Download Introduction to Algebra

Objective 11.1A
New Vocabulary
variable
variable expression
evaluating a variable expression
Discuss the Concepts
1. What does it mean to evaluate a variable expression?
2. What is the goal in evaluating a variable expression?
Concept Check
What are the steps involved in evaluating a variable expression?
(1) Write the variable expression. (2) Substitute the given values for the variables. (3) Simplify the resulting
numerical expression, using the Order of Operations Agreement, so that the final answer is one number.
Optional Student Activity
1. A community college charges each student a $15 student activity fee, which is added to the student’s
cost for tuition. Let T represent the tuition. Write a variable expression that represents the final bill after
the student activity fee has been added. T + 15
2. A phone company charges a flat fee of $25 to come to your house for repair work. This charge is added
to the repair bill. Let B represent the repair bill. Write a variable expression that represents the final bill
after the flat fee has been added. B + 25
3. The varsity baseball coach at a high school must order jerseys for the team. The price of one jersey is
$28. Let N represent the number of players on the baseball team. Write a variable expression that
represents the total cost for jerseys for the team. 28N
4. A community service organization must raise a total of $500. This amount is divided by the number of
members so that each member of the organization is responsible for raising the same amount of money.
Let N represent the number of people in the organization. Write a variable expression that represents
the amount of money each person is responsible for raising.
500
N
After each exercise in this activity, have students evaluate the variable expression written for some values
of the variable that you provide.
Objective 11.1B
New Vocabulary
terms of a variable expression
variable terms
constant term
numerical coefficient
variable part
like terms
simplifying a variable expression
Properties to Review
Commutative Property of Addition [1.2A]
Associative Property of Addition [1.2A]
Discuss the Concepts
Suppose you correctly simplify an expression and write the answer as x + 7, and another person writes the
answer as 7+ x. Are both answers correct?
Concept Check
1. Which of the following pairs of terms are like terms?
a. 3a and 3b
b. 7z2 and 7z3
c. 6ab and 3a
d. −4c2 and 6c2 d
2. Classify each statement below as illustrating the Commutative Property of Addition or the Associative
Property of Addition.
a. 5 + 3 = 3 + 5 The Commutative Property
b. (8 + 1) + 4 = 8 + (1 + 4) The Associative Property
c. x + 10 = 10 + x The Commutative Property
d. y + (1 + 7) = (y + 1) + 7 The Associative Property
Objective 11.1C
Properties to Review
Commutative Property of Multiplication [1.4A]
Associative Property of Multiplication [1.4A]
New Property
Distributive Property
Discuss the Concepts
When do we use the Distributive Property? What is its purpose? You might emphasize here that the
Distributive Property is used to remove parentheses from a variable expression and thus enables us to write
an expression equivalent to the original one. This is an important concept in simplifying expressions.
Concept Check
Classify each statement below as illustrating the Commutative Property of Multiplication or the Associative
Property of Multiplication.
a. 12(3) = 3(12) The Commutative Property
b. (3 * 6)9 = 3(6 * 9) The Associative Property
c. qr = rq The Commutative Property
d. x(yz) = (xy)z The Associative Property
Optional Student Activity
Operations such as addition and multiplication are called binary operations. The word binary means
“consisting of two parts.” Addition consists of two parts (the addends); multiplication consists of two parts
(the factors). Other binary operations can be defined. For instance, define ⊗ as a ⊗ b = (a * b) − (a + b).
Then 7 ⊗ 4 = (7 * 4) − (7 + 4) = 28 − 11 = 17.
a. Find 6 ⊗ 8. 34
b. Find
1
4
3
⊗
. 1
12
3
4
c. Does a ⊗ 0 = 0? No, for a ≠ 0
d. Does a ⊗ 1 = 1? No
e. Is ⊗ a commutative operation? Yes
f. Explain your answer to part e.
(a * b) − (a + b) is the same as (b * a) − (b + a) because the operations of multiplication and addition
are both commutative.
Answers to Writing Exercises
134. The simplification is incorrect because, by the Order of Operations Agreement, we must multiply first
and then add. The correct simplification is
2 + 3 (2x + 4)
= 2 + 6x + 12
= 6x + 2 + 12
= 6x + 14
Objective 11.2A
New Vocabulary
equation
solution of an equation
New Symbols
≠ (is not equal to)
Discuss the Concepts
1. What is a solution of an equation?
2. How can we determine whether a number is a solution of an equation?
Concept Check
Label each of the following as either an expression or an equation.
a. 3x + 7 = 9 Equation
b. 3x + 7 Expression
c. 4 − 6(y + 5) Expression
d. a + b = 8 Equation
e. a + b − 8 Expression
Optional Student Activity
Which numbers in parentheses are solutions of the equation?
1. x2 − 7x = 8
(−1, 2, 1, 8) −1, 8
2. y2 = 2y + 15
(−5, −3, 5, 3) −3, 5
3. 2x(x + 3) = x + 12
1
1 1 

