Section P.3
P.3
Algebraic Expressions
25
Algebraic Expressions
What you should learn:
• Identify the terms and coefficients of algebraic expressions
• Identify the properties of algebra
• Apply the properties of exponents to simplify algebraic
expressions
• Simplify algebraic expressions
by combining like terms and
removing symbols of grouping
• Evaluate algebraic expressions
Why you should learn it:
Algebraic expressions can help
you construct tables of values.
For instance, in Example 14 on
page 33, you can determine the
hourly wages of construction
workers using an expression and
a table of values.
Algebraic Expressions
A basic characteristic of algebra is the use of letters (or combinations of letters)
to represent numbers. The letters used to represent the numbers are variables, and
combinations of letters and numbers are algebraic expressions. Here are a few
examples.
3x,
x 2,
x
, 2x 3y
x2 1
Definition of Algebraic Expression
A collection of letters (called variables) and real numbers (called constants)
combined using the operations of addition, subtraction, multiplication, division, and exponentiation is called an algebraic expression.
The terms of an algebraic expression are those parts that are separated by
addition. For example, the algebraic expression x2 3x 6 has three terms: x2,
3x, and 6. Note that 3x is a term, rather than 3x, because
x2 3x 6 x2 3x 6.
Study Tip
It is important to understand
the difference between a term
and a factor. Terms are
separated by addition whereas
factors are separated by
multiplication. For instance,
the expression 4xx 2 has
three factors: 4, x, and
x 2.
Think of subtraction as a form of addition.
The terms x2 and 3x are called the variable terms of the expression, and 6 is
called the constant term of the expression. The numerical factor of a variable
term is called the coefficient of the variable term. For instance, the coefficient of
the variable term 3x is 3, and the coefficient of the variable term x2 is 1. (The
constant term of an expression is also considered to be a coefficient.)
Example 1 Identifying Terms and Coefficients
Identify the terms and coefficients in each algebraic expression.
(a) 5x 1
3
(b) 4y 6x 9
(c) x2y Solution
Terms
(a) 5x, 1
3
Coefficients
5, 1
3
(b) 4y, 6x, 9
4, 6, 9
1
(c) x2y, , 3y
x
1, 1, 3
Now try Exercise 3.
1
3y
x
26
Chapter P
Prerequisites
Properties of Algebra
Historical Note
The French mathematician François
Viète (1540–1603) was the first to
use letters to represent numbers.
He used vowels to represent
unknown quantities and consonants to represent known
quantities.
The properties of real numbers (see Section P.2) can be used to rewrite algebraic
expressions. The following list is similar to those given in Section P.2, except that
the examples involve algebraic expressions. In other words, the properties are true
for variables and algebraic expressions as well as for real numbers.
Properties of Algebra
Let a, b, and c be real numbers, variables, or algebraic expressions.
Property
Commutative Property of Addition
abba
Example
5x x2 x2 5x
Commutative Property of Multiplication
3 xx3 x33 x
ab ba
Associative Property of Addition
a b c a b c
x 6 3x2 x 6 3x2
Associative Property of Multiplication
abc abc
5x 4y6 5x4y 6
Distributive Properties
ab c ab ac
a bc ac bc
2x4 3x 2x 4 2x 3x
y 6y y y 6 y
Additive Identity Property
a00aa
4y2 0 0 4y2 4y2
Multiplicative Identity Property
a
11aa
5x31 15x3 5x3
Additive Inverse Property
a a 0
4x2 4x2 0
Multiplicative Inverse Property
a
1
a 1, a 0
x2 1
x
2
1
1
1
Because subtraction is defined as “adding the opposite,” the Distributive
Property is also true for subtraction. For instance, the “subtraction form” of
ab c ab ac is
ab c ab c
ab ac
ab ac.
Section P.3
Algebraic Expressions
27
In addition to these properties, the properties of equality, zero, and negation
given in Section P.2 are also valid for algebraic expressions. The next example
illustrates the use of a variety of these properties.
