E. The Product of Two Polynomials
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19
The F-O-I-L Method
By observing the product of two binomials in Example 13(a), we note a pattern that can
make the process more efficient. The product of two binomials can quickly be computed
using the First, Outer, Inner, Last (FOIL) method, an acronym giving the respective
position of each term in a product of binomials in relation to the other terms. We illustrate here using the product 12x Ϫ 12 13x ϩ 22.
WORTHY OF NOTE
Consider the product 1x ϩ 321x ϩ 22
in the context of area. If we view
x ϩ 3 as the length of a rectangle
(an unknown length plus 3 units),
and x ϩ 2 as its width (the same
unknown length plus 2 units), a
diagram of the total area would look
like the following, with the result
x2 ϩ 5x ϩ 6 clearly visible.
Combine like terms
6x2 ϩ x Ϫ 2
Inner
Outer
The first term of the result will always be the product of the first terms from each
binomial, and the last term of the result is the product of their last terms. We also note
that here, the middle term is found by adding the outermost product with the innermost
product. As you practice with the F-O-I-L process, much of the work can be done mentally and you can often compute the entire product without writing anything down
except the answer.
ᮣ
Multiplying Binomials Using F-O-I-L
Compute each product mentally:
a. 15n Ϫ 121n ϩ 22
b. 12b ϩ 32 15b Ϫ 62
a. 15n Ϫ 12 1n ϩ 22:
5n2 ϩ 9n Ϫ 2
product of
first two terms
S
ᮣ
S
Solution
10n ϩ (Ϫ1n) ϭ 9n
S
(x ϩ 3)(x ϩ 2) ϭ x2 ϩ 5x ϩ 6
EXAMPLE 14
S
6
S
2x
S
2
S
3x
S
x2
S
x
First Outer Inner Last
12x Ϫ 12 13x ϩ 22
S
3
6x2 ϩ 4x Ϫ 3x Ϫ 2
Last
First
S
x
The F-O-I-L Method for Multiplying Binomials
sum of
outer and inner
products
product of
last two terms
Ϫ12b ϩ 15b ϭ 3b
product of
first two terms
S
S
S
b. 12b ϩ 3215b Ϫ 62: 10b2 ϩ 3b Ϫ 18
sum of
outer and inner
products
E. You’ve just seen how
we can compute the product
of two polynomials
product of
last two terms
Now try Exercises 101 through 116 ᮣ
F. Special Polynomial Products
Certain polynomial products are considered “special” for two reasons: (1) the product
follows a predictable pattern, and (2) the result can be used to simplify expressions,
graph functions, solve equations, and/or develop other skills.
Binomial Conjugates
Expressions like x ϩ 7 and x Ϫ 7 are called binomial conjugates. For any given
binomial, its conjugate is found by using the same two terms with the opposite sign
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CHAPTER R A Review of Basic Concepts and Skills
between them. Example 15 shows that when we multiply a binomial and its conjugate,
the “outers” and “inners” sum to zero and the result is a difference of two squares.
EXAMPLE 15
ᮣ
Multiplying Binomial Conjugates
Compute each product mentally:
a. 1x ϩ 721x Ϫ 72
b. 12x Ϫ 5y2 12x ϩ 5y2
2
2
c. ax ϩ b ax Ϫ b
5
5
Ϫ7x ϩ 7x ϭ 0x
Solution
ᮣ
a. 1x ϩ 72 1x Ϫ 72 ϭ x2 Ϫ 49
difference of squares 1x2 2 Ϫ 172 2
10xy ϩ (Ϫ10xy) ϭ 0xy
b. 12x Ϫ 5y212x ϩ 5y2 ϭ 4x2 Ϫ 25y2
difference of squares: 12x2 2 Ϫ 15y2 2
Ϫ 52 x ϩ 25 x ϭ 0
2
2
4
c. ax ϩ b ax Ϫ b ϭ x2 Ϫ
5
5
25
2 2
difference of squares: x2 Ϫ a b
5
Now try Exercises 117 through 124 ᮣ
In summary, we have the following.
