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Precalculus

Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen, Dave Sobecki

Chapter 1

Equations and Inequalities - all with Video Answers

Educators

+ 1 more educators

Section 1

Linear Equations and Applications

00:00

Problem 1

What does it mean to solve an equation?

Melissa Stefan
Melissa Stefan
Numerade Educator
01:09

Problem 2

Describe the difference between an equation and an expression.

Erika Bustos
Erika Bustos
Numerade Educator
00:30

Problem 3

How can you tell if an equation is linear?

Erika Bustos
Erika Bustos
Numerade Educator
00:28

Problem 4

In one or two sentences, describe what parts $1-4$ in Theorem 1 say about working with equations.

Nick Johnson
Nick Johnson
Numerade Educator
00:38

Problem 5

How can you check your solution to an equation?

Erika Bustos
Erika Bustos
Numerade Educator
00:18

Problem 6

How do you check your solution to a word problem?

Nick Johnson
Nick Johnson
Numerade Educator
00:59

Problem 7

Explain why the following does not make sense: Solve the equation $P=2 l+2 w$.

Ernest Castorena
Ernest Castorena
Numerade Educator
00:27

Problem 8

Explain why the following does not make sense: Solve $\frac{y}{4}-\frac{y}{5}+1$

Nick Johnson
Nick Johnson
Numerade Educator
00:31

Problem 9

Solve each equation.
$$
10 x-7=4 x-25
$$

Erika Bustos
Erika Bustos
Numerade Educator
00:37

Problem 10

Solve each equation.
$$
11+3 y=5 y-5
$$

Ernest Castorena
Ernest Castorena
Numerade Educator
00:43

Problem 11

Solve each equation.
$$
3(x+2)=5(x-6)
$$

Erika Bustos
Erika Bustos
Numerade Educator
00:34

Problem 12

Solve each equation.
$$
3(y-4)+2 y=18
$$

Erika Bustos
Erika Bustos
Numerade Educator
00:55

Problem 13

Solve each equation.
$$
5+4(t-2)=2(t+7)+1
$$

Ernest Castorena
Ernest Castorena
Numerade Educator
01:12

Problem 14

Solve each equation.
$$
4-3(t+2)+t=5(t-1)-7 t
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:53

Problem 15

Solve each equation.
$$
5-\frac{3 a-4}{5}=\frac{7-2 a}{2}
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:35

Problem 16

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

Erika Bustos
Erika Bustos
Numerade Educator
01:06

Problem 17

Solve each equation.
$$
\frac{x+3}{4}-\frac{x-4}{2}=\frac{3}{8}
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:17

Problem 18

Solve each equation.
$$
\frac{x}{5}+\frac{3 x-1}{2}=\frac{6 x+5}{4}
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:05

Problem 19

Solve each equation.
$$
0.1(t+0.5)+0.2 t=0.3(t-0.4)
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:30

Problem 20

Solve each equation.
$$
0.1(w+0.5)+0.2 w=0.2(w-0.4)
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:35

Problem 21

Solve each equation.
$$
0.35(s+0.34)+0.15 s=0.2 s-1.66
$$

Erika Bustos
Erika Bustos
Numerade Educator
00:52

Problem 22

Solve each equation.
$$
0.35(u+0.34)-0.15 u=0.2 u-1.66
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:14

Problem 23

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

Erika Bustos
Erika Bustos
Numerade Educator
00:58

Problem 24

Solve each equation.
$$
\frac{3+w}{6 w}=\frac{1}{2 w}+\frac{4}{3}
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:53

Problem 25

Solve each equation.
$$
\frac{z}{z-1}=\frac{1}{z-1}+2
$$

Nick Johnson
Nick Johnson
Numerade Educator
00:34

Problem 26

Solve each equation.
$$
\frac{t}{t-1}=\frac{2}{t-1}+2
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:54

Problem 27

Solve each equation.
$$
\frac{y}{3}+\frac{y-10}{5}=\frac{2 y-2}{4}-3
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:36

