Download Numerical and Algebraic Expressions and Equations

Numerical and
Algebraic Expressions
and Equations
© Carnegie Learning
Sometimes
it's hard to tell
how a person is feeling
when you're not talking to
them face to face. People use
emoticons in emails and chat
messages to show different
facial expressions. Each
expression shows a different
kind of emotion. But you
probably already knew
that. ; )
5.1 It Must Be a Sign!
Operating with Rational Numbers.................................241
5.2 What’s It Really Saying?
Evaluating Algebraic Expressions. ............................... 257
5.3 Express Math
Simplifying Expressions Using Distributive Properties................................................ 265
5.4 Reverse Distribution
Factoring Algebraic Expressions................................... 271
5.5 Are They the Same or Different?
Verifying That Expressions Are Equivalent................... 279
5.6 It is time to justify!
Simplifying Algebraic Expressions Using Operations and Their Properties.................................. 287
239
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© Carnegie Learning
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It Must Be a Sign!
Operating with Rational
Numbers
Learning Goals
In this lesson, you will:






Model addition and subtraction on a number line.
Add and subtract integers.
Multiply and divide integers.
Sketch a Venn diagram to show relationships between sets of numbers.
Operate with rational numbers.
Solve real-world problems involving rational numbers.
O
pposites are all around us. If you move forward two spaces in a board game
and then move back in the opposite direction two spaces, you’re right back where
you started. When playing tug-of-war, if one team pulls on the rope exactly as hard
as the team on the opposite side, no one moves. If an atom has the same number
of positively charged protons as it does negatively charged electrons, then the
atom has no charge.
© Carnegie Learning
In what ways have you worked with opposites in mathematics?
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Problem 1 Adding on a Number Line
Recall that a number line can be used to model integer addition.
When adding a positive integer, move to the right on a number line.
When adding a negative integer, move to the left on a number line.
Example 1: The number line shows how to determine 3 1 6.
3
Step 1
6
Step 2
–10
–8
–6
–4
–2
0
2
4
6
8
10
Example 2: The number line shows how to determine 3 1 (26).
3
Step 1
–6
–8
–6
–4
–2
0
2
4
6
8
1. Compare the first steps in each example.
a. What distance is shown by the first term in each example?
b. Describe the graphical representation of the first term.
Where does it start and in which direction does it move?
10
Remember that
the absolute value
of a number is its
distance from 0.
© Carnegie Learning
–10
Step 2
c. What is the absolute value of the first term in each example?
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2. Compare the second steps in each example.
a. What distance is shown by the second term in each example?
b. Why did the graphical representation for the second terms both start at the
endpoints of the first terms but then continue in opposite directions? Explain your
reasoning.
c. What are the absolute values of the second terms?
3. Consider each expression in the worked examples. Which term in each expression
had the greater absolute value? How does the sign of that term relate to the sign of
the sum?
4. Use the number line to determine each sum. Show your work.
a. 22 1 8 5
© Carnegie Learning
–10
–8
–6
–4
–2
0
2
4
6
8
10
–8
–6
–4
–2
0
2
4
6
8
10
–8
–6
–4
–2
0
2
4
6
8
10
b. 2 1 (28) 5
–10
c. 22 1 (28) 5
–10
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d. 2 1 8 5
–10
–8
–6
–4
–2
0
2
5. In Question 4, what patterns do you notice when:
a. you are adding two positive numbers?
4
6
8
10
Can you see
how knowing the
absolute value is
important when adding
and subtracting signed
numbers?
b. you are adding two negative numbers?
c. you are adding a negative number and a
positive number?
6. The set of rational numbers includes integers, mixed numbers, fractions, and their
equivalent decimals.
Based on Questions 4 and 5, what patterns do you predict will occur when:
b. adding two negative fractions?
© Carnegie Learning
a. adding two positive mixed numbers?
c. adding a negative mixed number to a positive mixed number?
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Problem 2 Subtracting on a Number Line
Subtraction can mean to “take away” objects from a set. Subtraction can also mean
a comparison of two numbers, or the “difference between them.” Just as with integer
addition, a number line can be used to model integer subtraction.
Cara said, “Subtraction means to back up, or move in the opposite direction. Like in
football when a team is penalized or loses yardage, they have to move back.”
Analyze Cara’s examples.
Example 1: –5 – 2
–5
opposite of 2
–10 –8 –6 –4 –2
0
2
4
6
8
10
First, I moved from 0 to –5, and then I went in the opposite direction
of the 2 because I am subtracting. So, I went two units to the left and
ended up at –7.
So, –5 – 2 = –7.
Example 2: –5 – (–2)
–5
opposite of –2
–10
–8
–6
–4
–2
0
2
4
6
8
10
In this problem, I went from 0 to –5. Because I am subtracting –2, I went in the
opposite direction of the –2, or right two units, and ended up at –3.
So, –5 – (–2) = –3.
© Carnegie Learning
Example 3: 5 – (–2)
5
opposite of –2
–10 –8
–6
–4
–2
0
2
4
6
8
10
1. Read aloud Cara’s examples above to a classmate. Explain the model Cara created in
Example 3. Ask your teacher for help if needed.
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Example 4: 5 – 2
5
–10 –8 –6 –4 –2
0
2
opposite of 2
4
6
8
10
2. Explain the model Cara created in Example 4.
3. Consider each expression in Cara’s examples. Which term in each expression had
the greater absolute value? How does the sign of that term relate to the sign of
the difference?
Use Cara's
examples for
help.
4. Use the number line to complete each number sentence.
a. 7 2 (23) 5
–10
–8
–6
–4
–2
0
2
4
6
8
10
–6
–4
–2
0
2
4
6
8
10
–6
–4
–2
0
2
4
6
8
10
–10
–8
c. 22 2 2 5
–10
–8
© Carnegie Learning
b. 7 2 3 5
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d. 22 2 (22) 5
–10
–8
–6
–4
–2
0
2
4
6
8
10
For a subtraction expression, such as 29 2 (24), Cara’s method is to start at zero and go
to 29, and then go four spaces in the opposite direction of 24 to get 25.
Dava says, “I see another pattern. Since subtraction is the inverse of addition, you can
think of subtraction as adding the opposite number. That matches with Cara’s method of
going in the opposite direction.”
–9 – (–4) is the same as –9 + –(–4).
–9 + –(–4) = –9 + 4 = –5
opposite of –4 = –(–4)
–10
–8
–6
–4
–2
0
2
4
6
8
10
An example of Dava’s method is shown.
10 2 (26) 5 10 1 2(26)
5 10 1 6
5 16
© Carnegie Learning
5. Apply Dava’s method to determine each difference.
a. 23 2 (23)
b.28 2 5
c. 16 2 9
d.10 2 (220)
So, I can change any
subtraction problem to
show addition if I take
the opposite of the
number that follows the
subtraction sign.
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6. Tell whether these subtraction sentences are always true, sometimes true, or never
true. Give examples to explain your thinking.
a. positive 2 positive 5 positive
b. negative 2 positive 5 negative
c. positive 2 negative 5 negative
d. negative 2 negative 5 negative
7. If you subtract two negative integers, will the answer be greater than or less than the
number you started with? Explain your thinking.
9. What happens when a negative number is subtracted from zero?
© Carnegie Learning
8. What happens when a positive number is subtracted from zero?
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10. The set of rational numbers includes integers, mixed numbers, fractions, and their
equivalent decimals.
Decide whether each statement is true or false. Explain your reasoning.
a. Subtracting a positive rational number from a positive rational number always
results in a positive answer.
b. Subtracting a positive rational number from negative rational number always
results in a negative answer.
c. Subtracting a negative rational number from a positive rational number always
results in a negative answer.
d. Subtracting a negative rational number from a negative rational number always
© Carnegie Learning
results in a negative answer.
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11. Just by looking at the problem, how do you know if the sum of two rational numbers
is positive, negative, or zero?
12. How are addition and subtraction of rational numbers related?
Problem 3 Multiplying and Dividing Integers
When you multiply integers, you can think of multiplication as repeated addition.
Example 1
Consider the expression 3 3 (24).
As repeated addition, it means (24) 1 (24) 1 (24) 5 212.
You can think of 3 3 (24) as three groups of (24).
–
–
–
–
–
–
–
–
–
–
–
–
(–4)
–15
–12
–
5
–10
(–4)
–5
–
–
–
–
–
–
–
–
–
–
© Carnegie Learning
–
(–4)
0
5
10
15
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Example 2
Consider the expression: 4 3 (23).