Example 2 Identifying the Properties of Algebra
Identify the property of algebra illustrated in each statement.
(a) 5x23 35x2
(b) 3x2 x 3x2 x 0
(c) 3x 3y2 3x y2
(d) 5 x2 4x2 5 x2 4x2
(e) 5x 1
1, x 0
5x
(f ) y 63 y 6y y 63 y
Solution
(a) This statement illustrates the Commutative Property of Multiplication.
In other words, you obtain the same result whether you multiply 5x2 by 3, or
3 by 5x2.
(b) This statement illustrates the Additive Inverse Property. In terms of subtraction, this property simply states that when any expression is subtracted from
itself the result is zero.
(c) This statement illustrates the Distributive Property. In other words, multiplication is distributed over addition.
(d) This statement illustrates the Associative Property of Addition. In other
words, to form the sum
5 x2 4x2
it does not matter whether 5 and x2 are added first or x2 and 4x2 are added first.
(e) This statement illustrates the Multiplicative Inverse Property. Note that it is
important that x be a nonzero number. If x were zero, the reciprocal of x
would be undefined.
(f ) This statement illustrates the Distributive Property in reverse order.
ab ac ab c
Distributive Property
y 63 y 6y y 63 y
Note in this case that a y 6, b 3, and c y.
Now try Exercise 13.
EXPLORATION
Discovering Properties of Exponents Write each of the following as a single
power of 2. Explain how you obtained your answer. Then generalize your procedure
by completing the statement “When you multiply exponential expressions that have
the same base, you. . . .”
a. 22 23
b. 24 21
c. 25 22
d. 23 24
e. 21 25
28
Chapter P
Prerequisites
Properties of Exponents
Historical Note
Originally, Arabian mathematicians
used their words for colors to represent quantities (cosa, censa, cubo).
These words were eventually abbreviated to co, ce, cu. René Descartes
(1596–1650) simplified this even
further by introducing the symbols
x, x 2, and x 3.
Just as multiplication by a positive integer can be described as repeated addition,
repeated multiplication can be written in what is called exponential form (see
Section P.1). Let n be a positive integer and let a be a real number. Then the product of n factors of a is given by
an a a a . . . a.
a is the base and n is the exponent.
n factors
When multiplying two exponential expressions that have the same base, you
add exponents. To see why this is true, consider the product a3 a2. Because the
first expression represents a a a and the second represents a a, the product
of the two expressions represents a a a a a, as follows.
a3
a2 a a a a a a a a a a a32 a5
3 factors
Study Tip
The first and second properties
of exponents can be extended
to products involving three or
more factors. For example,
am
and
an
ak
amnk
abcm ambmcm.
2 factors
5 factors
Properties of Exponents
Let a and b represent real numbers, variables, or algebraic expressions, and
let m and n be positive integers.
Property
Example
1. aman amn
x5x 4 x54 x9
2. abm ambm
2x3 23x3 8x3
3. amn amn
x23 x2 3 x6
4.
am
amn, m > n, a 0
an
x6
x62 x4, x 0
x2
5.
ab
2x m
am
, b0
bm
3
x3
x3
23
8
Example 3 Illustrating the Properties of Exponents
(a) To multiply exponential expressions that have the same base, add exponents.
x2
x4 x x x x x x x x x x x x x24 x6
2 factors
4 factors
6 factors
(b) To raise the product of two factors to the same power, raise each factor to the
power and multiply the results.
3x3 3x 3x 3x 3 3 3 x x x 33 x3 27x3
3 factors
3 factors
3 factors
(c) To raise an exponential expression to a power, multiply exponents.
x32 x x x x x x x x x x x x x3 2 x6
3 factors
3 factors
Now try Exercise 29.