The Product of a Binomial and Its Conjugate
Given any expression that can be written in the form A ϩ B, the conjugate of the
expression is A Ϫ B and their product is a difference of two squares:
1A ϩ B2 1A Ϫ B2 ϭ A2 Ϫ B2
Binomial Squares
Expressions like 1x ϩ 72 2 are called binomial squares and are useful for solving many
equations and sketching a number of basic graphs. Note 1x ϩ 72 2 ϭ 1x ϩ 721x ϩ 72 ϭ
x2 ϩ 14x ϩ 49 using the F-O-I-L process. The expression x2 ϩ 14x ϩ 49 is called a
perfect square trinomial because it is the result of expanding a binomial square. If we
write a binomial square in the more general form 1A ϩ B2 2 ϭ 1A ϩ B21A ϩ B2 and
compute the product, we notice a pattern that helps us write the expanded form
more quickly.
LOOKING AHEAD
Although a binomial square can
always be found using repeated
factors and F-O-I-L, learning to
expand them using the pattern is
a valuable skill. Binomial squares
occur often in a study of algebra
and it helps to find the expanded
form quickly.
1A ϩ B2 2 ϭ 1A ϩ B2 1A ϩ B2
repeated multiplication
ϭ A ϩ AB ϩ AB ϩ B
F-O-I-L
ϭ A ϩ 2AB ϩ B
simplify (perfect square trinomial)
2
2
2
2
The first and last terms of the trinomial are squares of the terms A and B. Also, the
middle term of the trinomial is twice the product of these two terms: AB ϩ AB ϭ 2AB.
The F-O-I-L process shows us why. Since the outer and inner products are identical,
we always end up with two. A similar result holds for 1A Ϫ B2 2 and the process can be
summarized for both cases using the Ϯ symbol.
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Section R.2 Exponents, Scientific Notation, and a Review of Polynomials
21
The Square of a Binomial
Given any expression that can be written in the form 1A Ϯ B2 2,
1. 1A ϩ B2 2 ϭ A2 ϩ 2AB ϩ B2
2. 1A Ϫ B2 2 ϭ A2 Ϫ 2AB ϩ B2
ᮣ
CAUTION
EXAMPLE 16
ᮣ
Solution
ᮣ
Note the square of a binomial always results in a trinomial (three terms). In particular,
1A ϩ B2 2 A2 ϩ B2.
Find each binomial square without using F-O-I-L:
a. 1a ϩ 92 2
F. You’ve just seen how
we can compute special
products: binomial conjugates
and binomial squares
b. 13x Ϫ 52 2
a. 1a ϩ 92 2 ϭ a2 ϩ 21a # 92 ϩ 92
ϭ a2 ϩ 18a ϩ 81
b. 13x Ϫ 52 2 ϭ 13x2 2 Ϫ 213x # 52 ϩ 52
ϭ 9x2 Ϫ 30x ϩ 25
c. 13 ϩ 1x2 2 ϭ 9 ϩ 213 # 1x2 ϩ 1 1x2 2
ϭ 9 ϩ 6 1x ϩ x
c. 13 ϩ 1x2 2
1A ϩ B2 2 ϭ A 2 ϩ 2AB ϩ B 2
simplify
1A Ϫ B2 2 ϭ A 2 Ϫ 2AB ϩ B 2
simplify
1A ϩ B2 2 ϭ A 2 ϩ 2AB ϩ B 2
simplify
Now try Exercises 125 through 136 ᮣ
With practice, you will be able to go directly from the binomial square to the
resulting trinomial.
R.2 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section, if necessary.
1. The equation 1x2 2 3 ϭ x6 is an example of the
property of exponents.
2. The equation 1x3 2 Ϫ2 ϭ
property of
1
is an example of the
x6
exponents.
3. The sum of the “outers” and “inners” for 12x ϩ 52 2
is
, while the sum of the outers and inners
for 12x ϩ 52 12x Ϫ 52 is
.
ᮣ
4. The expression 2x2 Ϫ 3x Ϫ 10 can be classified as
a
of degree
, with a leading
coefficient of
.