You can think of this as four sets of (23), or (23) 1 (23) 1 (23) 1 (23) 5 212.
–
–
–
–
–
–
–
–
5
–
–
–
–
–
–
–
–
–
–
–
–
–
–
(–3)
–15
–
–
–12
(–3)
(–3)
–10
(–3)
–5
0
5
10
15
Example 3
Consider the expression: (23) 3 (24).
You know that 3 3 (24) means “three groups of (24)” and that 23 means “the
opposite of 3.” So, (23) 3 (24) means “the opposite of 3 groups of (24).”
© Carnegie Learning
Opposite of (–4)
Opposite of (–4)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
5
+
+
+
+
+
+
+
+
+
+
Opposite of (–4)
(+4)
–15
–10
–5
0
(+4)
5
(+4)
10
12
15
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1. What is the sign of the product of two integers when:
a. they are both positive?
b. they are both negative?
c. one is positive and one is negative?
d. one is zero?
2. What is the sign of the quotient of two integers when:
a. they are both positive?
b. they are both negative?
c. one is positive and one is negative?
d. the dividend is zero?
e. the divisor is zero?
your reasoning.
© Carnegie Learning
3. How do the answers to Question 2 compare to the answers to Question 1? Explain
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The set of integers is a subset of the rational numbers. The set of rational numbers also
includes mixed numbers, fractions, and their decimal equivalents. The rules to determine
the sign of a product or a quotient of integers also applies to mixed numbers, fractions,
and their decimal equivalents.
Remember that the set of rational numbers are all numbers that can be written as __
​ a ​,
b
where a and b are integers and b  0.
4. Sketch a Venn diagram to show all of the sets and subsets of numbers
that you know.
5. Use your Venn diagram to decide whether each statement is true or false. Explain
your reasoning.
a. An integer is sometimes a rational number.
b. A rational number is always an integer.
6. How will you determine the sign of a product if there are:
© Carnegie Learning
a. an even number of negative rational numbers?
b. an odd number of negative rational numbers?
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7. How will you determine the sign of a quotient if there are:
a. an even number of negative rational numbers?
b. an odd number of negative rational numbers?
8. Determine each product or quotient.
4
2  ​3 2​ __
 ​ 
a.​ __
5
3
254 
b.​ _____
 ​
218
1  ​
d. 26.3 3 3​ __
3
e. 16.5 4 26.6
7
f. 2​ ___  ​ 3 ___
​ 10 ​ 
10
7
88
g. 2​ ___ ​ 
44
5
1
h. 2​ __  ​4 2​ __ ​ 
5
6
© Carnegie Learning
c. 27 4 49
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Problem 4 Rational Number Operations in Context
1. Melissa has a total of $56 in her checking account. She uses her check card
to make a purchase of $27.42. How much money does Melissa have left in her
checking account?
2. Marco is an extreme biker. He is participating in an all-terrain biking event. During the
1 ​ miles during
1  ​miles. He rides a total of 31​ __
first leg of the event, he rides a total of 26​ __
5
2
the second leg. How many total miles did Marco ride during the biking event?
3. Serena plays the trumpet in her school’s marching band. She recently sold popcorn
as part of a fundraiser for the band’s next field trip. On Monday, she collected $45.
She collected $55.75 on Tuesday, and on Wednesday she collected $82.50. What is
the total amount of money that Serena collected for the fundraiser during the
three days?
4. A bus has a total of 14 passengers (not including the driver). At the next bus stop,
8 people get on the bus. At the stop after that, 11 people get off of the bus. How
© Carnegie Learning
many passengers remain on the bus?
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© Carnegie Learning
1  ​square feet. The length of the
5. The area of a rectangular playground is 325​ __
2
playground is 21 feet. What is the width of the playground?
Be prepared to share your solutions and methods.
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What’s It Really
Saying?
Evaluating Algebraic Expressions
Learning Goal
Key Terms
In this lesson, you will:
 variable
 algebraic expression
 Evaluate algebraic expressions.
 evaluate an algebraic
expression
D
o you have all your ducks in a row? That’s just a drop in the bucket!
That’s a piece of cake!
What do each of these statements have in common? Well, they are all idioms.
Idioms are expressions that have meanings which are completely different from
their literal meanings. For example, the “ducks in a row” idiom refers to asking if
someone is organized and ready to start a task. A person who uses this idiom is
not literally asking if you have ducks that are all lined up.
For people just learning a language, idioms can be very challenging to understand.
Usually if someone struggles with an idiom’s meaning, a person will say “that’s
just an expression,” and explain its meaning in a different way. Can you think of
© Carnegie Learning
other idioms? What does your idiom mean?
5.2 Evaluating Algebraic Expressions • 257
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Problem 1 Game Day Special
You volunteer to help out in the concession stand at your middle-school football game.
You must create a poster to display the Game Day Special: a hot dog, a bag of chips, and
a drink for $3.75.
1. Complete the poster by multiplying the number of specials by the cost of the special.
Game Day Special
1 hot dog, 1 bag of chips, and a drink
for $3.75
Number of Specials
Cost
1
2
3
4
5
6
In algebra, a variable is a letter or symbol that is used to represent a quantity. An
algebraic expression is a mathematical phrase that has at least one variable, and it can
contain numbers and operation symbols.
Whenever you perform the same mathematical process over and over again, an algebraic
expression is often used to represent the situation.
2. What algebraic expression did you use to represent the total cost on your poster?
3. The cheerleading coach wants to purchase a Game Day Special for every student on
the squad. Use your algebraic expression to calculate the total cost of purchasing
18 Game Day Specials.
© Carnegie Learning
Let s represent the number of Game Day Specials.
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Problem 2 Planning a Graduation Party
Your aunt is planning to
host your cousin’s high
school graduation party at
Lattanzi’s Restaurant and
Reception Hall. Lattanzi’s
has a flyer that describes
the Deluxe Graduation
Reception.
Deluxe Graduation Reception
Includes:
One salad (chef or Cæsar)
One entree (chicken, beef, or seafood)
Two side dishes
One dessert
Fee:
$105 for the reception hall plus $40 per guest
1. Write an algebraic expression to determine the cost of
the graduation party. Let g represent the number of
guests attending the party.
So, an equation
has an equals sign.
An expression
does not.
2. Determine the cost of the party for each number of attendees.
Show your work.
a. 8 guests attend
© Carnegie Learning
b. 10 guests attend
c. 12 guests attend
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Problem 3 Evaluating Expressions
In Problems 1 and 2, you worked with two expressions, 3.75s and (105 1 40g). You
evaluated those expressions for different values of the variable. To evaluate an algebraic
expression, you replace each variable in the expression with a number or numerical
expression and then perform all possible mathematical operations.
1. Evaluate each algebraic expression.
a. x 2 7
●
for x 5 28 ●
for x 5 211 ●
for x 5 16 Use parentheses
to show
multiplication
like -6(-3).
b. 26y
●
for y 5 23 ●
for y 5 0 ●
for y 5 7 c. 3b 2 5
●
for b 5 22 ●
for b 5 3
●
for b 5 9 ●
for n 5 25 ●
for n 5 0
●
for n 5 4 © Carnegie Learning
d. 21.6 1 5.3n
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Sometimes, it is more convenient to use a table to record the results when evaluating the
same expression with multiple values.
2. Complete each table.
a.
h
2h  7
2
1
8
7
b.
a
12
10
0
4
a
__
​    ​ 1 6
4
c.
x
x2  5
1
3
© Carnegie Learning
6
2
d.
y
5
1
0
15
1
2
__
​ __  ​ y
  1 3 ​    ​
5
5
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Problem 4 Evaluating Algebraic Expressions Using Given Values
1. Evaluate each algebraic expression for x 5 2, 23, 0.5, and 22​__
 1 ​.
3
a. 23x
b. 5x 1 10
c. 6 2 3x
d. 8x 1 75
© Carnegie Learning
Using tables may help
you evaluate these
expressions.
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1 ​ .
2. Evaluate each algebraic expression for x 5 27, 5, 1.5, and 21​ __
6
a. 5x
b. 2x 1 3x
c. 8x 2 3x
© Carnegie Learning
I'm noticing
something
similar about all of
these expressions.
What is it?
5.2 Evaluating Algebraic Expressions • 263
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5 ​ .
3. Evaluate each algebraic expression for x 5 23.76 and 221​ __
6
a.2.67x 2 31.85
3 ​x 1 56 ​__
 3 ​
b. 11​ __
4
8
Talk the Talk
1. Describe your basic strategy for evaluating any algebraic expression.
© Carnegie Learning
2. How are tables helpful when evaluating expressions?
Be prepared to share your solutions and methods.
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Express Math
Simplifying Expressions Using
Distributive Properties
Learning Goals
Key Terms
In this lesson, you will:




 Write and use the
distributive properties.
 Use distributive properties
Distributive Property of Multiplication over Addition
Distributive Property of Multiplication over Subtraction
Distributive Property of Division over Addition
Distributive Property of Division over Subtraction
to simplify expressions.
I
t once started out with camping out the night before the sale. Then, it evolved
to handing out wrist bands to prevent camping out. Now, it’s all about the
Internet. What do these three activities have in common?
For concerts, movie premieres, and highly-anticipated sporting events, the
distribution and sale of tickets have changed with computer technology.
Generally, hopeful ticket buyers log into a Web site and hope to get a chance to
buy tickets. What are other ways to distribute tickets? What are other things that
© Carnegie Learning
routinely get distributed to people?
5.3 Simplifying Expressions Using Distributive Properties • 265
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Problem 1 Fastest Math in the Wild West
Dominique and Sarah are checking
each other’s math homework. Sarah
Sarah is able to correctly multiply tells Dominique that she has a quick
7 by 230 in her head. She explains her
and easy way to multiply a one-digit
method to Dominique.
number by any other number in her
head. Dominique decides to test
Sarah by asking her to multiply
7 by 230.
1. Calculate the product 7 3 230.
Sarah
230 3 7
First, break 230 into the
sum of 200 and 30. Then,
multiply 7 x 200 to get a
product of 1400 and 7 x 30
to get a product of 210.
Finally, add 1400 and 210
together to get 1610.
2. Write an expression that shows the mathematical steps Sarah performed to calculate
the product.
Dominique makes the connection between Sarah’s quick mental calculation
and calculating the area of a rectangle. She draws the models shown to
Dominique
e as
Calculating 230 x 7 is the sam
tangle by
determining the area of a rec
width.
multiplying the length by the
tangle into
But I can also divide the rec
culate the
two smaller rectangles and cal
then add
area of each rectangle. I can
al.
the two areas to get the tot
© Carnegie Learning
represent Sarah’s thinking in a different way.
230
7
7
200
30
1400
210
1400 + 210 = 1610
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3. First, use Dominique’s method and sketch a model for each. Then, write an
expression that shows Sarah’s method and calculate.
a. 9(48)
b. 6(73)
I can draw at
least two different
models to determine
4(460).
c. 4(460)
© Carnegie Learning
Sarah’s and Dominique’s methods are both examples of the
Distributive Property of Multiplication over Addition, which
states that if a, b, and c are any real numbers, then
a  ( b 1 c) 5 a  b 1 a  c.
Including the Distributive Property of Multiplication over Addition,
there are a total of four different forms of the Distributive Property.
Another Distributive Property is the Distributive Property of
Multiplication over Subtraction, which states that if
a, b, and c are any real numbers, then a  ( b 2 c) 5 a  b 2 a  c.
5.3 Simplifying Expressions Using Distributive Properties • 267
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The Distributive Property also holds true for division over addition and division over
subtraction as well.
The Distributive Property of Division over Addition states that if a, b, and c are real
a b
b 
​ c ​.
​5 __
​ c ​1 __
numbers and c fi 0, then ______
​ a 1
c   
The Distributive Property of Division over Subtraction states that if a, b, and c are real
a b
a 2  b 
​ c ​.
​5 __
​ c ​2 __
numbers and c fi 0, then ​ ______
c  
4. Draw a model for each expression, and then simplify.
a. 6(x 1 9)
c. 2(4a 1 1)
b. 7(2b 2 5)
x 1  ​
15 
 
