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MATH
STUDENT BOOK
8th Grade | Unit 2
Unit 2 | Modeling Problems in Integers
Math 802
Modeling Problems in Integers
Introduction |3
1. Equations with Real Numbers
5
Translating Expressions and Equations |5
Solving One-Step Equations |10
Solving Two-Step Equations |16
SELF TEST 1: Equations with Real Numbers |21
2.Functions
23
Relations and Functions |23
Functions |29
Analyzing Graphs |35
SELF TEST 2: Functions |41
3.Integers
47
Addition of Integers |47
Subtraction of Integers |55
Multiplying and Dividing Integers |61
Evaluating Expressions |67
SELF TEST 3: Integers |71
4. Modeling with Integers
73
Graphing |73
One-Step Equations |80
Two-Step Equations |85
Problem Solving |89
SELF TEST 4: Modeling with Integers |93
5.Review
99
LIFEPAC Test is located in the
center of the booklet. Please
remove before starting the unit.
Section 1 |1
Modeling Problems in Integers | Unit 2
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2| Section 1
Unit 2 | Modeling Problems in Integers
Modeling Problems in Integers
Introduction
In this unit, functions are represented in a variety of ways. Students are asked
to interpret graphical models in practical situations. Variable, expression, and
equation are defined. Students solve one-step and two-step equations with
whole numbers first. After algebra tiles and the number line are used to establish
the “rules” for adding, subtracting, multiplying, and dividing integers, students
solve equations in the integers. The order of operations is revisited in this unit.
Students evaluate expressions using signed numbers. Finally, students are
guided through the process for using an equation.
Objectives
Read these objectives. The objectives tell you what you will be able to do when
you have successfully completed this LIFEPAC. When you have finished this
LIFEPAC, you should be able to:
zz Perform operations of integers.
zz Solve one-step and two-step equations, with real numbers and integers.
zz Translate contextual situations into one-step and two-step equations
before solving them.
zz Identify relations and functions in their many forms, including ordered
pairs, mapping diagrams, t-charts, and graphing.
zz Identify domains, ranges, independent variables, dependent variables,
and inputs and outputs.
zz Graph functions and read the graphs of functions.
Section 1 |3
Modeling Problems in Integers | Unit 2
Survey the LIFEPAC. Ask yourself some questions about this study and write your questions here.
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4| Section 1
Unit 2 | Modeling Problems in Integers
1. Equations with Real Numbers
Translating Expressions and Equations
Objectives
z Translate written statements into math symbols, expressions, and equations.
z Represent
a simple word problem as an equation.
Vocabulary
equation—a mathematical statement that shows two expressions are equal; it has an
equal sign
Translating Expressions
Have you ever read a word problem and had
to reread it a couple of times to try to figure
out what it is asking? Sometimes it feels like it
was written in a foreign language. In a way, it
was. It was written in mathematical language.
Similar to foreign languages, the mathematics
language also has rules of usage and
conventions. In order to understand it, all
you have to do is be able to translate it into a
math problem full of numbers, symbols, and
variables.
To be able to read mathematical statements,
you must become familiar with how to give
them meaning. You can learn to translate
from phrases and sentences written in English
to those written in math. You will learn the
language of math, just as you learned English.
You started simply by learning sounds of
letters. Then, you learned to use letters to
make words.
Numbers are the first part of the mathematical
language that you learned. You know how
to translate from a number written as a word
in English to its mathematical symbol, the
numeral. The word “five” is the numeral “5”
and you understand what it means in both
languages.
Figure 1 is a table of math symbols. In the
columns are some of the words that translate
from English into that math symbol. You may
be able to think of more words that you have
used previously to represent these.
Section 1 |5
Modeling Problems in Integers | Unit 2
+
add
addition
sum
more than
more
plus
added to
-
subtract
subtraction
difference
minus
from
less than
less
×,·,()
multiply
multiplied by
product
times
of
by
÷,—
divide
quotient
divided into
per
=
is
equals
equal to
is equal to
Figure 1| Math Symbols
Now when we see the these words, we can
translate them into the correct mathematical
symbol. Let’s move on to putting some words
together to form expressions.