Problem 28

Solve each equation.
$$
\frac{z+4}{7}+\frac{z}{6}=\frac{z+8}{3}+5
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:01

Problem 29

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

Nick Johnson
Nick Johnson
Numerade Educator
00:47

Problem 30

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

Nick Johnson
Nick Johnson
Numerade Educator
01:23

Problem 31

Solve each equation.
$$
\frac{6}{y+4}+1=\frac{5}{2 y+8}
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:01

Problem 32

Solve each equation.
$$
\frac{4 y}{y-3}+5=\frac{12}{y-3}
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:54

Problem 33

Solve each equation.
$$
\frac{3 a-1}{a^{2}+4 a+4}-\frac{3}{a^{2}+2 a}=\frac{3}{a}
$$

Erika Bustos
Erika Bustos
Numerade Educator
00:55

Problem 34

Solve each equation.
$$
\frac{1}{b-5}-\frac{10}{b^{2}-5 b+25}=\frac{1}{b+5}
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:21

Problem 35

Use a calculator to solve each equation to three significant digits.
$$
3.142 x-0.4835(x-4)=6.795
$$

Erika Bustos
Erika Bustos
Numerade Educator
01:21

Problem 36

Use a calculator to solve each equation to three significant digits.
$$
1.73 y+0.279(y-3)=2.66 y
$$

Erika Bustos
Erika Bustos
Numerade Educator
00:52

Problem 37

Use a calculator to solve each equation to three significant digits.
$$
\frac{2.32 x}{x-2}-\frac{3.76}{x}=2.32
$$

Yujie Wang
Yujie Wang
College of San Mateo
00:52

Problem 38

Use a calculator to solve each equation to three significant digits.
$$
\frac{2.34}{x}+5.67=\frac{5.67 x}{x+4}
$$

Yujie Wang
Yujie Wang
College of San Mateo
00:38

Problem 39

Solve for the indicated variable in terms of the other variables.
$a_{n}=a_{1}+(n-1) d$ for $d$ (arithmetic progressions)

Ernest Castorena
Ernest Castorena
Numerade Educator
00:34

Problem 40

Solve for the indicated variable in terms of the other variables.
$F=\frac{9}{5} C+32$ for $C($ temperature scale $)$

Ernest Castorena
Ernest Castorena
Numerade Educator
00:55

Problem 41

Solve for the indicated variable in terms of the other variables.
$\frac{1}{f}=\frac{1}{d_{1}}+\frac{1}{d_{2}}$ for $f$ (simple lens formula)

Ernest Castorena
Ernest Castorena
Numerade Educator
01:20

Problem 42

Solve for the indicated variable in terms of the other variables.
$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$ for $R_{1}$ (electric circuit)

Ernest Castorena
Ernest Castorena
Numerade Educator
00:44

Problem 43

Solve for the indicated variable in terms of the other variables.
$A=2 a b+2 a c+2 b c$ for $a$ (surface area of a rectangular solid)

Ernest Castorena
Ernest Castorena
Numerade Educator
00:52

Problem 44

Solve for the indicated variable in terms of the other variables.
$A=2 a b+2 a c+2 b c$ for $c$

Ernest Castorena
Ernest Castorena
Numerade Educator
01:14

Problem 45

Solve for the indicated variable in terms of the other variables.
$y=\frac{2 x-3}{3 x+5}$ for $x$

Ernest Castorena
Ernest Castorena
Numerade Educator
00:58

Problem 46

Solve for the indicated variable in terms of the other variables.
$x=\frac{3 y+2}{y-3}$ for $y$

Ernest Castorena
Ernest Castorena
Numerade Educator
01:21

Problem 47

Imagine that the indicated "solutions" were given to you by a student whom you were tutoring in this class. Is the solution right or wrong? If the solution is wrong, explain what is wrong and show a correct solution.
$$
\begin{aligned}
\frac{x}{x-3}+4 &=\frac{2 x-3}{x-3} \\
x+4 x-12 &=2 x-3 \\
x &=3
\end{aligned}
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:04