d.​ _______
5
Dividing by 5
is the same as
multiplying by what
number?
5. Use one of the Distributive Properties to rewrite each expression in
a. 3y(4y 1 2)
b. 12( x 1 3)
c. 24a(3b 2 5)
d. 27y(2y 2 3x 1 9)
6m 1  ​
12 
 
e.​ ________
22
22 2 4x
 
f.​ ________
 ​ 
2
© Carnegie Learning
an equivalent form.
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Problem 2 Simplifying and Evaluating
1. Simplify each expression. Show your work.
a. 26(3x 1 (24y))
b. 24(23x 2 8) 2 34
27.2 2
 6.4x
 
 ​ 
c.​ ____________
20.8
(  )(  ) (  )(  )
© Carnegie Learning
1 ​   ​​3__
1 ​   ​​22​ __
1 ​   ​
​ 1 ​   ​1 ​22​ __
d.​22​ __
2 4
2
4
(  )
1 ​   ​1 5y
​27​ __
2
___________
 ​ 
e.​ 
 
1 ​ 
2​ __
2
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2. Evaluate each expression for the given value. Then, use properties to simplify the
original expression. Finally, evaluate the simplified expression.
2  ​
a. 2x (23x 1 7) for x 5 21​ __
3
4.2x 27
b.​ ________
 