English Clause
the sum of sixteen and
seven
the quotient of fiftyfive and eleven
thirteen multiplied by
six
twenty less than ninety
Mathematical Expression
Look at the following table to see how we use
variables in math expressions.
16 + 7 or 7 + 16
English Clause
55 ÷ 11 or
13 × 6 or 13 · 6 or 13(6) or
6 × 13 or 6 · 13 or 6(13)
90 - 20
Keep in mind! There is a difference between
“less” and “less than.” Words like “than” and
“from” mean you need to switch the order of
the numbers in the expression.
Remember that addition and multiplication
are commutative, which means that it doesn’t
matter what order we write the numbers in.
Subtraction and division are not commutative,
so we have to be careful to write the numbers
in the correct order. “Twenty less than ninety”
cannot be written as 20 - 90.
Variables
Sometimes in math, we need to represent an
unknown quantity. Consider the statement,
“Somebody is three years old.” We do not
know who somebody is, so it is an unknown.
In math, we might say, “seventeen added to
a number.” In this case, we know seventeen
means 17, but “a number” could mean any
6| Section 1
number. Since we are unsure of the number,
we need a variable to represent it in the
expression.
fifteen added to a number
a number divided by nine
four subtracted from a
number
the product of a number and
nine
Mathematical
Expression
15 + x or x + 15
n ÷ 9 or
y-4
9k
You can see that there are different letters in
the expressions. There is no rule that says
what letter you have to use, but you may find
it easier to use a letter that makes sense for
the problem. People often use the first letter
of the name of the variable. For example, if
you are finding the cost of something, you
may want to use the variable c.
You might think that there is a mistake in the
expression 9k. Product means to multiply, but
there is not a multiplication symbol between
nine and the variable k. This is actually a
convention in math. When a number and a
variable, or two variables, are written next to
each other with no operation sign between
them, it means to multiply.
Another convention that is important to note
is that when a number and variable are
Unit 2 | Modeling Problems in Integers
written next to each other as a product, we
always write the number first. For example,
“a number times ten” would be written as 10x.
“The product of a number and seven” would
be written as 7x.
►
• Translation: 6n = 72
►
Writing Equations
In the English language, we find sentences
to be more useful than clauses. Similarly,
in the language of mathematics, we find
equations to be more useful than expressions.
Equations are similar to sentences because
they allow us to show a complete thought.
The sum of a number and three equals five
is a complete thought, which we can write
mathematically as x + 3 = 5. We now know
enough about our unknown number to find a
value. An equation gives us information for
each side of an equal sign. Look at the Figure
2 below for more examples of translating
equations.
Let’s look at more examples of translating
expressions and equations. Be sure to look
carefully for the difference between the
expressions and the equations.
Examples:
►
The quotient of a number and nine is
four.
• Translation:
►
=4
Sixteen more than a number.
• Translation: y + 16
A number multiplied by six is equal
to 72.
There are 27 people in the room,
and they line up into several rows,
each row having the same number
of people. Write an expression to
represent the number of people that
would be in each row.
• Translation: 27 ÷ r
►
Sally’s neighbor asks her to babysit.
She gets paid $5 an hour. Write an
expression that represents how
much money Sally would make for
babysitting h hours.
• Translation: 5h
►
Seven friends went to the movies last
weekend. Each paid the same ticket
price. Their total admission cost was
$49. Write an equation to represent
the cost c of each person.
• Translation: 7c = 49
►
The Language Club is going to a
French play. Tickets cost thirteen
dollars each. It will cost forty-five
dollars for bus rental. Write an
expression that will represent the
total cost for s students to go to the
play.
• Translation: 13s + 45
English Sentence
The product of seven and a number is 42.
Six more than twice a number equals 20.
Five times a number less twelve is three.
The quotient of four times a number and seven is four.
Mathematical Equation
7x = 42
6 + 2x = 20 or 2x + 6 = 20
5y - 12 = 3
or 4x ÷ 7 = 4
Figure 2 | Translating Equations
Section 1 |7
Modeling Problems in Integers | Unit 2
In English, you moved from learning the
alphabet to eventually writing essays. Our
goal in math is to be able to solve problems.