 4,  1 , 1 , 4  4, 1
2
2 2 

4. 2x2 − 6x = 5x + 15
1
1 1 

 3,  2 , 2 , 3  2 , 3
2
2 2 

Objective 11.2B
New Vocabulary
solving an equation
Properties to Review
Addition Property of Zero [1.2A]
New Properties
Addition Property of Equations
Discuss the Concepts
What is the solution of the equation x = 9? Use the answer to explain why the goal in solving the equations
in this objective is to get the variable alone on one side of the equation.
Concept Check
Which of the following are equations of the form x + a = b?
a. y + 7.8 = −9.2 Yes
b. 0.3 = z + 1.4 Yes
c. −9 = 3d No
d. −8 + n = −5.6 Yes
Optional Student Activity
Make equations using the numbers 4, 8, and 12 to fill the boxes in
x


a. What is the largest number solution possible? 0
b. What is the smallest number solution possible? −16
Objective 11.2C
Vocabulary to Review
reciprocal [2.7A]
Properties to Review
Multiplication Property of One [1.4A]
New Properties
Multiplication Property of Reciprocals
Multiplication Property of Equations
Concept Check
Which are equivalent equations?
a. 5x = −20
b. −2x = 8
c. 24 = 6x
3
d.
4
x = −3
e. −1 =