 ​ 
for x 5 1.26
1.4
© Carnegie Learning
c. Which form—simplified or not simplified—did you prefer to evaluate? Why?
Be prepared to share your solutions and methods.
270 • Chapter 5 Numerical and Algebraic Expressions and Equations
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Reverse Distribution
Factoring Algebraic Expressions
Learning Goals
Key Terms
In this lesson, you will:
 factor
 common factor
 greatest common
 Use the distributive properties to
factor expressions.
 Combine like terms to simplify expressions.
factor (GCF)
 coefficient
 like terms
 combining like
terms
M
any beginning drivers have difficulty with driving in reverse. They think that
they must turn the wheel in the opposite direction of where they want the back
end to go. But actually, the reverse, is true. To turn the back end of the car to the
left, turn the steering wheel to the left. To turn the back end to the right, turn the
wheel to the right.
Even after mastering reversing, most people would have difficulty driving that
way all the time. But not Rajagopal Kawendar. In 2010, Kawendar set a world
© Carnegie Learning
record for driving in reverse—over 600 miles at about 40 miles per hour!
5.4 Factoring Algebraic Expressions • 271
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Problem 1 Factoring
You can use the Distributive Property in reverse. Consider the expression:
7(26) 1 7(14)
Since both 26 and 14 are being multiplied by the same number, 7, the Distributive
Property says you can add the multiplicands together first, and then multiply their sum
by 7 just once.
7(26) 1 7(14) 5 7(26 1 14)
You have factored the original expression. To factor an expression means to rewrite the
expression as a product of factors.
The number 7 is a common factor of both 7(26) and 7(14). A common factor is a number or
an algebraic expression that is a factor of two or more numbers or algebraic expressions.
1. Factor each expression using a Distributive Property.
a. 4(33) 2 4(28)
b. 16(17) 1 16(13)
The Distributive Properties can also be used in reverse to factor algebraic expressions.
For example, the expression 3x 1 15 can be written as 3( x) 1 3(5), or 3( x 1 5). The factor,
3, is the greatest common factor to both terms. The greatest common factor (GCF) is the
largest factor that two or more numbers or terms have in common.
When factoring algebraic expressions, you can factor out the greatest common factor
from all the terms.
Consider the expression 12x 1 42. The greatest common
factor of 12x and 42 is 6. Therefore, you can rewrite the
It is important to pay attention to negative numbers. When factoring an expression
that contains a negative leading coefficient, or first term, it is preferred to factor out the
negative sign. A coefficient is the number that is multiplied by a variable in an
© Carnegie Learning
expression as 6(2x 1 7).
algebraic expression.
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How can you
check to make
sure you factored
correctly?
Look at the expression 22x 1 8. You can think about the
greatest common factor as being the coefficient of 22.
22x 1 8 5 (22)x 1 (22)(24)
5 22(x  4)
2. Rewrite each expression by factoring out the greatest common
factor.
a. 7x 1 14
b. 9x 2 27
c. 10y 2 25
d. 8n 1 28
e. 3x2 2 21x
f. 24a2 1 18a
g. 15mn 2 35n
h. 23x 2 27
© Carnegie Learning
i. 26x 1 30
So, when you
factor out a
negative number all the
signs will change.
5.4 Factoring Algebraic Expressions • 273
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Problem 2 Using the Distributive Properties to Simplify
So far, the Distributive Properties have provided ways to rewrite given algebraic
expressions in equivalent forms. You can also use the Distributive Properties to simplify
algebraic expressions.
Consider the algebraic expression 5x 1 11x.
1. What factors do the terms have in common?
2. Rewrite 5x 1 11x using the Distributive Property.
3. Simplify your expression.
The terms 5x and 11x are called like terms, meaning that their variable portions are the
same. When you add 5x and 11x together, you are combining like terms.
4. Simplify each expression by combining like terms.
a. 5ab 1 22ab
b. 32x2 2 44x2
5. Simplify each algebraic expression by combining like terms. If the expression is
a. 6x 1 9x
b. 213y 2 34y
c. 14mn 2 19mn
d. 8mn 2 5m
e. 6x2 1 12x2 2 7x2 f. 6x2 1 12x2 2 7x
g. 23z 2 8z 2 7
© Carnegie Learning
already simplified, state how you know.
h. 5x 1 5y
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Problem 3 More Factoring and Evaluating
1. Factor each expression.
a. 224x 1 16y 5
b. 24.4 2 1.21z 5
c. 227x 2 33 5
d. 22x 2 9y 5
e. 4x 1 (25xy) 2 3x 5
2. Evaluate each expression for the given value. Then factor the expression and evaluate
the factored expression for the given value.
b. 30x 2 140 for x 5 5.63
© Carnegie Learning
1  ​
a. 24x 1 16 for x 5 2​ __
2
c. Which form—simplified or not simplified—did you prefer to evaluate? Why?
5.4 Factoring Algebraic Expressions • 275
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Problem 4 Combining Like Terms and Evaluating
1. Simplify each expression by combining like terms.
a. 30x 2 140 2 23x 5
b. 25(22x 2 13) 2 7x 5
c. 24x 2 5(2x 2 y) 2 3y 5
d. 7.6x 2 3.2(3.1x 2 2.4) 5
( 
)
3 ​ ​4x 2 2​ __
2  ​x 2 1​ __
1 ​   ​5
e. 3​ __
7
3
4
expression and evaluate the simplified expression for the given value.
a. 25x 2 12 1 3x for x 5 2.4
© Carnegie Learning
2. Evaluate each expression for the given value. Then combine the like terms in each
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( 
)
1  ​x 2 1​ __
2 ​   ​for x = 21​ __
1 ​ 
2 ​ ​6x 1 2​ __
b. 22​ __
3
5
4
2
© Carnegie Learning
Why doesn't it
change the answer
when I simplify
first?
Be prepared to share your solutions and methods.
5.4 Factoring Algebraic Expressions • 277
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© Carnegie Learning
278 • Chapter 5 Numerical and Algebraic Expressions and Equations
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Are They the Same
or Different?
Verifying That Expressions
Are Equivalent
Learning Goals
In this lesson, you will:
 Simplify algebraic expressions.
 Verify that algebraic expressions are equivalent by graphing, simplifying, and
evaluating expressions.
B
art and Lisa are competing to see who can get the highest grades. But they
