Algebra and Trigonometry

Michael Sullivan

Chapter 1 Equations and Inequalities - all with Video Answers

Educators

Section 1

Linear Equations

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Problem 1

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

Tim Thornhill

Numerade Educator

00:54

Problem 2

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

Ankit Gupta

Numerade Educator

00:44

Problem 3

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

Ankit Gupta

Numerade Educator

00:32

Problem 4

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

Ankit Gupta

Numerade Educator

00:45

Problem 5

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

Ankit Gupta

Numerade Educator

00:32

Problem 6

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

Ankit Gupta

Numerade Educator

00:24

Problem 7

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

Ankit Gupta

Numerade Educator

00:28

Problem 8

True or False Some equations have no solution.

Ankit Gupta

Numerade Educator

00:36

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

Yujie Wang

College of San Mateo

00:33

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) $10+x$

Yujie Wang

College of San Mateo

00:26

Problem 11

In Problems 11-18, mentally solve each equation.
$$
7 x=21
$$

Prashansha Kaushik

Numerade Educator

00:25

Problem 12

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

Ankit Gupta

Numerade Educator

00:28

Problem 13

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

Ankit Gupta

Numerade Educator

00:26

Problem 14

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

Ankit Gupta

Numerade Educator

01:14

Problem 15

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

Sanchit Jain

Numerade Educator

00:26

Problem 16

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

Ankit Gupta

Numerade Educator

00:35

Problem 17

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

Prashansha Kaushik

Numerade Educator

00:38

Problem 18

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

Ankit Gupta

Numerade Educator

01:25

Problem 19

In Problems 19-68, solve each equation, if possible.
$$
3 x+4=x
$$

Prashansha Kaushik

Numerade Educator

01:30

Problem 20

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

Prashansha Kaushik

Numerade Educator

01:17

Problem 21

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

Prashansha Kaushik

Numerade Educator

01:30

Problem 22

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

Prashansha Kaushik

Numerade Educator

01:33

Problem 23

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

Prashansha Kaushik

Numerade Educator

01:17

Problem 24

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

Prashansha Kaushik

Numerade Educator

01:24

Problem 25

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

Prashansha Kaushik

Numerade Educator

01:16

Problem 26

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

Prashansha Kaushik

Numerade Educator

02:29

Problem 27

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

Prashansha Kaushik

Numerade Educator

01:52

Problem 28

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

Prashansha Kaushik

Numerade Educator

02:16

Problem 29

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

Prashansha Kaushik

Numerade Educator

01:12

Problem 30

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

Prashansha Kaushik

Numerade Educator

02:14

Problem 31

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

Prashansha Kaushik

Numerade Educator

01:12

Problem 32

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

Prashansha Kaushik

Numerade Educator

01:51

Problem 33

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

Prashansha Kaushik

Numerade Educator

01:22

Problem 34

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

Prashansha Kaushik

Numerade Educator

01:33

Problem 35

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

Prashansha Kaushik

Numerade Educator

01:45

Problem 36

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

Prashansha Kaushik

Numerade Educator

01:34

Problem 37

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

Prashansha Kaushik

Numerade Educator

01:06

Problem 38

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

Prashansha Kaushik

Numerade Educator

02:41

Problem 39

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

Prashansha Kaushik

Numerade Educator

00:46

Problem 40

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

Yujie Wang

College of San Mateo

02:17

Problem 41

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

Yujie Wang

College of San Mateo

02:16

Problem 42

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

Yujie Wang

College of San Mateo

01:13

Problem 43

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

Prashansha Kaushik

Numerade Educator

00:43

Problem 44

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

Yujie Wang

College of San Mateo

01:24

Problem 45

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

Prashansha Kaushik

Numerade Educator

01:33

Problem 46

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

Prashansha Kaushik

Numerade Educator

01:24

Problem 47

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

Yujie Wang

College of San Mateo

00:56

Problem 48

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

Yujie Wang

College of San Mateo

01:12

Problem 49

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

Yujie Wang

College of San Mateo

01:14

Problem 50

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

Yujie Wang

College of San Mateo

01:11

Problem 51

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

Manisha Sarker

Numerade Educator

00:32

Problem 52

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

Yujie Wang

College of San Mateo

00:49

Problem 53

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

Yujie Wang

College of San Mateo

00:45

Problem 54

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

Yujie Wang

College of San Mateo

01:45

Problem 55

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

Yujie Wang

College of San Mateo

01:47

Problem 56

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

Yujie Wang

College of San Mateo

00:42

Problem 57

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

Yujie Wang

College of San Mateo

00:33

Problem 58

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

Yujie Wang

College of San Mateo

01:13