5. Discuss/Explain why one of the following
expressions can be simplified further, while the
other cannot: (a) Ϫ7n4 ϩ 3n2; (b) Ϫ7n4 # 3n2.
6. Discuss/Explain why the degree of 2x2y3 is greater
than the degree of 2x 2 ϩ y 3. Include additional
examples for contrast and comparison.
DEVELOPING YOUR SKILLS
Determine each product using the product and/or power properties.
7.
2 2#
n 21n5
3
10. 1Ϫ1.5vy2 2 1Ϫ8v4y2
8. 24g5
# 3 g9
8
11. 1a2 2 4 # 1a3 2 2 # b2 # b5
9. 1Ϫ6p2q2 12p3q3 2
12. d 2 # d 4 # 1c5 2 2 # 1c3 2 2
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Simplify using the product to a power property.
13. 16pq2 2 3
14. 1Ϫ3p2q2 2
15. 13.2hk2 2 3
17. a
16. 1Ϫ2.5h5k2 2
p 2
b
2q
18. a
19. 1Ϫ0.7c 2 110c d 2
4 2
21.
R–22
CHAPTER R A Review of Basic Concepts and Skills
1 34x3y2 2
3 2 2
b 3
b
3a
5m2n3 2
b
2r4
45. a
5p2q3r4
2 2 3
22. 1 45x3 2 2
24. 1 23m2n2 2 # 1 12mn2 2
47.
25. Volume of a cube: The
3x2
formula for the volume of a
cube is V ϭ S3, where S is the
length of one edge. If the
length of each edge is 3x2,
3x2
a. Find a formula for volume
in terms of the variable x.
b. Find the volume of the cube if x ϭ 2.
49.
23. 1Ϫ38x2 2 116xy2 2
26. Area of a circle: The formula
for the area of a circle is
A ϭ r2, where r is the length
of the radius. If the radius is
given as 5x3,
a. Find a formula for area in
terms of the variable x.
b. Find the area of the circle if x ϭ 2.
3x2
5x3
Simplify using the quotient property or the property of
negative exponents. Write answers using positive
exponents only.
27.
Ϫ6w
Ϫ2w2
28.
8z
16z5
29.
Ϫ12a3b5
4a2b4
30.
5m3n5
10mn2
5
7
31. 1 23 2 Ϫ3
33.
2
hϪ3
32. 1 56 2 Ϫ1
34.
3
mϪ2
35. 1Ϫ22 Ϫ3
36. 1Ϫ42 Ϫ2
37.
38.
Ϫ3
1 Ϫ1
2 2
Ϫ2
1 Ϫ2
3 2
Simplify each expression using the quotient to a power
property.
39. a
2p4
41. a
0.2x2 3
b
0.3y3
q3
b
2
40. a
Ϫ5v4 2
b
7w3
42. a
Ϫ0.5a3 2
b
0.4b2
44. a
2
b
Ϫ2pq2r4
46. a
4p3
3x2y
b
3
9p3q2r3
12p5qr
3
b
2
Use properties of exponents to simplify the following.
Write the answer using positive exponents only.
20. 1Ϫ2.5a 2 13a b 2
3 2
43. a
51.
9p6q4
Ϫ12p4q6
20hϪ2
12h5
1a2 2 3
a4 # a5
aϪ3 # b Ϫ4
b
cϪ2
Ϫ612xϪ3 2 2
48.
50.
52.
53. a
54.
55.
56.
57.
10xϪ2
14aϪ3bc0
Ϫ713a2bϪ2c2 3
59. 40 ϩ 50
61. 2Ϫ1 ϩ 5Ϫ1
63. 30 ϩ 3Ϫ1 ϩ 3Ϫ2
65. Ϫ5x0 ϩ 1Ϫ5x2 0
58.