are in different classes. In the first week, Lisa took a quiz and got 9 out of 10
correct for a 90%. Bart took a test and got 70 out of 100 for a 70%. Looks like
Lisa won the first week!
The next week, Lisa took a test and got 35 out of 100 correct for a 35%. Bart took
a quiz and got 2 out of 10 correct for a 20%. Lisa won the second week also!
Over the two weeks, it looks like Lisa was the winner. But look at the total number
of questions and the total each of them got correct: Lisa answered a total of 110
questions and got a total of 34 correct for about a 31%. Bart answered a total of
© Carnegie Learning
110 questions and got a total of 72 correct for a 65%! Is Bart the real winner?
This surprising result is known as Simpson’s Paradox. Can you see how it works?
5.5 Verifying That Expressions Are Equivalent • 279
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Problem 1 Are They Equivalent?
Consider this equation:
24(3.2x 2 5.4) 5 12.8x 1 21.6
Keegan says that to tell if the expressions in an equation
So, equivalent is
not the same as
equal ? What's the
difference?
are equivalent (not just equal), you just need to evaluate each
expression for the same value of x.
1. Evaluate the expression on each side of the equals sign for x 5 2.
2. Are these expressions equivalent? Explain your reasoning.
Jasmine
t two expressions are not
Keegan’s method can prove tha
that they are equivalent.
equivalent, but it can’t prove
3. Explain why Jasmine is correct. Provide an example of two expressions that verify
Kaitlyn
t,
two expressions are equivalen
There is a way to prove that
to try and turn one expression
not just equal: use properties
algebraically into the other.
© Carnegie Learning
your answer.
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4. Reconsider again the equation 24(3.2x 2 5.4) 5 12.8x 1 21.6. Use the distributive
properties to simplify the left side and to factor the right side to try to determine if
these expressions are equivalent.
5. Are the expressions equivalent? Explain your reasoning.
6. Will Kaitlyn’s method always work? Explain your reasoning.
Jason
verify the equivalence of
There is another method to
then
if their graphs are the same,
expressions: graph them, and
ivalent.
the expressions must be equ
7. Graph each expression on your graphics calculator or other graphing technology.
Sketch the graphs.
If you are
using your graphing
calculator, don't
forget to set your
bounds.
© Carnegie Learning
y
36
32
28
24
20
16
12
8
4
–12 –10 –8
–6
–4
–2
–4
–8
–12
–16
–20
–24
–28
–32
–36
2
4
6
8
10
12 x
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8. Are the expressions equivalent? Explain your reasoning.
9. Which method is more efficient for this problem? Explain your reasoning.
Problem 2 Are These Equivalent?
Determine whether the expressions are equivalent using the specified method.
1. 3.1(22.3x 2 8.4) 1 3.5x 5 23.63x 1 (226.04) by evaluating for x 5 1.
© Carnegie Learning
Remember,
evaluating can
only prove that two
expressions are not
equivalent.
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( 
( 
)
)
1  ​  ​5 4​ __
3 ​ 5 22​ __
1  ​​23x 2 2​ ___
1  ​by simplifying each side.
1 ​​  4x 1 __
2. 3​ __
​ 2 ​   ​2 1​ __
5
3
4
2
4 
10
3. Graph each expression using graphing technology to determine if these are
equivalent expressions. Sketch the graphs.
y
36
32
28
24
20
16
12
8
4
© Carnegie Learning
–12 –10 –8
–6
–4
–2
–4
–8
–12
–16
–20
–24
–28
–32
–36
2
4
6
8
10
12 x
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Problem 3 What About These? Are They Equivalent?
For each equation, verify whether the expressions are equivalent using any of these
methods. Identify the method and show your work.
( 
)
( 
)
6 ​   ​1 12​ __
1 ​   ​1 5​ __
2 ​​  23x 1 1​ __
1 ​ 5 26​ __
2 ​ 
1  ​​6x 2 2​ __
1. 3​ __
3
3
3
5
5
3
© Carnegie Learning
2. 4.2(23.2x 2 8.2) 2 7.6x 5 215.02x 1 3(4.1 2 3x)
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( 
)
1 ​ x 2 2​ ___
1  ​  ​5 10x 2 3​( x 1 1 )​1 7.2
3. 22​23​ __
2
10
y
36
32
28
24
20
16
12
8
4
–6
–4
–2
–4
–8
–12
–16
–20
–24
–28
–32
–36
2
4
6
8
10
12 x
© Carnegie Learning
–12 –10 –8
Be prepared to share your solutions and methods.
5.5 Verifying That Expressions Are Equivalent • 285
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© Carnegie Learning
286 • Chapter 5 Numerical and Algebraic Expressions and Equations
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It is time to
justify!
Simplifying Algebraic
Expressions Using Operations
and Their Properties
Learning Goal
In this lesson, you will:
 Simplify algebraic expressions using operations and their properties.
H
ave you ever been asked to give reasons for something that you did? When
did this occur? Were your reasons accepted or rejected? Is it always important to
© Carnegie Learning
have reasons for doing something or believing something?
5.6 Simplifying Algebraic Expressions Using Operations and Their Properties • 287
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Problem 1 Justifying!!
One method for verifying that algebraic expressions are equivalent is to simplify the
expressions into two identical expressions.
For this equation, the left side is simplified completely to show that the
two expressions are equivalent.
( 
)
1 ​ x 1 3
1 ​​  1__
​ 1  ​x 2 __
22​ __
​ 2 ​   ​1 2 5 23​ __
5
2 3
3
Step
( 
Justification
)
1 1
2
__
22​ __  ​​  1__
​    ​x
  2 ​    ​  ​ 1 2 5
5
2 3
2 1 1
(  )(  ) (  )(  )
5 4
5
2
​ 2​ __ ​  ​​ __
​   ​ x  ​1 ​ 2__
​   ​   ​​ 2__
​    ​  ​1 2 5
5
2 3
2
1 1 1
Distributive Property of Multiplication over Addition
10
2​ ___ ​ x 1 1 1 2 5
3
Multiplication
1
  13
23​ __  ​x
3
Addition Yes, they are equivalent.
© Carnegie Learning
Given
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1. Use an operation or a property to justify each step and indicate if the expressions
are equivalent.
a. 24(23x 2 8) 2 4x 1 8 5 28x 1 40
Step
Justification
24(23x 2 8) 2 4x 1 8 5
12x 1 32 1 (24x) 1 8 5
12x 1 (24x) 1 32 1 8 5
28x 1 40 5
b. 22.1(23.2x 2 4) 1 1.2(2x 2 5) 5 9.16x 1 3.4
Step
Justification
22.1(23.2x 2 4) 1 1.2(2x 2 5) 5
6.72x 1 8.4 1 2.4x 1 (26) 5
© Carnegie Learning
6.72x 1 2.4x 1 8.4 1 (26) 5
9.16x 1 2.4 5
5.6 Simplifying Algebraic Expressions Using Operations and Their Properties • 289
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(  )
13
24x 2
9 
23x 1
7 
c.​ ________
 