Problem 59

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

Yujie Wang

College of San Mateo

01:11

Problem 60

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

Yujie Wang

College of San Mateo

02:48

Problem 61

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

Yujie Wang

College of San Mateo

03:04

Problem 62

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

Yujie Wang

College of San Mateo

01:18

Problem 63

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

Yujie Wang

College of San Mateo

01:04

Problem 64

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

Yujie Wang

College of San Mateo

04:20

Problem 65

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

Yujie Wang

College of San Mateo

05:11

Problem 66

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

Yujie Wang

College of San Mateo

02:35

Problem 67

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

Yujie Wang

College of San Mateo

01:45

Problem 67

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

Yujie Wang

College of San Mateo

02:24

Problem 68

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

Yujie Wang

College of San Mateo

01:18

Problem 69

In Problems 69-72, use a calculator to solve each equation. Round the solution to two decimal places.
$$
3.2 x+\frac{21.3}{65.871}=19.23
$$

Yujie Wang

College of San Mateo

01:30

Problem 70

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

Ankit Gupta

Numerade Educator

01:39

Problem 71

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

Ankit Gupta

Numerade Educator

01:53

Problem 72

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

Ankit Gupta

Numerade Educator

00:34

Problem 73

In Problems $73-76$, solve each equation. The letters $a$, $b$, and $c$ are constants.
$$
a x-b=c, \quad a \neq 0
$$

Yujie Wang

College of San Mateo

00:39

Problem 74

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

Ankit Gupta

Numerade Educator

00:54

Problem 75

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

Ankit Gupta

Numerade Educator

00:42

Problem 76

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

Ankit Gupta

Numerade Educator

00:50

Problem 77

Solve each equation. The letters $a$, $b$, and $c$ are constants. Find the number $a$ for which $x=4$ is a solution of the equation
$$
x+2 a=16+a x-6 a
$$

Yujie Wang

College of San Mateo

00:57

Problem 78

Solve each equation. The letters $a$, $b$, and $c$ are constants. Find the number $b$ for which $x=2$ is a solution of the equation
$$
x+2 b=x-4+2 b x
$$

Yujie Wang

College of San Mateo

01:09

Problem 79

Problems 79-84 list some formulas that occur in applications. Solve each formula for the indicated variable.
$$
\text { Electricity } \frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}} \text { for } R
$$

Prashansha Kaushik

Numerade Educator

00:44

Problem 80

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

Prashansha Kaushik

Numerade Educator

00:44

Problem 81

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

Prashansha Kaushik

Numerade Educator

00:31

Problem 82

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

Prashansha Kaushik

Numerade Educator

01:12

Problem 83

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

Prashansha Kaushik

Numerade Educator

00:42

Problem 84

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

Prashansha Kaushik

Numerade Educator

01:20

Problem 85

Finance A total of 20,000 dollar 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 dollar, how much will be invested in each type of investment?

Ankit Gupta

Numerade Educator

01:07

Problem 86

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

Ankit Gupta

Numerade Educator

01:30

Problem 87

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

Yujie Wang

College of San Mateo

03:12

Problem 88

Computing Hourly Wages 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 dollar for working 50 hours, 4 of which were on Sunday, what is her regular hourly rate?

Yujie Wang

College of San Mateo

01:34

Problem 89

Computing Grades 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 $80 ?$

Ankit Gupta

Numerade Educator

02:13

Problem 90

Computing Grades 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 $90 ?$

Ankit Gupta

Numerade Educator

01:49

Problem 91

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

Yujie Wang

College of San Mateo

02:14

Problem 92

Business: Discount Pricing 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 dollar, what was its list price? How much can be saved by purchasing last year's model?

Yujie Wang

College of San Mateo

00:51

Problem 93

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

Yujie Wang

College of San Mateo

01:03

Problem 94

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

Andrew Elmer

Numerade Educator

01:27

Problem 95

Business: Theater Attendance 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 dollar. The adult admission is 7.50 dollar, and the children's admission is 4.50 dollar. How many adult patrons were there?

Ankit Gupta

Numerade Educator

01:18

Problem 96

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

Yujie Wang

College of San Mateo

01:09

Problem 97

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

Ankit Gupta

Numerade Educator

00:59

Problem 98

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

Ankit Gupta

Numerade Educator

02:55

Problem 99

Counting Calories 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.

Lynn Larson

Numerade Educator

02:00

Problem 100

Counting Calories 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?

Ankit Gupta

Numerade Educator

02:54

Problem 101

Challenge Problem Sharing the cost of a Pizza Judy and Tom agree to share the cost of an 18 dollar 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.]

Yujie Wang

College of San Mateo

04:44

Problem 102

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

Yujie Wang

College of San Mateo

02:36

Problem 103

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

Yujie Wang

College of San Mateo

03:44

Problem 104

Challenge Problem 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.

Andrew Elmer

Numerade Educator

02:30

Problem 105

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

Suman Saurav Thakur

Numerade Educator

01:38

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.

Ankit Gupta

Numerade Educator

01:43

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.

Ankit Gupta

Numerade Educator