Prealgebra

Problem 1

A ________ is a letter (or symbol) that stands for a number.

Ali Soave

Numerade Educator

00:19

Problem 2

Variables and/or numbers can be combined with the operations of addition, subtraction, multiplication, and division to create algebraic _____.

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Numerade Educator

00:18

Problem 3

Phrases such as increased by and more than indicate the operation of _____. Phrases such as decreased by and less than indicate the operation of _____.

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Numerade Educator

Problem 4

The word product indicates the operation of _____. The word quotient indicates the operation of _____.

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Numerade Educator

00:26

a. Write an algebraic expression that is a combination of the number $10,$ the variable $x,$ and the operation of addition.
b. Write an algebraic expression that is a combination of the numbers 3 and $2,$ the variable $t,$ and the operations of multiplication and subtraction.

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Numerade Educator

00:22

Problem 6

The illustration below shows the commute to work (in miles) for two men, Mr. Lamb and Mr. Lopez, who work in the same office.
a. Who lives farther from the office?
b. How much farther?
CAN'T COPY THE GRAPH

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Numerade Educator

00:40

Problem 7

Match each algebraic expression to the correct phrase.
a. $c+2$ i. twice $c$
b. $2-c$ ii. $c$ increased by 2
c. $c-2$ iii. $c$ less than 2
d. $2 c$ iv. 2 less than $c$

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Numerade Educator

00:26

Problem 8

Fill in the blank to complete the translation.
a. 16 less than $m$
CAN'T COPY THE GRAPH
b. 16 is less than $m$
16 \quad m

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Numerade Educator

00:20

Problem 9

The knife shown below is 12 inches long. Write an algebraic expression that represents the length of the blade (in inches).
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

00:54

Problem 10

The following table shows the ages of three family members.
a. Who is the youngest person shown in the table?
b. Who is the oldest person listed in the table?
c. On whose age are the ages in the table based?
CAN'T COPY THE GRAPH

Ali Soave

Numerade Educator

00:32

Problem 11

Complete the table. Then fill in the blank.
CAN'T COPY THE TABLE
We $\quad$ the number of decades by 10 to find the number of years.

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Numerade Educator

00:31

Problem 12

Complete the table. Then fill in the blank.
CAN'T COPY THE TABLE
We $\quad$ the number of inches by 12 to find the number of feet.

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Numerade Educator

00:17

Problem 13

Write each algebraic expression in simpler form.
a. $x \cdot 8$
b. $5(t) \quad$
c. $10 \div g$

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Numerade Educator

Problem 14

Consider the phrase:
the product of 5 and $w$ increased by 30
Insert a comma in the phrase so that it translates to $5 w+30$

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Numerade Educator

00:06

Problem 15

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
The sum of the length $l$ and 15

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Numerade Educator

Problem 16

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
The difference of a number and 10

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Problem 17

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
The product of a number and 50

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Problem 18

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
Three-fourths of the population $p$

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Problem 19

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
The ratio of the amount won $w$ and lost $l$

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00:06

Problem 20

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
The $\operatorname{tax} t$ added to $c$

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Numerade Educator

Problem 21

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
$P$ increased by two-thirds of $p$

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Numerade Educator

Problem 22

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
21 less than the total height $h$

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Numerade Educator

Problem 23

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
The square of $k,$ minus $2,005$

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Numerade Educator

Problem 24

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
$s$ subtracted from $S$

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Numerade Educator

Problem 25

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
1 less than twice the attendance $a$

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Numerade Educator

Problem 26

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
$J$ reduced by 500

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Numerade Educator

Problem 27

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
$1,000$ split $n$ equal ways

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Problem 28

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
Exceeds the cost $c$ by $25,000$

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Numerade Educator

Problem 29

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
90 more than twice the current price $p$

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Numerade Educator

Problem 30

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
64 divided by the cube of $y$

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Numerade Educator

Problem 31

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
3 times the total of $35, h,$ and 300

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Numerade Educator

Problem 32

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
Decrease $x$ by $-17$

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Numerade Educator

Problem 33

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
680 fewer than the entire population $p$

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Numerade Educator

Problem 34

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
Triple the number of expected participants

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Numerade Educator

Problem 35

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
The product of $d$ and $4,$ decreased by 15

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Numerade Educator

Problem 36

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
The quotient of $y$ and $6,$ cubed

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Numerade Educator

Problem 37

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
Twice the sum of 200 and $t$

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Numerade Educator

Problem 38

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
The square of the quantity 14 less than $x$

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Numerade Educator

Problem 39

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
The absolute value of the difference of $a$ and 2

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Numerade Educator

Problem 40

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
The absolute value of $a$, decreased by 2

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Numerade Educator

Problem 41

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
One-tenth of the distance $d$

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Problem 42

Translate each phrase to an algebraic expression. If no variable is given, use x as the variable.
Double the difference of $x$ and 18

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Numerade Educator

Problem 43

Write an algebraic expression that represents the unknown quantity.
GARDENING The height of a hedge was $f$ feet before a gardener cut 2 feet off the top. Write an algebraic expression that represents the height of the trimmed hedge (in feet).