Problem 48

Imagine that the indicated "solutions" were given to you by a student whom you were tutoring in this class. Is the solution right or wrong? If the solution is wrong, explain what is wrong and show a correct solution.
$$
\begin{aligned}
\frac{x^{2}+1}{x-1} &=\frac{x^{2}+4 x-3}{x-1} \\
x^{2}+1 &=x^{2}+4 x-3 \\
x &=1
\end{aligned}
$$

Nick Johnson
Nick Johnson
Numerade Educator
00:54

Problem 49

Solve the equation.
$\frac{x-\frac{1}{x}}{1+\frac{1}{x}}=3$

Nick Johnson
Nick Johnson
Numerade Educator
00:53

Problem 50

Solve the equation.
$$
\frac{x-\frac{1}{x}}{x+1-\frac{2}{x}}=1
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:27

Problem 51

Solve the equation.
$$
\frac{x+1-\frac{2}{x}}{1-\frac{1}{x}}=x+2
$$

Nick Johnson
Nick Johnson
Numerade Educator
01:50

Problem 52

Solve for $y$ in terms of $x: \frac{y}{1-y}=\left(\frac{x}{1-x}\right)^{3}$

Brandon Cleary
Brandon Cleary
Numerade Educator
01:45

Problem 53

Solve for $x$ in terms of $y: y=\frac{a}{1+\frac{b}{x+c}}$

Brandon Cleary
Brandon Cleary
Numerade Educator
02:12

Problem 54

Let $m$ and $n$ be real numbers with $m$ larger than $n$. Then there exists a positive real number $p$ such that $m=n+p$. Find the fallacy in the following argument:
$$
\begin{aligned}
m &=n+p \\
(m-n) m &=(m-n)(n+p) \\
m^{2}-m n &=m n+m p-n^{2}-n p \\
m^{2}-m n-m p &=m n-n^{2}-n p \\
m(m-n-p) &=n(m-n-p) \\
m &=n
\end{aligned}
$$

Yujie Wang
Yujie Wang
College of San Mateo
01:05

Problem 55

Find a number so that 10 less than two-thirds the number is one-fourth the number.

Erika Bustos
Erika Bustos
Numerade Educator
00:46

Problem 56

Find a number so that 6 more than one-half the number is twothirds the number.

Erika Bustos
Erika Bustos
Numerade Educator
01:57

Problem 57

Find four consecutive even integers so that the sum of the first three is 2 more than twice the fourth.

Ernest Castorena
Ernest Castorena
Numerade Educator
01:56

Problem 58

Find three consecutive even integers so that the first plus twice the second is twice the third.

Ernest Castorena
Ernest Castorena
Numerade Educator
01:22

Problem 59

Find the perimeter of a triangle if one side is 16 feet, another side is two-sevenths the perimeter, and the third side is one-third the perimeter.

Brandon Cleary
Brandon Cleary
Numerade Educator
01:29

Problem 60

Find the perimeter of a triangle if one side is 11 centimeters, another side is two-fifths the perimeter, and the third side is onehalf the perimeter.

Yujie Wang
Yujie Wang
College of San Mateo
01:50

Problem 61

A new game show requires a playing field with a perimeter of 54 yards and length 3 yards less than twice the width. What are the dimensions?

Erika Bustos
Erika Bustos
Numerade Educator
02:10

Problem 62

A celebrity couple wants to have a rectangular pool put in the backyard of their vacation home. They want it to be 24 meters long, and they insist that it have at least as much area as the neighbor's pool, which is a square 12 meters on a side. Find the dimensions of the smallest pool that meets these criteria.

Yujie Wang
Yujie Wang
College of San Mateo
01:10

Problem 63

The sale price of an MP3 player after a $30 \%$ discount was $\$ 140$. What was the original price?

Erika Bustos
Erika Bustos
Numerade Educator
01:00

Problem 64

A sporting goods store marks up each item it sells $60 \%$ above wholesale price. What is the wholesale price on a snowboard that sells for \$ $144 ?$

Erika Bustos
Erika Bustos
Numerade Educator
01:47

Problem 65

One employee of a computer store is paid a base salary of $\$ 2,150$ a month plus an $8 \%$ commission on all sales over $\$ 7,000$ during the month. How much must the employee sell in 1 month to earn a total of $\$ 3,170$ for the month?