As you learn the language of mathematics
you will be able to move from solving simple
to more difficult word problems. Don’t expect
to skip the simple stuff and jump right into the
tough stuff.
Let’s Review
Before going on to the practice problems,
make sure you understand the main points
of this lesson.
„ You
can translate English expressions
that involve quantities into
mathematical statements.
„ A
mathematical expression can be a
constant, a variable, or terms combined
using mathematical operations.
„ An
equation is a mathematical
statement that says two expressions
have the same value.
Complete the following activities.
1.1
Match the mathematical expression with its translation.
__________ xy
the product of two numbers is six
six times a number equals y
__________ the sum of x and y
__________ x - y
the sum of two numbers is six
__________ x + y
x divided by y
__________ y - x
y divided by x
x subtracted from y
__________ y ÷ x
x minus y
__________ x + y = 6
six less than a number is y
__________ xy = 6
the product of two numbers
_______ 6x = y
__________ y = x - 6
1.2
Which of the following expressions represents “the product of a number and eight”?
8x
x+8
x÷8
8-x
…
…
…
…
1.3
Which expression would represent the cost of one CD, if the total cost for three of
them is $36?
3(36)
3 + 36
36 - 3
…
…
…
…
8| Section 1
Unit 2 | Modeling Problems in Integers
1.4
All of the following represent the same concept except _____.
47 take away 28
47 less than 28
…
…
47 minus 28
…
1.5
The expression could be translated as _____.
a number and 8
x divided by 8
…
…
the product of x and 8
…
1.6
47 subtract 28
…
8 divided by x
…
Each of the following correctly match an English phrase with a mathematical
expression except _____.
a number times seven, 7x
the sum of a number and seven,
…
…
7+x
a number subtracted from seven,
…
7-x
a number divided by 7,
…
1.7
It costs twelve dollars to get in to the fair. Tickets for rides cost extra and are d
dollars each. If Tyra buys 20 tickets, how much would it cost her for the fair?
$12 - 20d
$12d + 20
$12 + 20d
20d - $12
…
…
…
…
1.8
In order for an expression to be an equation, it must have _____.
an equal sign
more than one variable
…
…
a variable
…
1.9xy = 20 can be read as _____.
the product of two numbers is
…
twenty
the sum of two numbers is twenty
…
an addition sign
…
the quotient of two numbers is
…
twenty
the difference of two numbers is
…
twenty
1.10 Ike is five years older than his brother. If Ike’s age is represented by x, which of the
following would represent his brother’s age?
x+5
5x
5-x
x-5
…
…
…
…
Section 1 |9
Modeling Problems in Integers | Unit 2
Solving One-Step Equations
You and two of your friends have $55 saved
to buy concert tickets for your favorite band.
You heard that tickets cost $18 a piece. Will
you have enough money for all of you to
attend the concert?
This lesson will introduce you to ways to solve
word problems by translating them into math
problems.
Objectives
z Translate and solve one-step equations in context.
z Identify
the inverse operation needed to solve a one-step equation.
z Identify
the property of equality used to solve a one-step equation.
Vocabulary
inverse operations—opposite operations that undo one another
property of equality—what happens to one side of an equation must also happen to the
other side of the equation
solution—a value or values of the variable that make an algebraic sentence true
solve—find the solution(s) to an equation
Translating Word Problems to Equations
You can translate the word problem in the
introduction into an equation. The total
cost of the tickets is what we need to find.
We can use a variable to represent this
unknown. Because you are buying three
tickets at the same price, you can find the
total cost by adding 18 + 18 + 18. It can also
be found by multiplying 3 · 18. So, if we use
the variable c to represent the cost, then c
= 3 · 18, or c = 54. Since the total cost is $54,
then yes, you have enough money!
Granted, you probably could have solved this
problem without writing an equation. So, why
did we?