5m5n2
10m5n
5k3
20kϪ2
153 2 4
59
1pϪ4q8 2 2
p5qϪ2
18nϪ3
Ϫ813nϪ2 2 3
Ϫ312x3yϪ4z2 2
18xϪ2yz0
60. 1Ϫ32 0 ϩ 1Ϫ72 0
62. 4Ϫ1 ϩ 8Ϫ1
64. 2Ϫ2 ϩ 2Ϫ1 ϩ 20
66. Ϫ2n0 ϩ 1Ϫ2n2 0
Convert the following numbers to scientific notation.
67. In mid-2009, the U.S. Census Bureau estimated the
world population at nearly 6,770,000,000 people.
68. The mass of a proton is generally given as
0.000 000 000 000 000 000 000 000 001 670 kg.
Convert the following numbers to decimal notation.
69. The smallest microprocessors in common use
measure 6.5 ϫ 10Ϫ9 m across.
70. In 2009, the estimated net worth of Bill Gates, the
founder of Microsoft, was 5.8 ϫ 1010 dollars.
Compute using scientific notation. Show all work.
71. The average distance between the Earth and the
planet Jupiter is 465,000,000 mi. How many hours
would it take a satellite to reach the planet if it
traveled an average speed of 17,500 mi per hour?
How many days? Round to the nearest whole.
72. In fiscal terms, a nation’s debt-per-capita is the
ratio of its total debt to its total population. In the
year 2009, the total U.S. debt was estimated at
$11,300,000,000,000, while the population was
estimated at 305,000,000. What was the U.S. debtper-capita ratio for 2009? Round to the nearest
whole dollar.
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Section R.2 Exponents, Scientific Notation, and a Review of Polynomials
Identify each expression as a polynomial or
nonpolynomial (if a nonpolynomial, state why);
classify each as a monomial, binomial, trinomial, or
none of these; and state the degree of the polynomial.
73. Ϫ35w3 ϩ 2w2 ϩ 1Ϫ12w2 ϩ 14
74. Ϫ2x3 ϩ 23x2 Ϫ 12x ϩ 1.2
4
75. 5nϪ2 ϩ 4n ϩ 117 76. 3 ϩ 2.7r2 ϩ r ϩ 1
r
2
3
77. p Ϫ 5
78. q3 ϩ 2qϪ2 Ϫ 5q
Write each polynomial in standard form and name the
leading coefficient.
79. 7w ϩ 8.2 Ϫ w3 Ϫ 3w2
80. Ϫ2k2 Ϫ 12 Ϫ k
81. c3 ϩ 6 ϩ 2c2 Ϫ 3c
82. Ϫ3v3 ϩ 14 ϩ 2v2 ϩ 1Ϫ12v2
83. 12 Ϫ 23x2
84. 8 ϩ 2n ϩ 7n
2
96. 1s Ϫ 32 15s ϩ 42
97. 1x Ϫ 321x2 ϩ 3x ϩ 92
98. 1z ϩ 52 1z2 Ϫ 5z ϩ 252
99. 1b2 Ϫ 3b Ϫ 282 1b ϩ 22
100. 12h2 Ϫ 3h ϩ 82 1h Ϫ 12
101. 17v Ϫ 4213v Ϫ 52
103. 13 Ϫ m213 ϩ m2
102. 16w Ϫ 1212w ϩ 52
104. 15 ϩ n215 Ϫ n2
105. 1p Ϫ 2.521p ϩ 3.62 106. 1q Ϫ 4.921q ϩ 1.22
107. 1x ϩ 12 2 1x ϩ 14 2
109. 1m ϩ 34 2 1m Ϫ 34 2
108. 1z ϩ 13 21z ϩ 56 2
110. 1n Ϫ 25 21n ϩ 25 2
111. 13x Ϫ 2y212x ϩ 5y2 112. 16a ϩ b21a ϩ 3b2
113. 14c ϩ d213c ϩ 5d2 114. 15x ϩ 3y212x Ϫ 3y2
115. 12x2 ϩ 52 1x2 Ϫ 32 116. 13y2 Ϫ 2212y2 ϩ 12
For each binomial, determine its conjugate and find the
product of the binomial with its conjugate.