Ali Soave

Numerade Educator

00:20

Problem 44

Write an algebraic expression that represents the unknown quantity.
SHOPPING A married couple needed to purchase 21 presents for friends and relatives on their holiday gift list. If the husband purchased $g$ presents, write an algebraic expression that represents the number of presents that the wife needs to buy.

Ali Soave

Numerade Educator

Problem 45

Write an algebraic expression that represents the unknown quantity.
PACKAGING A restaurant owner purchased $s$ six-packs of cola. Write an algebraic expression that represents the number of cans that this would be.

Ali Soave

Numerade Educator

00:18

Problem 46

Write an algebraic expression that represents the unknown quantity.
NOISE The highest decibel reading during a rock concert was only 5 decibels shy of that of a jet engine. If a jet engine is normally $j$ decibels, write an algebraic expression that represents the decibel reading for the concert.

Ali Soave

Numerade Educator

Problem 47

Write an algebraic expression that represents the unknown quantity.
SUPPLIES A pad of yellow legal paper contains $p$ pages. If a lawyer uses 15 pages every day, write an algebraic expression that represents the number of days that one pad will last.

Ali Soave

Numerade Educator

Problem 48

Write an algebraic expression that represents the unknown quantity.
ACCOUNTING The projected cost $c$ (in dollars) of a freeway was too low by a factor of $10 .$ Write an algebraic expression that represents the actual cost of the freeway (in dollars).

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Numerade Educator

Problem 49

Write an algebraic expression that represents the unknown quantity.
RECYCLING A campus ecology club collected $t$ tons of newspaper. A Boy Scout troop then contributed an additional 2 tons. Write an algebraic expression that represents the number of tons of newspaper that were collected by the two groups.

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Numerade Educator

Problem 50

Write an algebraic expression that represents the unknown quantity.
GRADUATION A graduating class of $x$ people took buses that held 40 students each to an all-night graduation party. Write an algebraic expression that represents the number of buses that were needed to transport the class.

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Numerade Educator

Problem 51

Write an algebraic expression that represents the unknown quantity.
STUDYING A student will devote $h$ hours to study for a government final exam. She wants to spread the studying evenly over a four-day period. Write an algebraic expression that represents the number of hours that she should study each day.

Ali Soave

Numerade Educator

Problem 52

Write an algebraic expression that represents the unknown quantity.
BASEBALL TEAMS After all $c$ children complete a Little League tryout, the league officials decide that they have enough players for 8 teams of equal size. Write an algebraic expression that represents the number of players that will be on each team.

Ali Soave

Numerade Educator

Problem 53

Write an algebraic expression that represents the unknown quantity.
SCOTCH TAPE Suppose $x$ inches of tape have been used off the roll shown below. Write an algebraic expression that represents the number of inches of tape that are left on the roll.
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

Problem 54

Write an algebraic expression that represents the unknown quantity.
MODELING A model's skirt is $x$ inches long. The designer then lets the hem down 2 inches. Write an algebraic expression that represents the length of the altered skirt (in inches).

Ali Soave

Numerade Educator

Problem 55

There are two unknowns.
GEOMETRY The length of a rectangle is 6 inches longer than its width. Choose a variable to represent one of the unknown dimensions of the rectangle. Then write an algebraic expression that represents the other dimension.
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

Problem 56

There are two unknowns.
PLUMBING The smaller pipe shown below takes three times longer to fill the tank than does the larger pipe. Choose a variable to represent one of the unknown times it takes to fill the tank. Then write an algebraic expression that represents the other time.
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

Problem 57

There are two unknowns.
TRUCK REPAIR The truck radiator shown below was full of coolant. Then three quarts of coolant were drained from it. Choose a variable to represent one of the unknown amounts of coolant in the radiator. Then write an algebraic expression that represents the other amount.
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

Problem 58

There are two unknowns.
SALE PRICES During a sale, the regular price of a CD was reduced by 2 dollars. Choose a variable to represent one of the unknown prices of the CD. Then write an algebraic expression that represents the other price.

Ali Soave

Numerade Educator

Problem 59

There are two unknowns.
GEOGRAPHY Alaska is much larger than Vermont. To be exact, the area of Alaska is 380 square miles more than 50 times that of Vermont. Choose a variable to represent one area. Then write an algebraic expression that represents the other area.
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

Problem 60

There are two unknowns.
ROAD TRIPS On the second part of her trip, Tamiko drove 20 miles less than three times as far as the first part. Choose a variable to represent the number of miles driven on one part of her trip. Then write an algebraic expression that represents the number of miles driven on the other part.