 
​   ​  ​
 ​
1 ​ ________
 ​
5 23x 1 ​2___
6
2
3
Step
Justification
1
7 
2
9 
________
________
​ 24x
 23x
 
 ​ 1 ​ 
 ​ 5
2
3
(  )
9
7  ​ 5
____
____
 
 
2​ __  ​  ​ 1 ​ 23x
 
  __
​ 24x
 ​ 1 ​
 ​ 1 ​ 
2
2
3
3
(  )
9
7  ​ 5
22x 1 ​ 2​ __  ​  ​ 1 (2x) 1 ​ __
2
3
(  )
9
7  ​ 5
22x 1 (2x) 1 ​ 2​ __  ​  ​ 1 ​ __
2
3
(  )
27
14 ​ 5
 
23x 1 ​ 2​ ___ ​  ​ 1 ​ ___
6
6
(  )
© Carnegie Learning
13
23x 1 ​ 2 ​ ___ ​  ​ 5
6
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2. For each equation, simplify the left side completely using the given operation or
property that justifies each step and indicate if the expressions are equivalent.
26
___
6x 2 ​
7 
 5 2​   ​ x 1 __
​ 7 ​ 
a. 24x 2 ​ _______
5
5
5
Step
Justification
Given
Distributive Property of Division over Subtraction
Division
Addition of Like Terms Yes, they are equivalent.
b. 24x 2 3(3x 2 6) 1 8(2.5x 1 3.5) 5 7x 1 46
Step
Justification
Given
Distributive Property of Multiplication over Addition
© Carnegie Learning
Addition
Commutative Property of Addition
Addition of Like Terms Yes, they are equivalent.
5.6 Simplifying Algebraic Expressions Using Operations and Their Properties • 291
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Problem 2 Simplify and Justify!
For each equation, simplify the left side completely to determine if the two expressions
are equivalent. Use an operation or a property to justify each step and indicate if the
expressions are equivalent.
1. 24x 1 3(27x 1 3) 2 2(23x 1 4) 5 219x 1 1
Step
Justification
I think it's
easier to simplify
first and then come
up with the reasons
when I'm done.
What do you think?
2. 25(x 1 4.3) 2 5(x 1 4.3) 5 210x 2 43
Justification
© Carnegie Learning
Step
292 • Chapter 5 Numerical and Algebraic Expressions and Equations
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For each equation, simplify the left side and the right side completely to determine if the
two expressions are equivalent. Use an operation or a property to justify each step and
indicate if the expressions are equivalent.
3. 23(2x 1 17) 1 7(2x 1 30) 28x 5 26x 1 3(2x 1 50) 1 9
Left side
Step
Justification
Right side
Justification
© Carnegie Learning
Step
5.6 Simplifying Algebraic Expressions Using Operations and Their Properties • 293
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1  ​(4x 1 6) 1 1​ __
1 ​ (26x 2 3) 2 4x 5 25​ __
1 ​ x 2 13 1 2​ __
1 ​ (26x 1 1) 2 5​ __
1 ​ x 2 14​ __
1 ​ 
4. 23​ __
2
3
2
2
2
2
Left side
Step
Justification
Right side
Justification
© Carnegie Learning
Step
Be prepared to share your solutions and methods.
294 • Chapter 5 Numerical and Algebraic Expressions and Equations
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Chapter 5 Summary
Key Terms
Properties









 Distributive Property of Multiplication
variable (5.2)
over Addition (5.3)
algebraic expression (5.2)
 Distributive Property of Multiplication
evaluate an algebraic expression (5.2)
over Subtraction (5.3)
factor (5.4)
 Distributive Property of Division over
common factor (5.4)
Addition (5.3)
greatest common factor (GCF) (5.4)
 Distributive Property of Division over
coefficient (5.4)
Subtraction (5.3)
like terms (5.4)
combining like terms (5.4)
Modeling Integer Addition and Integer Subtraction on a Number Line
A number line can be used to model integer addition. When adding a positive integer,
move to the right on the number line. When adding a negative integer, move to the left on
the number line. Subtraction means to move in the opposite direction on the number line.
When subtracting a positive integer, move to the left on the number line. When subtracting
a negative integer, move to the right on the number line.
Examples
28 1 3 5 25
–8
3
© Carnegie Learning
–15
–10
–5
0
5
10
15
0
5
10
15
210 2 (26) 5 24
(–10)
–(–6)
–15
–10
–5
Chapter 5 Summary • 295
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Adding and Subtracting Integers
When adding two integers with the same sign, add the integers and keep the sign. When
adding integers with opposite signs, subtract the integers and keep the sign of the integer
with the greater absolute value. Because subtraction is the inverse of addition, it is the
same as adding the opposite number.
Examples
29 1 (212) 5 2(9 1 12) 5 221
7 1 (215) 5 28
27 2 19 5 27 1 (219) 5 226
12 2 21 5 12 1 (221) 5 29
Adding and Subtracting Rational Numbers
When adding and subtracting positive and negative rational numbers, follow the same
rules as when adding and subtracting integers. When adding rational numbers with the
same sign, add the numbers and keep the sign. When the rational
sign of the number with the greater absolute value. Because
subtraction is the inverse of addition, it is the same as
adding the opposite number.
Examples
28.54 1 (23.4) 5 2(8.54 1 3.4) 5 211.94
3 ​ ) 5 2(10​ __
3 ​ 2 5​ __
1  ​1 (210​ __
2 ​ ) 5 25​ __
1 ​ 
5​ __
2
4
4
4
4
5 ​ ) 5 27​ __
5 ​ 5 3​ __
3 ​ 
1  ​2 (210​ __
2 ​ 1 10​ __
27​ __
4
8
8
8
8
28.5 2 3.4 5 28.5 1 (23.4)5 211.9
Having trouble
remembering something
new? Your brain learns
by association so create a
mnemonic, a song, or a story
about the information
and it will be easier to
remember!
© Carnegie Learning
numbers have different signs, subtract the numbers and keep the
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Multiplying Integers
When multiplying integers, multiplication can be thought of as repeated addition.
Example
Consider the expression 3 3 (23). As repeated addition, it means (23) 1 (23) 1 (23) 5 29.
The expression 3 3 (23) can be thought of as three groups of (23). =
(-3)
-15
-10 -9
(-3)
-5
(-3)
0
5
10
15
Determining the Sign of a Product or Quotient
The sign of a product or quotient of two integers depends on the signs of the two integers
© Carnegie Learning
being multiplied or divided. The product or quotient will be positive when both integers
have the same sign. The product or quotient will be negative when one integer is positive
and the other is negative.
Examples
3 3 2 5 6
64253
(23) 3 (22) 5 6
(26) 4 (22) 5 3
3 3 (22) 5 26
6 4 (22) 5 23
(23) 3 2 5 26
(26) 4 2 5 23
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Picturing the Rational Numbers
The Venn diagram represents relationships between natural numbers, whole numbers,
integers, and rational numbers. From the Venn diagram, you can see that natural numbers
are a subset of whole numbers. Natural numbers and whole numbers are subsets of
integers. Natural numbers, whole numbers, and integers are subsets of rational numbers.
The set of rational numbers is all numbers that can be written as __
​ a ​, where a and b are
b
integers and b fi 0. Rational numbers include mixed numbers, fractions, and their
equivalent decimals.
Rational
Integers
Whole
Natural
Multiplying and Dividing Rational Numbers
The rules used to determine the sign of a product or quotient of two integers also apply
when multiplying and dividing rational numbers.
Example
The product or quotient of each are shown following the rules for determining the sign
for each.
4
1 ​ 5 ___
1 ​ 
1 ​ 3 5​ __
​ 13 ​ 3 ___
​ 16 ​ 5 ___
​ 13 ​ 3 __
​ 4 ​ 5 ___
​ 52 ​ 5 17​ __
3​ __
4
3
4
3
1
3
3
3
1
9
2
1
5
9 2
18
27 ___
27
3 ​ 4 1​ __
3 ​ 
7 ​ 5 2​ ___
26​ __
 ​ 4 ​ 15 ​ 5 2​ ___ ​ 3 ___
​  8  ​ 5 2​ __ ​ 3 __
​   ​ 5 2​ ___ ​ 5 23​ __
5
4
4
1 5
5
4
8
8
15
258.75 4 26.25 5 9.40
© Carnegie Learning
12.1 3 25.6 5 267.76
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Writing Algebraic Expressions
When a mathematical process is repeated over and over, a mathematical phrase, called
an algebraic expression, can be used to represent the situation. An algebraic expression
is a mathematical phrase involving at least one variable and sometimes numbers and
operation symbols.
Example
The algebraic expression 2.49p represents the cost of p pounds of apples.
One pound of apples costs 2.49(1), or $2.49. Two pounds of apples costs 2.49(2),
or $4.98.
Evaluating Algebraic Expressions
To evaluate an algebraic expression, replace each variable in the expression with a
number or numerical expression and then perform all possible mathematical operations.
Example
The expression 2x 2 7 has been evaluated for these values of x: 9, 2, 23, and 4.5.
2(9) 2 7 5 18 2 7
5 11
2(2) 2 7 5 4 2 7
2(23) 2 7 5 26 2 7
5 23
2(4.5) 2 7 5 9 2 7
5 213
52
Using the Distributive Property of Multiplication over
Addition to Simplify Numerical Expressions
The Distributive Property of Multiplication over Addition states that if a, b, and c are any
real numbers, then a • (b 1 c) 5 a • b 1 a • c.
© Carnegie Learning
Example
A model is drawn and an expression written to show how the Distributive Property of
Multiplication over Addition can be used to solve a multiplication problem.
6(820) 800 20
6 4800
120
6(800 1 20)
5 6(800) 1 6(20)
5 4800 1 120
5 4920
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Using the Distributive Properties to Simplify and
Evaluate Algebraic Expressions
Including the Distributive Property of Multiplication over Addition, there are a total of four
different forms of the Distributive Property.
Another Distributive Property is the Distributive Property of Multiplication over Subtraction,
which states that if a, b, and c are any real numbers, then a • (b 2 c) 5 a • b 2 a • c.
The Distributive Property of Division over Addition states that if a, b, and c are real
a ​1 __
b 
​5 __
​ c
​ b
numbers and c fi 0, then ______
​ a 1
c   
c ​.
The Distributive Property of Division over Subtraction states that if a, b, and c are real
a ​2 __
a 2  b 
​5 __
​ c
​ b
numbers and c fi 0, then ​ ______
c  
c ​.
Example
The Distributive Properties have been used to simplify the algebraic expression. The
simplified expression is then evaluated for x 5 2.
4(6x 2 7) 1 10 ______________
1
10  
______________
    