117. 4m Ϫ 3
118. 6n ϩ 5
85. 13p3 Ϫ 4p2 ϩ 2p Ϫ 72 ϩ 1p2 Ϫ 2p Ϫ 52
119. 7x Ϫ 10
120. c ϩ 3
121. 6 ϩ 5k
122. 11 Ϫ 3r
87. 15.75b2 ϩ 2.6b Ϫ 1.92 ϩ 12.1b2 Ϫ 3.2b2
123. x ϩ 16
124. p Ϫ 12
Find the indicated sum or difference.
86. 15q2 Ϫ 3q ϩ 42 ϩ 1Ϫ3q2 ϩ 3q Ϫ 42
88. 10.4n2 ϩ 5n Ϫ 0.52 ϩ 10.3n2 Ϫ 2n ϩ 0.752
Find each binomial square.
90. 1 59n2 ϩ 4n Ϫ 12 2 Ϫ 1 23n2 Ϫ 2n ϩ 34 2
127. 14g ϩ 32 2
89. 1 34x2 Ϫ 5x ϩ 22 Ϫ 1 12x2 ϩ 3x Ϫ 42
91. Subtract q5 ϩ 2q4 ϩ q2 ϩ 2q from q6 ϩ 2q5 ϩ
q4 ϩ 2q3 using a vertical format.
92. Find x4 ϩ 2x3 ϩ x2 ϩ 2x decreased by
x4 Ϫ 3x3 ϩ 4x2 Ϫ 3x using a vertical format.
Compute each product.
93. Ϫ3x1x2 Ϫ x Ϫ 62
94. Ϫ2v2 1v2 ϩ 2v Ϫ 152
95. 13r Ϫ 52 1r Ϫ 22
ᮣ
23
125. 1x ϩ 42 2
126. 1a Ϫ 32 2
129. 14p Ϫ 3q2 2
130. 15c ϩ 6d2 2
131. 14 Ϫ 1x2 2
128. 15x Ϫ 32 2
132. 1 1x ϩ 72 2
Compute each product.
133. 1x Ϫ 321y ϩ 22
134. 1a ϩ 321b Ϫ 52
135. 1k Ϫ 521k ϩ 621k ϩ 22
136. 1a ϩ 621a Ϫ 121a ϩ 52
WORKING WITH FORMULAS
137. Medication in the bloodstream: M ؍0.5t 4 ؉ 3t 3 ؊ 97t 2 ؉ 348t
If 400 mg of a pain medication are taken orally, the number of milligrams in the bloodstream is modeled by the
formula shown, where M is the number of milligrams and t is the time in hours, 0 Յ t 6 5. Construct a table of
values for t ϭ 1 through 5, then answer the following.
a. How many milligrams are in the bloodstream after 2 hr? After 3 hr?
b. Based on part a, would you expect the number of milligrams in the bloodstream after 4 hr to be less or more? Why?
c. Approximately how many hours until the medication wears off (the number of milligrams in the bloodstream is 0)?
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Aa
138. Amount of a mortgage payment: M ؍
r n
r
b a1 ؉ b
12
12
r n
b ؊1
12
The monthly mortgage payment required to pay off (or amortize) a loan is given by the formula shown, where M is
the monthly payment, A is the original amount of the loan, r is the annual interest rate, and n is the term of the loan
in months. Find the monthly payment (to the nearest cent) required to purchase a $198,000 home, if the interest rate
is 6.5% and the home is financed over 30 yr.
ᮣ
a1 ؉
APPLICATIONS
139. Attraction between particles: In electrical theory,
the force of attraction between two particles P and
kPQ
Q with opposite charges is modeled by F ϭ 2 ,
d
where d is the distance between them and k is a
constant that depends on certain conditions. This is
known as Coulomb’s law. Rewrite the formula
using a negative exponent.
140. Intensity of light: The intensity of illumination
from a light source depends on the distance from
k
the source according to I ϭ 2 , where I is the
d
intensity measured in footcandles, d is the distance
from the source in feet, and k is a constant that
depends on the conditions. Rewrite the formula
using a negative exponent.