Ali Soave

Numerade Educator

Problem 61

There are two unknowns.
DESSERTS The number of calories in a slice of pie is 100 more than twice the calories in a scoop of ice cream. Choose a variable to represent the number of calories in one type of dessert. Then write an algebraic expression that represents the number of calories in the other type.

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Numerade Educator

Problem 62

There are two unknowns.
WASTE DISPOSAL A waste disposal tank buried in the ground holds 15 gallons less than four times what a tank mounted on a truck holds. Choose a variable to represent the number of gallons that one type of tank holds. Then write an algebraic expression that represents the number of gallons the other tank holds.

Ali Soave

Numerade Educator

00:34

Problem 63

Two approaches are used to represent the unknowns.
LANDSCAPING
a. Let $b$ represent the height of the birch tree (in feet) that is shown in the next column. Write an algebraic expression that represents the height of the elm tree (in feet).
b. Let $e$ stand for the height of the elm tree (in feet). Write an algebraic expression that represents the height of the birch tree (in feet).
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

00:26

Problem 64

Two approaches are used to represent the unknowns.
BUILDING MATERIALS
a. Let $b=$ the length of the beam shown below (in feet). Write an algebraic expression that represents the length of the pipe.
b. Let $p=$ the length of the pipe (in feet). Write an algebraic expression that represents the length of the beam.
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

00:33

Problem 65

Two approaches are used to represent the unknowns.
MARINE SCIENCE
a. Let $s$ represent the length (in feet) of the great white shark shown below. Write an algebraic expression that represents the length (in feet) of the orca (killer whale).
b. Let $w$ represent the length (in feet) of the orca (killer whale) shown below. Write an algebraic expression that represents the length (in feet) of the great white shark.
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

00:50

Problem 66

Two approaches are used to represent the unknowns.
WEIGHTS AND MEASURES
a. Refer to the scale shown on the next page. Which mixture is heavier, A or B? How much heavier is it?
b. Let $a$ represent the weight (in ounces) of mixture A. Write an algebraic expression that represents the weight (in ounces) of mixture B.
c. Let $b$ represent the weight (in ounces) of mixture B. Write an algebraic expression that represents the weight (in ounces) of mixture A.
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

00:25

Problem 67

There are three unknowns.
INVENTIONS The digital clock was invented 11 years before the automatic teller machine (ATM). The camcorder was invented 15 years after the ATM. Write algebraic expressions to represent the ages (in years) of each of these inventions. (Source: Wikipedia)

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Numerade Educator

Problem 68

There are three unknowns.
FAMOUS TOMS Tom Petty was born 6 years before Tom Hanks. Tom Cruise was born 6 years after Tom Hanks. Write algebraic expressions to represent the ages of each of these celebrities. (Source: celebritybirthdaylist.com )

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Numerade Educator

00:29

Problem 69

There are three unknowns.
NEW YORK ARCHITECTURE The Woolworth Building was completed 18 years before the Empire State Building. The United Nations Building was completed 21 years after the Empire State Building. Write algebraic expressions to represent the ages of each of these buildings. (Source: emporis.com)

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Numerade Educator

00:32

Problem 70

There are three unknowns.
CHILDREN'S BOOKS The Tale of Peter Rabbit was first published 24 years before Winnie-the-Pooh. The Cat in the Hat was first published 31 years after Winnie-the-Pooh. Write algebraic expressions to represent the ages of each of these books. (Source: Wikipedia)

Ali Soave

Numerade Educator

Problem 71

Use a table to help answer Problems.
Write an algebraic expression that represents the number of seconds in $m$ minutes.

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Numerade Educator

Problem 72

Use a table to help answer Problems.
Write an algebraic expression that represents the number of minutes in $h$ hours.

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Numerade Educator

Problem 73

Use a table to help answer Problems.
Write an algebraic expression that represents the number of inches in $f$ feet.

Ali Soave

Numerade Educator

Problem 74

Use a table to help answer Problems.
Write an algebraic expression that represents the number of feet in $y$ yards.

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Numerade Educator

00:21

Problem 75

Use a table to help answer Problems.
Write an algebraic expression that represents the number of centuries in $y$ years.

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Numerade Educator

Problem 76

Use a table to help answer Problems.
Write an algebraic expression that represents the number of decades in $y$ years.

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Numerade Educator

Problem 77

Use a table to help answer Problems.
Write an algebraic expression that represents the number of dozen eggs in $e$ eggs.