It takes practice and skill building to get
comfortable with problem solving. Some
problems do not have answers that are
obvious or arrived at easily. Try to compare
it to reading. Your teacher did not give you a
10| Section 1
200-page novel to read in second grade. As
you went through school, teachers asked you
to read more-challenging books. In math, you
will start out with fairly simple problems, and
they will get more difficult as you move on.
Some of the problems in this lesson may
seem too simple and you won’t want to bother
with writing out the work. Try to keep in mind
that what you learn now will prepare you to
tackle the more difficult problems that you will
face later on. There is a system for problem
solving using an equation that you will need to
practice.
Let’s go back to the problem about the
concert tickets. What if the problem said that
you spent $81 on concert tickets for yourself
and two of your friends? Your friends are
paying you back and you need to tell each
one how much they owe you. How much did
each ticket cost?
Unit 2 | Modeling Problems in Integers
We can write an equation: total cost = number
of tickets · price of a ticket. Putting in the
known values gives us $81 = 3 · p. We can
simplify this equation to $81 = 3p or 3p = $81.
The variable p represents the unknown, the
price of a ticket.
Reminder! Two things touching, like the 3
and p, mean multiplication.
This problem differs from the previous
problem because the variable is not all by
itself on one side of the equation. We don’t
have p equal to something. We actually need
to solve for the value of p.
To solve an equation means to find the value
of the unknown, the variable. In order to do
this, we must get the variable all by itself on
one side of the equal sign. It does not matter
which side.
Think About It! Why doesn’t it matter which
side of the equal sign the variable is on?
So, what is the process used to solve an
equation?
We can think of an equation as if it were
a balance scale. The two sides must be in
balance at all times. For example, 5 = 5 is
an equation. If you added two to one side of
the equation and not the other, the equation
Property
addition property of equality
subtraction property of equality
multiplication property of equality
division property of equality
would be “off balance.” It would say 7 = 5. In
other words, the two sides would no longer
be equal to each other. If we add two to both
sides of the equation though, we would have
7 = 7. This equation is still balanced.
You can keep an equation “balanced” by
doing the same thing to both sides of the
equation. We call this idea a property of
equality. Figure 3 shows the properties of
equality that you will use in this lesson.
The properties of equality really just say that
whatever operation you do to one side of an
equation you must do the same operation
to the other side of the equation to keep it
balanced.
So let’s return to the problem of buying 3
concert tickets for $81. The equation
was 3p = 81.
We want to solve the equation by solving
for the unknown p. We need to get p by
itself on one side of the equation. Since p is
multiplied by 3, we need to have an operation
that will “undo” multiplication. Division is the
inverse operation, or opposite operation, of
multiplication. To solve this equation, we must
divide both sides by 3. When there is only
one operation done to a variable that needs
to be “undone,” the equation is a “one-step”
equation.
Definition
If the same value is added to both sides of an
equation, the results are equal.
If the same value is subtracted from both sides of
an equation, the results are equal.
If both sides of an equation are multiplied by the
same value, the results are equal.
If both sides of an equation are divided by the
same value, the results are equal.
Figure 3 | Properties of Equality
Section 1 |11
Modeling Problems in Integers | Unit 2
p = 27
We have solved for the value of p. We call 27
a solution of the equation, because it makes
the two sides of the equation equal. We can
check our solution by replacing 27 for the
value of p in the original equation. If we are
correct, both sides will be equal.
3(27) = 81
81 = 81
We just solved a one-step equation, because
there was only one inverse operation needed
to solve it. However, we solved the equation
in a series of four steps. You should always
follow these steps to solve a one-step
equation. The steps are:
1. Locate the variable.
2. Identify the operation being done to
the variable.
3. Do the inverse operation on both
sides of the equation.
4. Check the solution.
Reminder!! When solving an equation that
has a variable, the goal is to “undo” what was
done to the variable.
Here are some examples using the above
steps.
Example:
►
Solve x - 12 = 61.
Solution:
12 is being subtracted from
x - 12 = 61 x. The inverse operation is
x - 12 + 12 = 61 + 12
addition.
Use the addition property
of equality.
x = 73 Complete the addition.