6 factors
Section P.3
Study Tip
In the expression x 5, the
coefficient of x is understood to
be 1. Similarly, the power (or
exponent) of x is also understood to be 1. So
x4
29
Example 4 Illustrating the Properties of Exponents
(a) To divide exponential expressions that have the same base, subtract
exponents.
5 factors
x5 x x x x x
x52 x3
x2
xx
x x2 x412 x7.
Note such occurrences in
Examples 5(a) and 6(b).
Algebraic Expressions
2 factors
(b) To raise the quotient of two expressions to the same power, raise each expression to the power and divide the results.
3 factors
x
3
3
x
3
x
x
xxx
x3
x3
3
3 3 3 27
333
3 factors
Now try Exercise 35.
Example 5 Applying Properties of Exponents
Use the properties of exponents to simplify each expression.
(a) x2 y 43x
(b) 2 y23
(c) 2y 23
(d) 3x25x3
Solution
(a) x2 y 43x 3x 2
x y 4 3x21y 4 3x3y 4
(b) 2 y23 2 y2 3 2y6
(c) 2y23 23 y23 8y2 3 8y6
(d) 3x25x3 353x2
x3 3125x23 375x5
Now try Exercise 63.
Example 6 Applying Properties of Exponents
Use the properties of exponents to simplify each expression.
(a)
14a5b3
7a2b2
2yx 2 3
(b)
(c)
x n y 3n
x2y4
(d)
2a2b32
a3b2
Solution
(a)
14a5b3
2a52b32 2a3b
7a2b2
(b)
2yx (c)
x n y 3n
xn2 y 3n4
x 2 y4
(d)
2a2b32 22a2 2b3 2 4a4b6
3 2 4a43b62 4ab4
a3b2
a3b2
ab
2 3
x23
x2 3
x6
3 3 3
3
2y
2y
8y
Now try Exercise 65.
30
Chapter P
Prerequisites
Simplifying Algebraic Expressions
One common use of the basic properties of algebra is to rewrite an algebraic
expression in a simpler form. To simplify an algebraic expression generally
means to remove symbols of grouping such as parentheses or brackets and combine like terms.
Two or more terms of an algebraic expression can be combined only if they
are like terms. In an algebraic expression, two terms are said to be like terms if
they are both constant terms or if they have the same variable factor(s). For
example, the terms 4x and 2x are like terms because they have the same variable factor, x. Similarly, 2x2y, x2 y, and 12x 2 y are like terms because they have
the same variable factor, x2y. Note that 4x2y and x 2 y 2 are not like terms because
their variable factors are different.
To combine like terms in an algebraic expression, simply add their respective
coefficients and attach the common variable factor. This is actually an application
of the Distributive Property, as shown in Example 7.
Example 7 Combining Like Terms
Study Tip
Simplify each expression by combining like terms.
As you gain experience with the
properties of algebra, you may
want to combine some of the
steps in your work. For instance,
you might feel comfortable listing only the following steps to
solve Example 7(c).
5x 3y 4x
5x 4x 3y
x 3y
Group like
terms.
Combine
like terms.
(a) 2x 3x 4
(b) 3 5 2y 7y
(c) 5x 3y 4x
Solution
(a) 2x 3x 4 2 3x 4
Distributive Property
5x 4
Simplest form
(b) 3 5 2y 7y 3 5 2 7y
2 5y
(c) 5x 3y 4x 5x 4x 3y
5x 4x 3y
5 4x 3y
x 3y
Distributive Property
Simplest form
Commutative Property
Associative Property
Distributive Property
Simplest form
Now try Exercise 93.
Example 8 Combining Like Terms
(a) 7x 7y 4x y 7x 4x 7y y
3x 6y
(b) 2x2 3x 5x2 x 2x2 5x2 3x x
3x2
2x
(c) 3xy2 4x2y2 2xy2 xy2
3xy 2 2xy 2 4x 2y 2 x 2 y 2
5xy2 3x2 y2
Now try Exercise 99.
