a) identity
(b) solution
(c) degree
(d) model

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Problem 10

Multiple Choice
A chemist mixes 10 liters of a $20 \%$ solution with $x$ liters of a $35 \%$ solution. Which of the following expressions represents the total number of liters in the mixture?
(a) $x$
(b) $20-x$
(c) $\frac{35}{x}$
(d) $10+x$

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Problem 11

Mentally solve each equation.
$7 x=21$

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Problem 12

Mentally solve each equation.
$6 x=-24$

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Problem 13

Mentally solve each equation.
$$3 x+15=0$$

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Problem 14

Mentally solve each equation.
$$6 x+18=0$$

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Problem 15

Mentally solve each equation.
$$2 x-3=0$$

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Problem 16

Mentally solve each equation.
$$3 x+4=0$$

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Problem 17

Mentally solve each equation.
$$\frac{1}{4} x=\frac{7}{20}$$

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Problem 18

Mentally solve each equation.
$$\frac{2}{3} x=\frac{9}{2}$$

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Problem 19

Solve each equation, if possible.
$$3 x+4=x$$

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Problem 20

Solve each equation, if possible.
$$2 x+9=5 x$$

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Problem 21

Solve each equation, if possible.
$$2 t-6=3-t$$

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Problem 22

Solve each equation, if possible.
$$5 y+6=-18-y$$

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Problem 23

Solve each equation, if possible.
$6-x=2 x+9$

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Problem 24

Solve each equation, if possible.
$3-2 x=2-x$

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Problem 25

Solve each equation, if possible.
$$
3+2 n=4 n+7
$$

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Problem 26

Solve each equation, if possible.
$$6-2 m=3 m+1$$

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Problem 27

Solve each equation, if possible.
$$3(5+3 x)=8(x-1)$$

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Problem 28

Solve each equation, if possible.
$$3(2-x)=2 x-1$$

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Problem 29

Solve each equation, if possible.
$$8 x-(3 x+2)=3 x-10$$

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Problem 30

Solve each equation, if possible.
$$7-(2 x-1)=10$$

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Problem 31

Solve each equation, if possible.
$$\frac{3}{2} x+2=\frac{1}{2}-\frac{1}{2} x$$

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Problem 32

Solve each equation, if possible.
$$\frac{1}{3} x=2-\frac{2}{3} x$$

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Problem 33

Solve each equation, if possible.
$$\frac{1}{2} x-5=\frac{3}{4} x$$

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Problem 34

Solve each equation, if possible.
$$1-\frac{1}{2} x=6$$

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Problem 35

Solve each equation, if possible.
$$\frac{2}{3} p=\frac{1}{2} p+\frac{1}{3}$$

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Problem 36

Solve each equation, if possible.
$$\frac{1}{2}-\frac{1}{3} p=\frac{4}{3}$$

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Problem 37

Solve each equation, if possible.
$$0.2 m=0.9+0.5 m$$

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Problem 38

Solve each equation, if possible.
$$0.9 t=1+t$$

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Problem 39

Solve each equation, if possible.
$$\frac{x+1}{3}+\frac{x+2}{7}=2$$

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Problem 40

Solve each equation, if possible.
$$\frac{2 x+1}{3}+16=3 x$$

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Problem 41

Solve each equation, if possible.
$$\frac{5}{8}(p+3)-2=\frac{1}{4}(2 p-3)+\frac{11}{16}$$

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Problem 42

Solve each equation, if possible.
$$\frac{1}{3}(w+1)-3=\frac{2}{5}(w-4)-\frac{2}{15}$$

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Problem 43

Solve each equation, if possible.
$$\frac{2}{y}+\frac{4}{y}=3$$

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Problem 44

Solve each equation, if possible.
$$\frac{4}{y}-5=\frac{5}{2 y}$$

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Problem 45

Solve each equation, if possible.
$$\frac{1}{2}+\frac{2}{x}=\frac{3}{4}$$

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Problem 46

Solve each equation, if possible.
$$\frac{3}{x}-\frac{1}{3}=\frac{1}{6}$$

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Problem 47

Solve each equation, if possible.
$$(x+7)(x-1)=(x+1)^{2}$$

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Problem 48

Solve each equation, if possible.
$$(x+2)(x-3)=(x+3)^{2}$$

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Problem 49

Solve each equation, if possible.
$$x(2 x-3)=(2 x+1)(x-4)$$

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Problem 50

Solve each equation, if possible.
$$x(1+2 x)=(2 x-1)(x-2)$$

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Problem 51

Solve each equation, if possible.
$$p\left(p^{2}+3\right)=12+p^{3}$$

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Problem 52

Solve each equation, if possible.
$$w\left(4-w^{2}\right)=8-w^{3}$$

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Problem 53

Solve each equation, if possible.
$$\frac{x}{x-2}+3=\frac{2}{x-2}$$

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Problem 54

Solve each equation, if possible.
$$\frac{2 x}{x+3}=\frac{-6}{x+3}-2$$

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Problem 55

Solve each equation, if possible.
$$\frac{2 x}{x^{2}-4}=\frac{4}{x^{2}-4}-\frac{3}{x+2}$$