12| Section 1
Check:
Replace x with 73 in the
73 - 12 = 61 original equation.
61 = 61 Check for balance.
Example:
►
Solve 5p = 80.
Solution:
p is being multiplied by 5. The inverse
5p = 80 operation is division.
Use the division property of equality.
p = 16 Complete the division.
Check:
5(16) = 80 Replace p with 16 to check the solution.
80 = 80 Check for balance.
Example:
►
Find n for
= 16.
Solution:
n is being divided by 8. The inverse
= 16 operation is multiplication.
8·
Use the multiplication property of
= 16 · 8 equality.
n = 128 Complete the multiplication.
Check:
Replace n with 128 to check the
= 16 solution.
16 = 16 Check for balance.
Now let’s solve some word problems by
writing and solving one-step equations. We
will need to combine the skill of translating
with the skill of solving equations.
1. Use a variable to represent the
unknown in the problem. (What are
we asked to find?)
2. Write an equation.
3. Locate the variable in the equation.
Unit 2 | Modeling Problems in Integers
4. Identify the operation being done to
the variable.
5. Do the inverse operation on both
sides of the equation. (The goal is
to “undo” what was done to the
variable.)
6. Check the solution both in the equation and in the context of the word
problem. In other words, does it
make sense?
7. State the answer to the problem.
Write and solve an equation for the
following problem:
Example:
►
Movie tickets are $6 for admission
to the Sunday matinee. Martha paid
$48 for her family to go to the movie.
How many people are in Martha’s
family?
Solution:
►
Each ticket costs $6, so the total for a
number (n) of tickets is $6 times n, or
$6n. For Martha’s family the total is
equal to $48. So, the equation we can
write is $48 = $6n.
n is being multiplied by $6. The
$48 = $6n inverse operation is division.
Check:
=
8=n
Use the division property of
equality.
=
n=8
Notice that each equation has a solution of
8. This is one nice thing about the language
of math. We can approach problems in
different ways as long as we are careful in the
translation and solution.
Note that in this lesson we are not solving
equations that have a negative variable, such
as 5 - x = 3. Later on, we will solve equations
like this. The process you are learning now
will work all of the time, even if we have
fractions or negative numbers.
Before you move on to the practice, it’s
important to remind you once again that you
may feel like you want to skip the work or rush
through the problems because they seem
easy. Remember, we are doing each step
now so that when the problems become more
difficult, you will have the skills in place to
tackle them.
Let’s Review
Before going on to the practice problems,
make sure you understand the main points
of this lesson.
„ You
can translate a contextual problem
into an equation.
„ You
Replace n with 8 to check the
$48 = $6(8) solution.
can solve one-step equations using
inverse operations and properties of
equality.
There are 8 people in Martha’s
„ The
It is important to note that you could also
solve this problem by writing the following
equation:
„ The
$48 = $48 family.
$6n = $48
If the total cost is $48, dividing by the
cost of one ticket would give us how
many tickets were purchased.
Again, we can switch the sides of the
equation.
We can switch the sides of an
equation.
inverse operation for addition is
subtraction (and vice versa).
inverse operation for multiplication
is division (and vice versa).
„ The
inverse operation tells you which
property of equality you need to use.
Section 1 |13
Modeling Problems in Integers | Unit 2
Complete the following activities.
1.11 x + 127 = 185
x=
1.14
1.1216p = 208
p=
1.15523x = 523
x=
n=
= 300
1.13 144 = m – 98
m=
1.16 Henry purchased compact disks for $112. The compact discs cost $16 each.
Which of the following equations could you use to find how many compact disks, x,
Henry purchased?
$16 = $112x
…
$112 =
…
$112 = $16x
…
$112 = x - $16
…
1.17 Which property of equality would be used to solve the equation 59 = x - 26?
addition
multiplication
…
…
subtraction
…
division
…
1.18 Which of the following values is the solution of
= 8?