Group like terms.
Combine like terms.
Group like terms.
Combine like terms.
Section P.3
Algebraic Expressions
31
A set of parentheses preceded by a minus sign can be removed by changing
the sign of each term inside the parentheses. For instance,
8x 5x 4 8x 5x 4 3x 4.
A set of parentheses preceded by a plus sign can be removed without changing
the signs of the terms inside the parentheses. For instance,
8x 5x 4 8x 5x 4 13x 4.
Example 9 Removing Symbols of Grouping
Simplify 3x 5 2x 7.
Solution
3x 5 2x 7 3x 15 2x 7
3x 2x 15 7
x8
Distributive Property
Group like terms.
Combine like terms.
Now try Exercise 105.
Example 10 Geometry: Perimeter and Area of a Region
Write and simplify an expression for the perimeter and for the area of each region
shown in Figure P.10.
x
Solution
(a) Perimeter of a rectangle 2
3x − 5
(a)
2x + 4
2x
x+5
length 2 width
23x 5 2x
Substitute.
6x 10 2x
Distributive Property
6x 2x 10
Group like terms.
8x 10
Combine like terms.
Area of a rectangle length width
(b)
3x 5x
Substitute.
Figure P.10
Distributive Property
3x2
5x
(b) Perimeter of a triangle sum of the three sides
Area of a triangle 1
2
2x 2x 4 x 5
Substitute.
2x 2x x 4 5
Group like terms.
5x 9
Combine like terms.
base height
12x 52x
Substitute.
xx 5
Commutative Property
of Multiplication
Multiply.
x2 5x
Distributive Property
1
2 2xx
5
Now try Exercise 137.
When removing symbols of grouping, combine like terms within the
innermost symbols of grouping first, as shown in the next example.
32
Chapter P
Prerequisites
Example 11 Removing Symbols of Grouping
Simplify each expression by combining like terms.
(a) 5x 2x3 2x 7
(b) 3x5x 4 2x5
Solution
(a) 5x 2x3 2x 7 5x 2x3 2x 14
5x 4x2 22x
Remove parentheses.
Combine like
terms in brackets.
Remove brackets.
4x2 27x
Combine like terms.
5x 2x2x 11
(b) 3x5x4 2x5 15x5 25x5
Properties of exponents
15x5 32x5
Evaluate power.
Combine like terms.
17x5
Now try Exercise 113.
Evaluating Algebraic Expressions
To evaluate an algebraic expression, substitute numerical values for each of the
variables in the expression. Here are some examples.
Expression
Value of Variable
Substitute
Value of Expression
3x 2
x2
32 2
628
x 1
41 21 1 4 2 1 1
x 2
222 4
4x2
2x 1
2xx 4
2
222 8
Example 12 Evaluating Algebraic Expressions
Evaluate each algebraic expression when x 2 and y 5.
(a) 2y 3x
(b) 5 x2
(c) 5 x2
Solution
(a) When x 2 and y 5, the expression 2y 3x has a value of
2y 3x 25 32
Substitute for x and y.
10 6
Simplify.
16.
Simplify.
(b) When x 2, the expression 5 x2 has a value of
5 x2 5 22
Substitute for x.
54
Simplify.
9.
Simplify.
(c) When x 2, the expression 5 x2 has a value of
5 x2 5 22
Substitute for x.
54
Simplify.
1.
Simplify.
Now try Exercise 121.
Section P.3
Algebraic Expressions
33
Example 13 Evaluating Algebraic Expressions
Evaluate each algebraic expression when x 2 and y 1.
(a) x2 2xy y2
(b) y x
(c)
2xy
5x y
Solution
(a) When x 2 and y 1, the expression x2 2xy y2 has a value of
x2 2xy y2 22 221 12
Substitute for x and y.
4 4 1 9.