Ernest Castorena
Ernest Castorena
Numerade Educator
05:24

Problem 66

A second employee of the computer store in Problem 65 is paid a base salary of $\$ 1,175$ a month plus a $5 \%$ commission on all sales during the month.
(A) How much must this employee sell in 1 month to earn a total of $\$ 3,170$ for the month?
(B) Determine the sales level where both employees receive the same monthly income. If employees can select either of these payment methods, how would you advise an employee to make this selection?

Ernest Castorena
Ernest Castorena
Numerade Educator
01:50

Problem 67

In 1970 , Russian geologists began drilling a very deep borehole in the Kola Peninsula. Their goal was to reach a depth of 15 kilometers, but high temperatures in the borehole forced them to stop in 1994 after reaching a depth of 12 kilometers. They found that below 3 kilometers the temperature $T$ increased $2.5^{\circ} \mathrm{C}$ for each additional 100 meters of depth.
(A) If the temperature at 3 kilometers is $30^{\circ} \mathrm{C}$ and $x$ is the depth of the hole in kilometers, write an equation using $x$ that will give the temperature $T$ in the hole at any depth beyond 3 kilometers.
(B) What would the temperature be at 12 kilometers?
(C) At what depth (in kilometers) would they reach a temperature of $200^{\circ} \mathrm{C} ?$

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
01:51

Problem 68

An earthquake emits a primary wave and a secondary wave. Near the surface of the Earth the primary wave travels at about 5 miles per second, and the secondary wave travels at about 3 miles per second. From the time lag between the two waves arriving at a given seismic station, it is possible to estimate the distance to the quake. Suppose a station measures a time difference of 12 seconds between the arrival of the two waves. How far is the earthquake from the station? (The epicenter can be located by obtaining distance bearings at three or more stations.)

Yujie Wang
Yujie Wang
College of San Mateo
01:58

Problem 69

The kangaroo rat is an endangered species native to California. In order to keep track of their population size in a state nature preserve, a conservation biologist trapped, tagged, and released 80 individuals from the population. After waiting 2 weeks for the animals to mix back in with the general population, she again caught 80 individuals and found that 22 of them were tagged. Assuming that the ratio of tagged animals to total animals in the second sample is the same as the ratio of all tagged animals to the total population in the preserve, estimate the total number of kangaroo rats in the preserve.

Yujie Wang
Yujie Wang
College of San Mateo
01:55

Problem 70

Repeat Problem 69 with a first (marked) sample of 70 and a second sample of 30 with only 11 marked animals.

Yujie Wang
Yujie Wang
College of San Mateo
01:30

Problem 71

How many gallons of distilled water must be mixed with 50 gallons of $30 \%$ alcohol solution to obtain a $25 \%$ solution?

Ernest Castorena
Ernest Castorena
Numerade Educator
01:37

Problem 72

How many gallons of hydrochloric acid must be added to 12 gallons of a $30 \%$ solution to obtain a $40 \%$ solution?

Yujie Wang
Yujie Wang
College of San Mateo
01:34

Problem 73

A chemist mixes distilled water with a $90 \%$ solution of sulfuric acid to produce a $50 \%$ solution. If 5 liters of distilled water are used, how much $50 \%$ solution is produced?

Ernest Castorena
Ernest Castorena
Numerade Educator
03:31

Problem 74

A fuel oil distributor has 120,000 gallons of fuel with $0.9 \%$ sulfur content, which exceeds pollution control standards of $0.8 \%$ sulfur content. How many gallons of fuel oil with a $0.3 \%$ sulfur content must be added to the 120,000 gallons to obtain fuel oil that will comply with the pollution control standards?