4
24
40
…
…
…
256
…
1.19 Angelica is buying 5 tickets to the school musical. Tickets cost $7 each.
If c represents Angelica’s total cost for tickets, which equation is correct?
c=5
c = 5 + $7
c = (5)($7)
c = $7
…
…
…
…
14| Section 1
Unit 2 | Modeling Problems in Integers
1.20 There are 32 desks in a room. If x represents the number of rows of desks, which
expression would equal the number of desks in each row?
32 - x
32 + x
32x
…
…
…
…
1.21 Which property of equality would be used to solve 3x = 81?
addition
multiplication
…
…
subtraction
…
division
…
1.22 Solve the word problem using the steps given in the lesson. Show your equation
and each step you take to solve it.
Neil and Tom love to collect baseball cards. Neil has 83 more baseball cards than
Tom. Neil has 517 baseball cards. How many baseball cards does Tom have?
Section 1 |15
Modeling Problems in Integers | Unit 2
Solving Two-Step Equations
To get to your friend’s house you needed
to travel north for 2 blocks, turn west for
1 block, and then north again for the last
3 blocks. What is the easiest way for you
to figure out how to get home from their
house?
Objectives
z Solve two-step equations using real
numbers.
z Translate
word problems into two-step
equations and then solve.
z Check
solutions for reasonableness.
Solving Two-Step Equations
You know that the easiest way for you to
get home is to just reverse the directions
you used to get there, Take another look
at the map, to get home, you will want to
travel 3 blocks south, east for one block,
and then two more blocks south.
You use a similar approach when solving
equations with variables. You just reverse
the math operations done to a variable to
get the variable by itself on one side of an
equation.
Before you start this lesson, you need to
be aware that most of the problems in it
are fairly simple. You may be able to do
them in your head. The key here is to learn
the concepts and a process for solving
equations that will always work. When the
equations become more complicated, you
will have the skills necessary to solve them.
Even though showing work may be boring
or seem like a waste of time right now, it
will be worth it later on.
To solve an equation, you first identify
the math operations done to the variable.
When you figure that out, then you
16| Section 1
can “work backwards,” or use inverse
operations, to undo the operations and get
the variable by itself. When there are two
operations to undo, the equation is a twostep equation.
You must always keep in mind that an
equation must stay balanced, like a scale.
What you do to one side of the equal sign,
you must do to the other side of the equal
sign.
Example:
►
Solve: 2x - 5 = 13.
►
The variable x is multiplied by 2, and
then 5 is subtracted from it. The
result is 13. So, let’s work backwards
to undo it.
Unit 2 | Modeling Problems in Integers
Solution:
Solution:
To undo the subtraction of 5,
2x - 5 = 13 add 5.
Use the addition property of
Use the subtraction
+ 29 - 29 property of equality.
87 - 29 =
2x - 5 + 5 = 13 + 5 equality.
2x = 18 Complete the addition.
Use the division property of equality.
=
x = 9 Complete the division.
Does this value of 9 make sense? We can
check a possible solution by putting it back
into the original equation. Simplify each
side where necessary by using the order of
operations. If both sides are equal, then we
have a solution.
Check:
►
2(9) - 5 = 13
►
18 - 5 = 13
►
13 = 13
Use the multiplication
(2)58 =
(2) property of equality.
Complete the
Now we need to undo the multiplying
by 2, so we need to divide by 2. Use
the division property of equality.
Check:
►
Complete the subtraction.
58 =
116 = x multiplication.
►
87 =
+ 29
►
87 = 58 + 29
►
87 = 87
Example:
►
Solve: = 12.
Solution:
2x is divided by 5. Use the inverse
= 12 operation to solve.
Use the multiplication property of
(5)
= 12(5) equality.
x is multiplied by 2. Use the
2x = 60 inverse operation to solve.
Use the division property of
Keep in mind! Use parentheses when
substituting a value into an equation. Getting
into the habit now will make it easier when
you work with negative numbers.
Example:
►
Solve: 87 =
+ 29.
►
What has happened to the variable?
►
The variable, x, has been divided by
2 and then had 29 added to it. We
will need to work backwards using
inverse operations to undo this.
=
equality.
x = 30 Complete the division.