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Problem 56

Solve each equation, if possible.
$$\frac{x}{x^{2}-9}+\frac{4}{x+3}=\frac{3}{x^{2}-9}$$

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Problem 57

Solve each equation, if possible.
$$\frac{x}{x+2}=\frac{3}{2}$$

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Problem 58

Solve each equation, if possible.
$$\frac{3 x}{x-1}=2$$

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Problem 59

Solve each equation, if possible.
$$\frac{7}{3 x+10}=\frac{2}{x-3}$$

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Problem 60

Solve each equation, if possible.
$$\frac{-4}{x+4}=\frac{-3}{x+6}$$

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Problem 61

Solve each equation, if possible.
$$\frac{6 t+7}{4 t-1}=\frac{3 t+8}{2 t-4}$$

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Problem 62

Solve each equation, if possible.
$$\frac{8 w+5}{10 w-7}=\frac{4 w-3}{5 w+7}$$

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Problem 63

Solve each equation, if possible.
$$\frac{4}{x-2}=\frac{-3}{x+5}+\frac{7}{(x+5)(x-2)}$$

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Problem 64

Solve each equation, if possible.
$$\frac{-4}{2 x+3}+\frac{1}{x-1}=\frac{1}{(2 x+3)(x-1)}$$

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Problem 65

Solve each equation, if possible.
$$\frac{2}{y+3}+\frac{3}{y-4}=\frac{5}{y+6}$$

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Problem 66

Solve each equation, if possible.
$$\frac{5}{5 z-11}+\frac{4}{2 z-3}=\frac{-3}{5-z}$$

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Problem 67

Solve each equation, if possible.
$$\frac{x}{x^{2}-9}-\frac{x-4}{x^{2}+3 x}=\frac{10}{x^{2}-3 x}$$

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Problem 68

Solve each equation, if possible.
$$\frac{x+1}{x^{2}+2 x}-\frac{x+4}{x^{2}+x}=\frac{-3}{x^{2}+3 x+2}$$

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Problem 69

Use a calculator to solve each equation. Round the solution to two decimal places.
$$3.2 x+\frac{21.3}{65.871}=19.23$$

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Problem 70

Use a calculator to solve each equation. Round the solution to two decimal places.
$$6.2 x-\frac{19.1}{83.72}=0.195$$

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Problem 71

Use a calculator to solve each equation. Round the solution to two decimal places.
$$14.72-21.58 x=\frac{18}{2.11} x+2.4$$

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Problem 72

Use a calculator to solve each equation. Round the solution to two decimal places.
$$18.63 x-\frac{21.2}{2.6}=\frac{14}{2.32} x-20$$

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Problem 73

Solve each equation. The letters $a, b,$ and $c$ are constants.
$a x-b=c, \quad a \neq 0$

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Problem 74

Solve each equation. The letters $a, b,$ and $c$ are constants.
$1-a x=b, \quad a \neq 0$

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Problem 75

Solve each equation. The letters $a, b,$ and $c$ are constants.
$\frac{x}{a}+\frac{x}{b}=c, a \neq 0, b \neq 0, a \neq-b$

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Problem 76

Solve each equation. The letters $a, b,$ and $c$ are constants.
$\frac{a}{x}+\frac{b}{x}=c, \quad c \neq 0$

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Problem 77

Find the number $a$ for which $x=4$ is a solution of the equation.
$$x+2 a=16+a x-6 a$$

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Problem 78

Find the number $b$ for which $x=2$ is a solution of the equation.
$$x+2 b=x-4+2 b x$$

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Problem 79

List some formulas that occur in applications. Solve each formula for the indicated variable.
Electricity $\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$ for $R$

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Problem 80

List some formulas that occur in applications. Solve each formula for the indicated variable.
Finance $A=P(1+r t)$ for $r$

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Problem 81

List some formulas that occur in applications. Solve each formula for the indicated variable.
Mechanics $F=\frac{m v^{2}}{R}$ for $R$

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Problem 82

List some formulas that occur in applications. Solve each formula for the indicated variable.
Chemistry $P V=n R T$ for $T$

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Problem 83

List some formulas that occur in applications. Solve each formula for the indicated variable.
Mathematics $S=\frac{a}{1-r}$ for $r$

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Problem 84

List some formulas that occur in applications. Solve each formula for the indicated variable.
Mechanics $v=-g t+v_{0} \quad$ for $t$

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Problem 85

A total of $\$ 20,000$ is to be invested, some in bonds and some in certificates of deposit (CDs). If the amount invested in bonds is to exceed that in CDs by $\$ 3000$, how much will be invested in each type of investment?