Technology
Most graphing utilities can be
used to evaluate an algebraic
expression for several values of
x and display the results in a
table. For instance, to evaluate
2x2 3x 2 when x is 0, 1, 2,
3, 4, 5, and 6, you can use the
steps below.
3. Set the table step or table
increment to 1.
4. Display the table.
The results are shown below.
Consult the user’s guide for
your graphing utility for specific
instructions.
(b) When x 2 and y 1, the expression y x has a value of
y x 1 2
3 3.
Substitute for x and y.
Simplify.
(c) When x 2 and y 1, the expression 2xy5x y has a value of
2xy
221
5x y 52 1
1. Enter the expression into the
graphing utility.
2. Using the table feature, set
the minimum value of the
table to 0.
Simplify.
Substitute for x and y.
4
4
.
10 1
9
Simplify.
Now try Exercise 131.
Example 14 Using a Mathematical Model
From 1997 to 2004, the average hourly wage for construction workers in the
United States can be modeled by the expression
0.0231t2 1.011t 9.66,
7 ≤ t ≤ 14
where t represents the year, with t 7 corresponding to 1997. Create a table that
shows the average hourly wages for these years. (Source: U.S. Bureau of
Labor Statistics)
Solution
To create a table of values that shows the average hourly wages for the years 1997
to 2004, evaluate the expression
0.0231t2 1.011t 9.66
for each integer value of t from t 7 to t 14.
Year
1997
1998
1999
2000
t
7
8
9
10
Hourly wage
$15.61
$16.27
$16.89
$17.46
Year
2001
2002
2003
2004
t
11
12
13
14
Hourly wage
$17.99
$18.47
$18.90
$19.29
Now try Exercise 145.
34
Chapter P
P.3
Prerequisites
Exercises
VOCABULARY CHECK: Fill in the blanks.
1. A collection of ________ and ________ combined using the operations of addition, subtraction,
multiplication, division, and exponentiation is called an ________.
2. The ________ of an algebraic expression are those parts that are separated by addition.
3. The numerical factor of a variable term is called the ________ of the variable term.
4. Repeated multiplication can be written in ________.
5. You ________ an algebraic expression by combining like terms and removing symbols of grouping.
6. To ________ an algebraic expression, substitute numerical values for each of the variables in the expression.
In Exercises 1–8, identify the terms of the algebraic
expression.
1. 10x 5
3. 3y 2y 8
4.
25z3
5. 4x2 3y2 5x 2y
6. 14u2 25uv 3v2
7. x2 2.5x 1
x
8.
(b) Commutative Property of Addition
6x 6 4.8z
2
25. (a) Commutative Property of Multiplication
6xy 3
4
6
t2
t
In Exercises 9–12, identify the coefficient of the term.
9. 5y3
6x 6 48
2.
16t2
2
24. (a) Distributive Property
(b) Associative Property of Multiplication
6xy 26. (a) Additive Identity Property
3ab 0 10. 4x6
3
11. 4t2
12. 8.4x
In Exercises 13–22, identify the property of algebra that is
illustrated by the statement.
(b) Commutative Property of Addition
3ab 0 27. (a) Additive Inverse Property
4t2 4t2 13. 4 3x 3x 4
(b) Commutative Property of Addition
14. 10 x y 10 x y
15. 52x 5
2x
16. x 23 3x 2
1
17. x 5 1, x 5
x5
4t2 4t2 28. (a) Associative Property of Addition
3 6 9 (b) Additive Inverse Property
9 9 18. x2 1 x2 1 0
19. 5 y3 3 5y3 5
20.
10x3y
0
3
10x3y
21. 16t 4 1 16t 4
In Exercises 29–38, write the expression as a repeated
multiplication.
x4
22. 32u2 3u 32u2 96u
29. x3
In Exercises 23–28, use the property to rewrite the
expression.
33. 2x
23. (a) Distributive Property
5x 6 (b) Commutative Property of Multiplication
5x 6 31. 2y3
5y 6
37. x
4
z5
32. 4t4
3
35.