Check:
►
= 12
►
= 12
►
12 = 12
Section 1 |17
Modeling Problems in Integers | Unit 2
Translating and Solving Two-Step
Equations
►
Let’s break the equation building into
smaller steps to help you build it.
Now let’s see if we can solve some word
problems that require us to write a two-step
equation.
►
Let b equal Sarah’s brother’s age.
►
Then, 2b is twice her brother’s age.
►
2b minus 5 is five less than twice her
brother’s age.
►
2b minus 5 equals 17, or Sarah’s age.
►
Solve: 2b - 5 = 17.
Example:
►
►
►
Justin paid $5 to rent a video game. The
store charged a $3 per day late fee. If
Justin had to pay $17 total, how many
days late was he returning the game?
We first need to translate the
problem into an equation. The
unknown variable is the number of
days Justin was late returning the
game. Let’s make that x. Since each
day there is a charge of $3, the late
fee is 3x. The total amount ($17) is
equal to the original fee ($5) plus
the late fee (3x). That means the
equation is 17 equals 5 plus 3x.
Solve: 17 = 5 + 3x.
Solution:
17 = 5 + 3x
17 - 5 = 5 + 3x - 5
5 is added to 3x. Use the
inverse operation.
Use the subtraction property
of equality.
12 = 3x Complete the subtraction.
=
Using the division property of
equality, divide by 3 on both
sides.
4 = x The game was 4 days late.
Check:
►
17 = 3(4) + 5
►
17 = 12 + 5
►
17 = 17
Example:
►
Sarah’s age is five less than twice her
brother’s age. If Sarah is 17, how old
is her brother?
18| Section 1
Solution:
Twice her brother’s age minus
2b - 5 = 17 5 equals 17, or Sarah’s age.
2b - 5 + 5 = 17 + 5 Add 5 to both sides.
Twice her brother’s age
2b = 22 equals 22.
Divide both sides by 2.
=
b = 11 Her brother is 11 years old.
Check:
►
2(11) - 5 = 17
►
22 - 5 = 17
►
17 = 17
We can use equations to model and solve
word problems. Every problem is unique,
but you can approach each problem in a
systematic way.
1. Use a variable to represent the
unknown in the problem.
2. Use key words to identify the math
operations for the equation.
3. Use the properties of equality to solve
for the unknown.
4. Check your solution to see if it makes
sense in the problem.
Unit 2 | Modeling Problems in Integers
Let’s Review
Before going on to the practice problems,
make sure you understand the main points
of this lesson.
„ You
can translate word problems into
two-step equations and then solve.
„ You
should check solutions for
reasonableness.
„ You
can solve two-step equations using
inverse operations and properties of
equality.
Complete the following activities.
1.23 Based on the lesson, identify the first operation needed to solve the following
equation. 2x + 3 = 11
addition
multiplication
…
…
subtraction
…
division
…
1.24 Based on the lesson, identify the first operation needed to solve the following
equation. - 3 = 11
addition
…
subtraction
…
multiplication
…
division
…
1.25 Based on the lesson, identify the first operation needed to solve the equation.
11x + 3 = 3
addition
multiplication
…
…
subtraction
…
division
…
1.26 Based on the lesson, identify the first operation needed to solve the following
-2=3
equation.
addition
…
subtraction
…
multiplication
…
division
…
1.27 What are the steps, in order, needed to solve 3x + 4 = 13?
Divide by 3, and then subtract 4.
Subtract 4, and then divide by 3.
…
…
Multiply by 3, and then add 4.
…
Add 4, and then multiply by 3.
…
Section 1 |19
Modeling Problems in Integers | Unit 2
1.28 Solve for b in the following equation.
b = 36
b = 12
…
…
-2=4
b = 24
…
b=1
…
1.29 Six more than the product of seven and a number is 55. Which equation and
solution correctly represent this sentence?
… + 6 = 55; x = 7
7x + 6 = 55; x = 7
…
… + 6 = 55; x = 49
7x + 6 = 55; x = 49
…
1.30 What are the steps, in order, needed to solve - 5 = 20?
Add 5, and then divide by 2.
Add 5, and then multiply by 2.
…
…
Subtract 5, and then multiply by 2.