1
4
x
f. x = 4
g. x = −4
a, b, d, and g are equivalent equations. c, e, and f are equivalent equations.
Optional Student Activity
Match each numbered equation with a lettered question that can be used to solve the equation.
1. x + 3 = 8 d
2. x − 5 = 20 b
3. 4x = 16 a
4.
x
7
1 e
5. 99 = −9x c
a. 4 times what number is equal to 16?
b. What number minus 5 is equal to 20?
c. 99 is equal to −9 times what number?
d. What number plus 3 is equal to 8?
e. What number divided by 7 is equal to 1?
Objective 11.2D
New Vocabulary
formula
Optional Student Activity
Match each numbered problem with a lettered equation. Then solve the problem.
1. Since 9 A.M., the temperature has risen 16°F. It is now 4°F. What was the temperature at 9 A.M.?
c. −12°F
2. There are 16 members in the Math Club. After graduation next month, there will be 4 members in the
club. How many members are graduating next month? a. 12 members
3. Your roommate, who borrowed $4 from you last week, repays you the $4. You put it in your pocket. You
now have $16 in your pocket. How much money was in your pocket before you added the $4 to it?
d. $12
4. A couple has given their child the same allowance every week for 4 weeks. The child has saved it all and
now has $16. What amount was given to the child each week? b. $4
a. 16 - x = 4
b. 4x = 16
c. x + 16 = 4
d. x + 4 = 16
Answers to Writing Exercises
108. No, none of the numbers 2, −2, 0, 3, 6, or 10 is a solution of the equation. There is no solution of the
equation because there is no number that is equal to itself plus 4.
111a. Students should rephrase the Addition Property of Equations (the same number or variable
expression can be added to each side of an equation without changing the solution of the equation).
b. Students should rephrase the Multiplication Property of Equations (each side of an equation can be
multiplied by the same nonzero number without changing the solution of the equation).
Objective 11.3A
Properties to Review
Addition Property of Equations [11.2B]
Multiplication Property of Equations [11.2C]
Discuss the Concepts
Explain the steps you would take to solve each equation.
a. 5y + 1 = 11
b. 2z − 9 = 11
c. 12 = 2 + 5a
d. −7v + 6 = −8
e. 8 − 5x = −12
f. 9 = 15 − 2y
Concept Check
Match each equation with the first step in solving that equation.
1. 3x − 7 = 5 a
2. 4x + 7 = −5 c
3. 7x − 5 = 2 b
4. −7x + 5 = −2 d
a. Add 7 to each side.
b. Add 5 to each side.
c. Subtract 7 from each side.
d. Subtract 5 from each side.
Optional Student Activity
1. Create a three-question quiz on the material in this objective. Provide solutions for all three questions.
2. Make up an equation of the form:
a. x + a = b that has −5 as a solution
b. ax = b that has −4 as a solution
c. ax + b = c that has −2 as a solution
Objective 11.3B
New Formula
F = 1.8C + 32
Optional Student Activity
Two people decide to open a business to recondition toner cartridges for copy machines. They rent a
building for $7000 per year and estimate that the building maintenance, taxes, and insurance will cost
$6500 per year. Each person wants to make $12 per hour in the first year and will work 10 hours per day for
260 days of the year. Assume that it costs $28 to restore a cartridge and that they can sell the restored
cartridge for $45.
a. How many cartridges must they restore and sell annually to break even, not including the hourly
wage they wish to earn? Approximately 794 cartridges
b. How many cartridges must they restore and sell annually just to earn the hourly wage they desire?
Approximately 3671 cartridges
c. Suppose the entrepreneurs are successful in their business and are restoring and selling 25
cartridges each day of the 260 days they are open. What would be their hourly wage for the year?
Approximately $18.65 per hour
Answers to Writing Exercises
105. No, the sentence “Solve 3x + 4(x − 3)” does not make sense because 3x + 4(x − 3) is an expression,
and you cannot solve an expression. You can solve an equation.
Objective 11.4A
Properties to Review
Addition Property of Equations [11.2B]
Multiplication Property of Equations [11.2C]
Concept Check
When solving an equation of the form ax + b = cx + d, would it always be correct to start by adding or
subtracting the variable term
a. on the right side of the equation? Yes
b. on the left side of the equation? Yes
c. with the smaller coefficient? Yes
Optional Student Activity
1. Rick weighs 76 lb plus half his weight. How much does Rick weigh? 152 lb
2. At the local gym, you can either pay $75 for a year’s membership and $5 per visit, or pay $150 for a
year’s membership plus $2 per visit. How many times during the year must you go to the gym for the two
plans to be equal in price? 25 visits
3. The gauge on a water tank shows that the tank is
tank, the gauge shows that it is
1
4
5
8
full. After 18 more gallons are drained from the
full. How many gallons of water were in the tank when it was
24 gallons
Properties to Review
Distributive Property [11.1C]
Discuss the Concepts
What are the steps involved in solving an equation containing parentheses?
Optional Student Activity
Solve.
1. 3(2x − 1) − (6x − 4) = −9 No solution
2. 2(5x − 6) − 3(x − 4) = 7x + 14 No solution
1
2
full?
Optional Student Activity
1. I am thinking of a number. When I subtract 3 from the number and then take 400% of the result, it is
equal to the original number. What is the original number? 4
2. If s = 4x − 3 and t = x + 5, find the value of x for which s = 2t − 1. 6
3. The population of the town of Danville increased by 5000 people during the 1990s. In the first decade of
the new millennium, the population of Danville decreased by 10%, at which time the town had 3000
more people than it had at the beginning of the 1990s. Find Danville’s population at the beginning of the
1990s. 15,000 people
Answers to Writing Exercises
105. Students should explain that the solution of the original equation is x = 0. Therefore, the fourth line,
where each side of the equation is divided by x, involved division by zero, which is not defined.