30. z2
34. 5y4
2z 3
38. t
3
36.
4
5
Section P.3
In Exercises 39–44, write the expression using exponential
notation.
89.
r ns m
rs3
39. 5x5x5x5x
90.
xx yy 41. y y y y y y y
4 3
In Exercises 91–102, simplify the expression by combining
like terms.
42. x x x y y y
43. zzzzzzz
91. 3x 4x
44. 9t9t9t9t9t9t
92. 2x2 4x2
In Exercises 45–90, use the properties of exponents to
simplify the expression.
x2
46. y3
y4
93. 9y 5y 4y
94. 8y 7y y
95. 3x 2y 5x 20y
47. a24
48. x53
96. 2a 13b 7a b
x7
49. 3
x
z8
50. 5
z
98. 5y3 3y 6y2 8y3 y 4
51.
a2
b
53. 33y4
2
52.
y2
x3
y2
54. 62x3
97. 8z2 32z 52z2 10
99. 2uv 5u2v2 uv uv2
3
100. 3m2n2 4mn n5m 2mn2
x5
55. 4x2
56. 4x3
57. 58. 59. 60. 61.
3
x x
2xy3x2 y3
37x5
33x3
2xy5
6xy3
5y2y4
5z45z4
2a22a2
2x4y2
2x3y
6a3b3
3ab2
x2y3
xy2
2x 4 2
5y
n1
x
xn
x n4
x n x3
62.
5z2 3
a4b4 4
63.
65.
67.
69.
71.
73.
75.
77.
79.
81.
83.
85.
87.
35
2n m 2
40. 2y2y2y2y2y
45. x5
Algebraic Expressions
5z3 2
2
z z3
5a2b32ab4
24y5
22 y3
43ab6
4ab2
3y32y2
6n3n2
2a28a
3a5b7
a33b2
3c5d 32
2cd 3
x22z8
x3z7
3a3 3
2b5
m3
a
a3
a3k
y m y2
y2z5 3
64.
66.
68.
70.
72.
74.
76.
78.
80.
82.
84.
86.
88.
101. 5ab2 2ab 4ab
102. 3xy xy 8
In Exercises 103–120, simplify the algebraic expression.
103. 10x 3 2x 5
104. 3x2 1 x2 6
105. 4x 1 2x 2
106. 7y2 5 8y2 4
107. 3z2 2z 4 z2 z 2
108. t2 10t 3 5t 8
109. 3 y2 3y 1 2 y 5
110. 5a 6 4a2 2a 1
111. 45 3x2 10
112. 25x2 x3 5
113. 23b 5 b2 b 3
114. 4t 1 t2 2t 5
115. y2 y 1 y y2 1
116. z2z4 z2 4z3z 1
117. xxy2 y 2xyxy 1
118. 2abb2 3 abb2 2
119. 2a3a23 120. 5y3 9a8
3a
4y5
7yy2
2y2
36
Chapter P
Prerequisites
In Exercises 121–136, evaluate the algebraic expression for
each specified value of the variable(s). If not possible, state
the reason.
139.
140.
1
b+
1
2
b+1
(a) x 5
(b) x 23
122. 32x 2
(a) x 6
(b) x 3
123. 10 x
(a) x 3
(b) x 3
124. x 6
(a) x 4
(b) x 5
(a) x 1
(b) x 13
(a) x 3
1
2
x7
126. 2x2 5x 3
Expression
(b) x (a) x 0
3
128. 5 x
(a) x 0
129. 3x 2y
(a) x 1,
130. 4x y
(b) x 6
y5
y 9
(b) x 2,
y 5
(a) x 1,
y2
(a) x 2,
y 3
y 1
(b) x 3,
133. y x
(a) x 2,
y 2
y5
(b) x 2,
134. x y
135. rt
y 2
(a) x 0,
y 10
(b) x 4,
y4
(a) r 40,
t 514
(b) r 35,
t4
(a) P $5000,
136. Prt
(b) P $750,
r 0.085,
r 0.07,
Geometry
In Exercises 137–140, (a) write and simplify an
expression for the perimeter of the region and (b) write and
simplify an expression for the area of the region.