  
 ​
5 ​ 24x 2 28
 ​
​ 
3
3
18 
5 _________
​ 24x 2 ​
 
3
8x 2 6 5 8(2) 2 6
18
24x
 2 ​ ___ ​ 
 ​ 
5 ​ ____
3
3
5 8x 2 6
5 16 2 6
5 10
Using the Distributive Properties to Factor Expressions
The Distributive Properties can be used in reverse to rewrite an expression as a product
of factors. When factoring expressions, it is important to factor out the greatest common
factor from all the terms. The greatest common factor (GCF) is the largest factor that two
Example
The expression has been rewritten by factoring out the greatest common factor.
24x3 1 3x2 2 9x 5 3x(8x2) 1 3x(x) 2 3x(3)
5 3x(8x2 1 x 2 3)
© Carnegie Learning
or more numbers or terms have in common.
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Combining Like Terms to Simplify Expressions
Like terms are terms whose variable portions are the same. When you add like terms
together, you are combining like terms. You can combine like terms to simplify algebraic
expressions to make them easier to evaluate.
Example
Like terms have been combined to simplify the algebraic expression. The simplified
expression is then evaluated for x 5 5.
7x 2 2(3x 2 4) 5 7x 2 6x 1 8
5x18
x185518
5 13
Determining If Expressions Are Equivalent by Evaluating
Evaluate the expression on each side of the equal sign for the same value of x. If the
results are the same, then the expressions are equivalent.
Example
The expressions are equivalent.
(x 1 12) 1 (4x 2 9) 5 5x 1 3 for x 5 1
(1 1 12) 1 (4 ? 1 2 9) 0 5 ? 1 1 3
13 1 (25) 0 5 1 3
8 5 8
Determining If Expressions Are Equivalent by Simplifying
The Distributive Properties and factoring can be used to determine if the expressions on
© Carnegie Learning
each side of the equal sign are equivalent.
Example
The expressions are equivalent.
( 
)
1
​ 2 ​   ​5 2​ __ ​ (45x 1 6)
5 ​23x 2 __
3
5
215x 2 2 5 215x 2 2
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Determining If Two Expressions Are Equivalent by Graphing
Expressions can be graphed to determine if the expressions are equivalent. If the graph of
each expression is the same, then the expressions are equal.
Example
3(x 1 3) 2 x 5 3x 1 3 is not true because the graph of each expression is not the same.
y
14
3(x
+
16
3)x
18
12
3x+3
10
8
6
4
2
2
4
6
8
10
12
14
16
18
x
Simplifying Algebraic Expressions Using Operations and
Their Properties
Simplify each side completely to determine if the two expressions are equivalent. Use an
operation or a property to justify each step and indicate if the expressions are equivalent.
Example
Step
Justification
23(22x 1(29)) 2 3x 2 7 5
Given
6x 1 27 1 (23x) 1 (27) 5
Distributive Property of Multiplication over Addition
6x 1 (23x) 1 27 1 (27) 5
Commutative Property of Addition
3x 1 20 5
Addition of Like Terms Yes, they are equivalent.
© Carnegie Learning
23(22x 2 9) 2 3x 2 7 5 3x 1 20
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