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Problem 86

A total of $\$ 10,000$ is to be divided between Sean and George, with George to receive $\$ 3000$ less than Sean. How much will each receive?

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Problem 87

Kim is paid time-and-a-half for hours worked in excess of 40 hours and had gross weekly wages of $\$ 910$ for 48 hours worked. What is her regular hourly rate?

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Problem 88

Leigh is paid time-and-a-half for hours worked in excess of 40 hours and double-time for hours worked on Sunday. If Leigh had gross weekly wages of $\$ 1083$ for working 50 hours, 4 of which were on Sunday, what is her regular hourly rate?

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Problem 89

Going into the final exam, which will count as two tests, Brooke has test scores of 80,83,71,61 , and $95 .$ What score does Brooke need on the final in order to have an average score of $80 ?$

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Problem 90

Going into the final exam, which will count as two-thirds of the final grade, Mike has test scores of $86,80,84,$ and $90 .$ What minimum score does Mike need on the final in order to earn a B, which requires an average score of $80 ?$ What does he need to earn an $A$, which requires an average of $90 ?$

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Problem 91

A store sells refurbished iPhones that cost $12 \%$ less than the original price. If the new price of a refurbished iPhone is $\$ 572,$ what was the original price? How much is saved by purchasing the refurbished phone?

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Problem 92

A car dealer, at a year-end clearance, reduces the list price of last year's models by $15 \%$. If a certain four-door model has a discounted price of $\$ 18,000,$ what was its list price? How much can be saved by purchasing last year's model?

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Problem 93

A movie theater marks up the candy it sells by $275 \%$. If a box of candy sells for $\$ 4.50$ at the theater, how much did the theater pay for the box?

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Problem 94

The suggested list price of a new car is $\$ 24,000$. The dealer's cost is $85 \%$ of list. How much will you pay if the dealer is willing to accept $\$ 300$ over cost for the car?

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Problem 95

The manager of the Coral Theater wants to know whether the majority of its patrons are adults or children. One day in July, 5200 tickets were sold and the receipts totaled $\$ 29,961 .$ The adult admission is $\$ 7.50$, and the children's admission is $\$ 4.50 .$ How many adult patrons were there?

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Problem 96

A pair of leather boots, discounted by $30 \%$ for a clearance sale, has a price tag of $\$ 399 .$ What was the original price?

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Problem 97

The perimeter of a rectangle is 60 feet. Find its length and width if the length is 8 feet longer than the width.

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Problem 98

The perimeter of a rectangle is 42 meters. Find its length and width if the length is twice the width.

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Problem 99

Herschel uses an app on his smartphone to keep track of his daily calories from meals. One day his calories from breakfast were 125 more than his calories from lunch, and his calories from dinner were 300 less than twice his calories from lunch. If his total caloric intake from meals was $2025,$ determine his calories for each meal.

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Problem 100

Tyshira tracks her net calories (calories taken in minus calories burned) as part of her fitness program. For one particular day, her net intake was 1480 calories. Her lunch calories were half her breakfast calories, and her dinner calories were 200 more than her breakfast calories. She ate 120 less calories in snacks than for breakfast, and she burned 700 calories by exercising on her elliptical. How many calories did she take in from snacks?

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Problem 101

Judy and Tom agree to share the cost of an $\$ 18$ pizza based on how much each ate. If Tom ate $\frac{2}{3}$ the amount that Judy ate, how much should each pay? [Hint: Some pizza may be left.]

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Problem 102

Find the largest perimeter of an isosceles triangle whose sides are of lengths $4 x+10,2 x+40$, and $3 x+18$.

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Problem 103

Solve:
$\frac{3}{4} x-\frac{1}{5}\left(\frac{1}{2}-3 x\right)+1=\frac{1}{4}\left(\frac{1}{20} x+6\right)-\frac{4}{5}$

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Problem 104

A regular hexagon is inscribed in a circle. Find the radius of the circle if the perimeter of the hexagon is 10 inches more than the radius.

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Problem 105

One step in the following list contains an error. Identify it and explain what is wrong.
$$\begin{aligned}x &=2 \quad (1) \\3 x-2 x &=2 \quad (2) \\3 x &=2 x+2 \quad (3) \\x^{2}+3 x &=x^{2}+2 x+2\quad (4) \\x^{2}+3 x-10 &=x^{2}+2 x-8\quad (5) \\(x-2)(x+5) &(x2(x+4)\quad(6)\\x+5 &=x+4\quad(7) \\1 &=0\quad(8)\end{aligned}$$

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Problem 106

The equation
$$\frac{5}{x+3}+3=\frac{8+x}{x+3}$$
has no solution, yet when we go through the process of solving it, we obtain $x=-3 .$ Write a brief paragraph to explain what causes this to happen.

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Problem 107

Make up an equation that has no solution and give it to a fellow student to solve. Ask the fellow student to write a critique of your equation.

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