106. Many beginning algebra students do not differentiate between an equation that has no solution and an
equation whose solution is zero. Students should explain that zero is a (real) number and that the
solution of the equation 2x + 3 = 3 is the (real) number zero. However, there is no solution to the
equation x = x + 1 because there is no (real) number that is equal to itself plus 1.
Objective 11.5A
New Vocabulary
See the list of verbal phrases that translate into mathematical operations.
Discuss the Concepts
Is the statement true or false?
1. “Five less than n” can be translated as “5 − n.” F
2. A variable expression contains an equals sign. F
3. The words quotient and ratio both indicate division. T
4. The expressions 7y − 8 and (7y) − 8 are equivalent. T
5. “Four times the difference between x and 3” can be translated as “4x − 3.” F
Optional Student Activity
1. Given that x is equal to −3, what is the value of x raised to the 2x power? −27
2. Given that y is equal to 3x and z is equal to 2y, find the sum of x, y, and z in terms of x. 10x
3. At a small college, there are three times as many commuters as boarders and four times as many
boarders as faculty members. Using C to represent the number of commuters, write an expression in
simplest form for the total number of commuters, boarders, and faculty at the college.
17
12
C
Objective 11.5B
Optional Student Activity
Write a variable expression. Use decimals for constants and coefficients.
1. In football, the number of points awarded for a touchdown is three times the number of points awarded
for a safety. Express the number of points awarded for a touchdown in terms of the number of points
awarded for a safety. 3s
2. According to USA Today, in Norway, annual spending per person for books is $40 more than it is in the
United States. Express the annual spending per person for books in Norway in terms of the annual
spending per person for books in the United States. B + 40
3. The cost of renting a car for a day is $39.95 plus 15¢ per mile driven. Express the cost of a one-day
rental in terms of the number of miles driven. 39.95 + 0.15m
4. In 2004, first-class mail cost 37¢ for the first ounce and 23¢ for each additional ounce. Express the cost
of mailing a package first class in terms of the weight of the package. 0.37 + 0.23(w − 1)
Answers to Writing Exercises
47. Students will provide different explanations of how variables are used. Look for the idea that a variable
is used to represent a number that is unknown or a number that can change, or vary.
Objective 11.6A
New Vocabulary
See the list of words and phrases that mean “equals.”
Optional Student Activity
1. One bleepet plus one-fifth of a bleepet equals twenty-one. What is the value of a bleepet?
17
1
2
2. The difference between twelve times a number and seven is the same as the difference between four
times a number and twenty-three. Find the number. 22
3. x is four more than y, and y is ten more than eight times ten. Find the mean of x, y, and (2x + 3y). 214
4. The mean of x and five times x is 30. Find the value of x. 10
5. One-half of a number plus three-fourths of the same number is 40. Find the number. 32
6. One-half of a number equals two-thirds of the same number. Find the number. 0
7. The sum of three numbers is one hundred twenty. The second number is ten more than the first, and the
third number is three times the first. Find the three numbers. 22, 32, and 66
Objective 11.6B
Concept Check
1. A Chinese restaurant charges $11.95 for the adult buffet and $7.25 for the children’s buffet. One family’s
bill came to $79.35. If there were three adults in the family, how many children were there? 6 children
2. A local feed store sells a 75-pound bag of feed for $10.90. If a customer buys more than one bag, each
additional bag costs $10.50. A customer bought $84.40 worth of feed. How many 75-pound bags of feed
did this customer purchase? 8 bags
Optional Student Activity
1. A rope is cut in half. Then one of the two pieces is cut so that the ratio of the pieces created from it is
2:1. The total length of the shortest of the three pieces and the longest of the three pieces is 6 ft. Find
the length of the original piece of rope. 9 ft
2. Aaron decided to quit smoking. He started by cutting back two cigarettes a day each day for a week. He
smoked 119 cigarettes during the week. How many cigarettes did he smoke the day before he started to
cut back? 25 cigarettes
Answers to Writing Exercises
51. The problem states that a 4-quart mixture of fruit juice is made from apple juice and cranberry juice.
There are 6 more quarts of apple juice than of cranberry juice. If we let x = the number of quarts of
cranberry juice, then x + 6 = the number of quarts of apple juice. The total number of quarts is 4.
Therefore, we can write the equation x + (x + 6) = 4.
x + (x + 6) = 4
2x + 6 = 4
2x = −2
x = −1
Since x = the number of quarts of cranberry juice, there are −1 qt of cranberry juice in the mixture. We
cannot add −1 qt to a mixture. The solution is not reasonable.
We see from the original problem that the answer will not be reasonable. If the total number of quarts
in the mixture is 4, we cannot have more than 6 qt of apple juice in the mixture.
52. Students should provide you with information concerning their majors, as well as two formulas from the
field of study and explanations of the variables used in each formula.
Answers to Focus on Problem Solving—From Concrete to Abstract
1. 20d nickels
2.
100d
c
minutes
3. mg miles
4.
5.
60h
x
25 q
5n
miles
photocopies
Answers to Projects and Group Activities—Averages
1.
2.
3.
4.
5.
6.
7.
Answers will vary.
Answers will vary.
Answers will vary.
0
Steps 2 and 3: Answers will vary. Step 4: 0
The sums are the same.
If the activity were conducted again, the outcome would be the same.
Students should describe the average of a set of data as a number that represents the “middle” of the
data. Therefore, the data values above the average and the data values below the average will “cancel
each other out.”