combinations of letters and numbers are algebraic expressions. Here are ... The
terms of an algebraic expression are those parts that are separated by addition.
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?