141. Rewriting an expression: In advanced mathematics,
negative exponents are widely used because they are
easier to work with than rational expressions.
5
3
2
Rewrite the expression 3 ϩ 2 ϩ 1 ϩ 4 using
x
x
x
negative exponents.
142. Swimming pool hours: A swimming pool opens at
8 A.M. and closes at 6 P.M. In summertime, the
ᮣ
number of people in the pool at any time can be
approximated by the formula S1t2 ϭ Ϫt 2 ϩ 10t,
where S is the number of swimmers and t is the
number of hours the pool has been open (8 A.M.:
t ϭ 0, 9 A.M.: t ϭ 1, 10 A.M.: t ϭ 2, etc.).
a. How many swimmers are in the pool at 6 P.M.?
Why?
b. Between what times would you expect the
largest number of swimmers?
c. Approximately how many swimmers are in the
pool at 3 P.M.?
d. Create a table of values for t ϭ 1, 2, 3, 4, . . .
and check your answer to part b.
143. Maximizing revenue: A sporting goods store finds
that if they price their video games at $20, they
make 200 sales per day. For each decrease of $1,
20 additional video games are sold. This means the
store’s revenue can be modeled by the formula
R ϭ 120 Ϫ 1x2 1200 ϩ 20x2, where x is the number
of $1 decreases. Multiply out the binomials and use
a table of values to determine what price will give
the most revenue.
144. Maximizing revenue: Due to past experience, a
jeweler knows that if they price jade rings at $60,
they will sell 120 each day. For each decrease of
$2, five additional sales will be made. This means
the jeweler’s revenue can be modeled by the
formula R ϭ 160 Ϫ 2x2 1120 ϩ 5x2, where x is the
number of $2 decreases. Multiply out the
binomials and use a table of values to determine
what price will give the most revenue.
EXTENDING THE CONCEPT
145. If 13x2 ϩ kx ϩ 12 Ϫ 1kx2 ϩ 5x Ϫ 72 ϩ
12x2 Ϫ 4x Ϫ k2 ϭ Ϫx2 Ϫ 3x ϩ 2, what is the
value of k?
1 2
1
b ϭ 5, then the expression 4x2 ϩ 2
2x
4x
is equal to what number?
146. If a2x ϩ
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R.3
Solving Linear Equations and Inequalities
LEARNING OBJECTIVES
In a study of algebra, you will encounter many families of equations, or groups of
equations that share common characteristics. Of interest to us here is the family of
linear equations in one variable, a study that lays the foundation for understanding
more advanced families. This section will also lay the foundation for solving a formula for
a specified variable, a practice widely used in science, business, industry, and research.
In Section R.3 you will review how to:
A. Solve linear equations
B.
C.
D.
E.
F.
using properties of
equality
Recognize equations that
are identities or
contradictions
Solve linear inequalities
Solve compound
inequalities
Solve basic applications
of linear equations and
inequalities
Solve applications of
basic geometry
CAUTION
A. Solving Linear Equations Using Properties of Equality
An equation is a statement that two expressions are
equal. From the expressions 31x Ϫ 12 ϩ x and
Ϫx ϩ 7, we can form the equation
31x Ϫ 12 ϩ x ϭ Ϫx ϩ 7,
Table R.1
x
31x ؊ 12 ؉ x
؊x ؉ 7
Ϫ2
Ϫ11
9
Ϫ1
Ϫ7
8
which is a linear equation in one variable (the
0
exponent on any variable is a 1). To solve an equa1
tion, we attempt to find a specific input or x-value
2
that will make the equation true, meaning the left3
hand expression will be equal to the right. Using
4
Table R.1, we find that 31x Ϫ 12 ϩ x ϭ Ϫx ϩ 7 is a
true equation when x is replaced by 2, and is a false
equation otherwise. Replacement values that make
the equation true are called solutions or roots of the equation.
ᮣ
Ϫ3
7
1
6
5
5
9
4
13
3
From Section R.1, an algebraic expression is a sum or difference of algebraic terms.