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Numerade Educator

Problem 78

Use a table to help answer Problems.
Write an algebraic expression that represents the number of days in $h$ hours.

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Numerade Educator

00:05

Problem 79

Translate each algebraic expression into words. (Answers may vary.)
$$\frac{3}{4} r$$

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Numerade Educator

00:06

Problem 80

Translate each algebraic expression into words. (Answers may vary.)
$$\frac{2}{3} d$$

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Numerade Educator

Problem 81

Translate each algebraic expression into words. (Answers may vary.)
$$t-50$$

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Numerade Educator

Problem 82

Translate each algebraic expression into words. (Answers may vary.)
$$c+19$$

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Numerade Educator

Problem 83

Translate each algebraic expression into words. (Answers may vary.)
$$x y z$$

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Problem 84

Translate each algebraic expression into words. (Answers may vary.)
$$10 a b$$

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Problem 85

Translate each algebraic expression into words. (Answers may vary.)
$$2 m+5$$

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Numerade Educator

Problem 86

Translate each algebraic expression into words. (Answers may vary.)
$$2 s-8$$

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Numerade Educator

00:18

Problem 87

A man sleeps $x$ hours per day. Write an algebraic expression that represents
a. the number of hours that he sleeps in a week.
b. the number of hours that he sleeps in a year (non-leap year).

Ali Soave

Numerade Educator

Problem 88

A store manager earns $d$ dollars an hour. Write an algebraic expression that represents
a. the amount of money he will earn in an 8 -hour day.
b. the amount of money he will earn in a 40 -hour week.

Ali Soave

Numerade Educator

Problem 89

A secretary earns an annual salary of $s$ dollars. Write an algebraic expression that represents
a. her salary per month.
b. her salary per week.

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Numerade Educator

Problem 90

Write an algebraic expression that represents the number of miles in $f$ feet. (Hint: There are $5,280$ feet in one mile.)

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Numerade Educator

Problem 91

ELECTIONS In $1960,$ John F. Kennedy was elected President of the United States with a popular vote only $118,550$ votes more than that of Richard M. Nixon. Choose a variable to represent the number of votes received by one candidate. Then write an algebraic expression that represents the number of votes received by the other candidate.

Ali Soave

Numerade Educator

Problem 92

THE BEATLES According to music historians, sales of the Beatles' second most popular single, Hey Jude, trail the sales of their most popular single, $I$ Want to Hold Your Hand, by $2,000,000$ copies. Choose a variable to represent the number of copies sold of one song. Then write an algebraic expression that represents the number of copies sold of the other song.

Ali Soave

Numerade Educator

00:23

Problem 93

COMPUTER COMPANIES IBM was founded 80 years before Apple Computer. Dell Computer Corporation was founded 9 years after Apple. Let $x$ represent the age (in years) of one of the companies. Write algebraic expressions to represent the ages (in years) of the other two companies.

Ali Soave

Numerade Educator

Problem 94

VEHICLE WEIGHTS Refer to the illustration below. The car is $1,000$ pounds lighter than the van. Choose a variable to represent the weight (in pounds) of one of the vehicles. Then write an algebraic expression that represents the weight (in pounds) of the other vehicle.
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

Problem 95

WITH/AGAINST THE WIND On a flight from Dallas to Miami, a jet airliner, which can fly 500 mph in still air, has a tail wind of $x$ mph. The tail wind increases the speed of the jet. On the return flight to Dallas, the airliner flies into a head wind of the same strength. The head wind decreases the speed of the jet. Use this information to complete the table.
CAN'T COPY THE TABLE

Ali Soave

Numerade Educator

Problem 96

SUB SANDWICHES Refer to the illustration below. Write an algebraic expression that represents the length (in inches) of the second piece of the sandwich.
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

Problem 97

SAVINGS ACCOUNTS A student inherited 5,000 dollars and deposits $x$ dollars in American Savings.
Write an algebraic expression that represents the amount of money (in dollars) left to deposit in a City Mutual account.
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

Problem 98

a. MIXING SOLUTIONS Solution 1 is poured into solution 2. Write an algebraic expression that represents the number of ounces in the mixture.
CAN'T COPY THE FIGURE
b. SNACKS Cashews were mixed with $p$ pounds of peanuts to make 100 pounds of a mixture. Write an algebraic expression that represents the number of pounds of cashews that were used.
CAN'T COPY THE FIGURE

Ali Soave

Numerade Educator

Problem 99

Explain how variables are used in this section.

Ali Soave

Numerade Educator

Problem 100

Explain the difference between the phrases greater than and is greater than.

Erika Bustos

Numerade Educator

Problem 101

Suppose in an application problem you were asked to find the unknown height of a building. Explain what is wrong with the following start.
Let $x=$ building

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Numerade Educator