138.
3
+
2h
x
2
2x
h+3
x
4
x2
143. h 12
144. y 20
h
2y
5
5
4
+4
h + 10
1y
2
Using a Model
In Exercises 145 and 146, use the model
which approximates the per capita consumption (in pounds
per person) of yogurt in the United States from 1996
through 2003 (see figure). In the model, t represents the
year, with t 6 corresponding to 1996. (Source:
USDA/Economic Research Service)
y
t3
2
3x
0.058t2 0.77t 8.4, 6 ≤ t ≤ 13
t 10
137.
142. x 3
b
y0
(b) x 6,
132. x2 xy y2
141. b 15
(b) x 3
(a) x 2,
131. x2 3xy y2
In Exercises 141–144, write an expression for
the area of the figure. Then evaluate the expression for the
given value of the variable.
b−3
(b) x 6,
Geometry
Values
x
127. 2
x 1
5
2h
Consumption
(in pounds per person)
125.
3x2
t
Values
121. 5 3x
t
t
b
Expression
1
3
8.5
8.0
Creamy
7.5
7.0
6.5
6.0
5.5
t
6
7
8
9
10
11
12
13
Year (6 ↔ 1996)
145. Use the graph to determine the per capita
consumption of yogurt in 1999. Then use the model
to approximate the per capita consumption of
yogurt in 1999, Compare your results.
146. Use the graph to determine the per capita
consumption of yogurt in 2002. Then use the model
to approximate the per capita consumption of
yogurt in 2002. Compare your results.
Section P.3
147. Geometry The area of a trapezoid with parallel
bases of lengths b1 and b2, and height h, is
1
2 b1 b2 h. Use the Distributive Property to show
that the area can also be written as
b1h 12b2 b1h.
148. Geometry Use both formulas given in Exercise
147 to find the area of a trapezoid with b1 7,
b2 12, and h 3.
149. Area of a Rectangle The figure shows two
adjoining rectangles. Demonstrate the Distributive
Property by filling in the blanks to write the total
area of the two rectangles in two ways.
b
c
37
Algebraic Expressions
154. Write, from memory, the properties of exponents.
Explain the difference between 2x3 and
155. Writing
2x3.
156. (a) Complete the table by evaluating 2x 5.
x
1
0
1
2
3
4
2x 5
(b) From the table in part (a), determine the
increase in the value of the expression for each
one-unit increase in x.
(c) Complete the table by using the table feature of
a graphing utility to evaluate 3x 2.
x
1
0
1
2
3
4
3x 2
a
(d) From the table in part (c), determine the
increase in the value of the expression for each
one-unit increase in x.
b+c
150. Area of a Rectangle The figure shows two
adjoining rectangles. Demonstrate the “subtraction
version” of the Distributive Property by filling in
the blanks to write the area of the left rectangle in
two ways.
(e) Use the results in parts (a) through (d) to make
a conjecture about the increase in the value of
the algebraic expression 7x 4 for each
one-unit increase in x. Use the table feature of
a graphing utility to confirm your result.
(f ) Use the results in parts (a) through (d) to make
a conjecture about the increase (or decrease) in
the value of the algebraic expression 3x 1
for each one-unit increase in x. Use the table
feature of a graphing utility to confirm your
result.
b
a
b−c
c
Synthesis
True or False? In Exercises 151 and 152, determine
whether the statement is true or false. Justify your answer.
151. 3ab 3ab
152. 2x5 y 2x5 2xy
153. Writing Explain the difference between constants
and variables in an algebraic expression.
(g) In general, what does the coefficient of the x-term
represent in the expression ax b?