Yujie Wang
Yujie Wang
College of San Mateo
03:23

Problem 75

An old computer can do the weekly payroll in 5 hours. A newer computer can do the same payroll in 3 hours. The old computer starts on the payroll, and after 1 hour the newer computer is brought on-line to work with the older computer until the job is finished. How long will it take both computers working together to finish the job? (Assume the computers operate independently.)

Yujie Wang
Yujie Wang
College of San Mateo
03:09

Problem 76

One pump can fill a gasoline storage tank in 8 hours. With a second pump working simultaneously, the tank can be filled in 3 hours. How long would it take the second pump to fill the tank operating alone?

Yujie Wang
Yujie Wang
College of San Mateo
03:52

Problem 77

The cruising speed of an airplane is 150 miles per hour (relative to the ground). You plan to hire the plane for a 3 -hour sightseeing trip. You instruct the pilot to fly north as far as she can and still return to the airport at the end of the allotted time.
(A) How far north should the pilot fly if the wind is blowing from the north at 30 miles per hour?
(B) How far north should the pilot fly if there is no wind?

Yujie Wang
Yujie Wang
College of San Mateo
02:06

Problem 78

Suppose you are at a river resort and rent a motor boat for 5 hours starting at 7 A.M. You are told that the boat will travel at 8 miles per hour upstream and 12 miles per hour returning. You decide that you would like to go as far up the river as you can and still be back at noon. At what time should you turn back, and how far from the resort will you be at that time?

Ernest Castorena
Ernest Castorena
Numerade Educator
02:51

Problem 79

A two-woman rowing team can row 1,200 meters with the current in a river in the same amount of time it takes them to row 1,000 meters against that same current. In each case, their average rowing speed without the effect of the current is 3 meters per second. Find the speed of the current.

Yujie Wang
Yujie Wang
College of San Mateo
03:46

Problem 80

The winners of the men's 1,000 -meter double sculls event in the 2008 Olympics rowed at an average of 11.3 miles per hour. If this team were to row this speed for a half mile with a current in $80 \%$ of the time they were able to row that same distance against the current, what would be the speed of the current?

Yujie Wang
Yujie Wang
College of San Mateo
01:17

Problem 81

A major chord in music is composed of notes whose frequencies are in the ratio 4: 5: 6 . If the first note of a chord has a frequency of 264 hertz (middle $C$ on the piano), find the frequencies of the other two notes.

Yujie Wang
Yujie Wang
College of San Mateo
01:28

Problem 82

A minor chord is composed of notes whose frequencies are in the ratio 10: 12: 15 . If the first note of a minor chord is $\mathrm{A}$, with a frequency of 220 hertz, what are the frequencies of the other two notes?

Yujie Wang
Yujie Wang
College of San Mateo
View

Problem 83

In an experiment on motivation, Professor Brown trained a group of rats to run down a narrow passage in a cage to receive food in a goal box. He then put a harness on each rat and connected it to an overhead wire attached to a scale. In this way, he could place the rat different distances from the food and measure the pull (in grams) of the rat toward the food. He found that the relationship between motivation (pull) and position was given approximately by the equation
$$
p=-\frac{1}{5} d+70 \quad 30 \leq d \leq 170
$$
where pull $p$ is measured in grams and distance $d$ in centimeters. When the pull registered was 40 grams, how far was the rat from the goal box?

Danielle Fairburn
Danielle Fairburn
Numerade Educator
00:31

Problem 84

Professor Brown performed the same kind of experiment as described in Problem $83,$ except that he replaced the food in the goal box with a mild electric shock. With the same kind of apparatus, he was able to measure the avoidance strength relative to the distance from the object to be avoided. He found that the avoidance strength $a$ (measured in grams) was related to the distance $d$ that the rat was from the shock (measured in centimeters) approximately by the equation
$$
a=-\frac{4}{3} d+230 \quad 30 \leq d \leq 170
$$
If the same rat were trained as described in this problem and in Problem $83,$ at what distance (to one decimal place) from the goal box would the approach and avoidance strengths be the same? (What do you think the rat would do at this point?)

Sari Ogami
Sari Ogami
Numerade Educator