…
Subtract 5, and then divide by 2.
…
1.31 You buy one shirt for $22 and two pairs of shorts. Your total cost is $56. Assuming
both pairs of shorts cost the same amount, what is the price of one pair of shorts?
What is the correct equation and solution for this problem?
2x - $22 = $56; x = $34
2x - $22 = $56; x = $17
…
…
2x + $22 = $56; x = $34
…
2x + $22 = $56; x = $17
…
1.32 One bucket of popcorn and two sodas cost $13. The popcorn was $7. What is the
price per soda?
$6
$3
$10
$13
…
…
…
…
1.33 Solve for x in the following equation.
x=3
x = 12
…
…
=4
x = 28
…
x = 16
…
1.34 Solve the word problem using the steps given in the lesson.
Skylar needs $56 to buy a new video game. If he doubles the amount of money he
has now, he will have enough to buy the game with $8 left over. How much money
does Skylar have right now?
Review the material in this section in preparation for the Self Test. The Self Test
will check your mastery of this particular section. The items missed on this Self Test will
indicate specific areas where restudy is needed for mastery.
20| Section 1
Unit 2 | Modeling Problems in Integers
SELF TEST 1: Equations with Real Numbers
Complete the following activities (6 points, each numbered activity).
1.01 X - 6 translates as “a number
subtracted from 6.”
True
{
False
{
1.04 “Five times a number” can be written
as 5x.
True
{
False
{
1.02 One hundred is the solution of the
equation m - 61 = 39.
True
{
False
{
1.05 To solve 38 = 6x - 16, you would
subtract 16 and then divide by 6.
True
{
False
{
1.034b + 19 can be translated as “nineteen
more than the product of 4 and a
number.”
True
{
False
{
1.06 Which of the following expressions represents “5 less than 3 times the number m”?
5 - 3m
3m - 5
5m - 3
3 - 5m
…
…
…
…
1.07 Which of the following steps could be used to solve the equation 93 = x + 58?
Add 58 to both sides.
Multiply both sides by 58.
…
…
Subtract 58 from both sides.
…
Divide both sides by 58.
…
1.08 Maria paid $48 for herself and two of her friends to go to a concert. The tickets are
all the same price. What equation could be used to find the price of one ticket?
p = 3 • $48
p = $48 - 3
3 = $48p
$48 = 3p
…
…
…
…
1.09 Which of the following is a solution for the equation 7x + 21 = 105?
105
84
15
…
…
…
12
…
1.010 The senior class had a car wash to raise money for the senior trip. It cost them
$35 for supplies. They washed 75 cars and charged $6 for each car. Which of the
following expressions represents the money they raised for their trip?
$35 + 75 • $6
75 • $6 - $35
…
…
$35 - 75 • $6
…
75 - $35 • $6
…
Section 1 |21
Modeling Problems in Integers | Unit 2
1.011 All of the following represent the same expression except _____.
19 less than 25
19 subtracted from 25
…
…
25 less than 19
…
25 minus 19
…
1.012 This model shows the solution to an equation.
x - 34 = 13
What property of equality was used to solve the equation?
x - 34 + 34 = 13 + 34
x = 47
addition
multiplication
…
…
subtraction
…
division
…
1.013 The cost to rent a video game for 3 days was $10. It costs the same amount each
day. How could one find the cost to rent the video for one day?
The cost to rent for one day is the
The cost to rent for one day is the
…
…
product of $10 and 3.
quotient of $10 and 3.
The cost to rent for one day is the
…
sum of $10 and 3.
The cost to rent for one day is the
…
difference of $10 and 3.
1.014 Clarence solved the equation 3x + 15 = 33 and showed the following work.
3x + 15 = 33
3x + 15 - 15 = 33
3x = 33
=
x = 11
Which of the following is true?
x is 11.
…
x should be 6.
…
x should be 16.
…
x should be 99.
…
1.015 All of the following have the same solution except _____.
50 = 5m
b+2=8
x-6=4
…
…
…
72
22| Section 1
90
SCORE
TEACHER
… = 2
initials
date
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