Algebraic expressions can be simplified, evaluated or written in an equivalent form, but
cannot be “solved,” since we’re not seeking a specific value of the unknown.
Solving equations using a table is too time consuming to be practical. Instead we
attempt to write a sequence of equivalent equations, each one simpler than the one before, until we reach a point where the solution is obvious. Equivalent equations are those
that have the same solution set, and can be obtained by using the distributive property to
simplify the expressions on each side of the equation. The additive and multiplicative
properties of equality are then used to obtain an equation of the form x ϭ constant.
The Additive Property of Equality
The Multiplicative Property of Equality
If A, B, and C represent algebraic
expressions and A ϭ B,
If A, B, and C represent algebraic
expressions and A ϭ B,
then A ϩ C ϭ B ϩ C
then AC ϭ BC and
B
A
ϭ , 1C
C
C
02
In words, the additive property says that like quantities, numbers, or terms can be
added to both sides of an equation. A similar statement can be made for the multiplicative property. These properties are combined into a general guide for solving linear
equations, which you’ve likely encountered in your previous studies. Note that not all
steps in the guide are required to solve every equation.
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CHAPTER R A Review of Basic Concepts and Skills
Guide to Solving Linear Equations in One Variable
• Eliminate parentheses using the distributive property, then combine any like terms.
• Use the additive property of equality to write the equation with all variable terms
on one side, and all constants on the other. Simplify each side.
• Use the multiplicative property of equality to obtain an equation of the form
x ϭ constant.
• For applications, answer in a complete sentence and include any units of measure
indicated.
For our first example, we’ll use the equation 31x Ϫ 12 ϩ x ϭ Ϫx ϩ 7 from our
initial discussion.
EXAMPLE 1
ᮣ
Solving a Linear Equation Using Properties of Equality
Solve for x: 31x Ϫ 12 ϩ x ϭ Ϫx ϩ 7.
Solution
ᮣ
31x Ϫ 12 ϩ x ϭ Ϫx ϩ 7
3x Ϫ 3 ϩ x ϭ Ϫx ϩ 7
4x Ϫ 3 ϭ Ϫx ϩ 7
5x Ϫ 3 ϭ 7
5x ϭ 10
xϭ2
original equation
distributive property
combine like terms
add x to both sides (additive property of equality)
add 3 to both sides (additive property of equality)
multiply both sides by 15 or divide both sides by 5
(multiplicative property of equality)
As we noted in Table R.1, the solution is x ϭ 2.
Now try Exercises 7 through 12
ᮣ
To check a solution by substitution means we substitute the solution back into the
original equation (this is sometimes called back-substitution), and verify the lefthand side is equal to the right. For Example 1 we have:
31x Ϫ 12 ϩ x ϭ Ϫx ϩ 7
original equation
312 Ϫ 12 ϩ 2 ϭ Ϫ2 ϩ 7
substitute 2 for x
3112 ϩ 2 ϭ 5
5 ϭ 5✓
simplify
solution checks
If any coefficients in an equation are fractional, multiply both sides by the least
common denominator (LCD) to clear the fractions. Since any decimal number can be
written in fraction form, the same idea can be applied to decimal coefficients.
EXAMPLE 2
ᮣ
Solution
ᮣ
A. You’ve just seen how
we can solve linear equations
using properties of equality
Solving a Linear Equation with Fractional Coefficients
Solve for n: 14 1n ϩ 82 Ϫ 2 ϭ 12 1n Ϫ 62.
1
4 1n ϩ 82
1
4n ϩ 2
Ϫ 2 ϭ 12 1n Ϫ 62
Ϫ 2 ϭ 12n Ϫ 3
1
1
4n ϭ 2n Ϫ 3
1
41 4n2 ϭ 41 12n Ϫ 32
n ϭ 2n Ϫ 12
Ϫn ϭ Ϫ12
n ϭ 12
original equation
distributive property
combine like terms
multiply both sides by LCD ϭ 4
distributive property
subtract 2n
multiply by Ϫ1
Verify the solution is n ϭ 12 using back-substitution.
Now try Exercises 13 through 30
ᮣ