Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models
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EXAMPLE 7
䊳
Solving a Compound Inequality
Solve the compound inequality, then write the solution in interval notation:
or
Solution
䊳
Begin with the statement as given:
or
original statement
or isolate variable term
or solve for
x
, reverse first inequality symbol
The solution or is better understood by graphing each
interval separately, then selecting both intervals (the union).
Interval notation:
Now try Exercises 59 and 60
䊳
EXAMPLE 8
䊳
Solving a Compound Inequality
Solve the compound inequality, then write the solution in interval notation:
and
Solution
䊳
Begin with the statement as given:
and
original statement
and subtract five
and divide by 3
The solution and can best be understood by graphing each interval
separately, then noting where they intersect.
Interval notation:
Now try Exercises 61 through 72
䊳
x 僆 16, 22.
11234678 5 3254 60
11234678 5 3254 60
x 2:
x 6 and x 2:
)
)
)
11234678 5 3254 60
)
x 6:
x 6 2x 7 6
x 6 2 x 7 6
3 x 6 6 3 x 7 18
3 x 5 6 1 3 x 5 7 13
3x 5 6 1.3x 5 7 13
x 僆 aq,
9
4
b ´ 11, q2.
)
)
11234678 5 2 3 4 5 60
x
or x 1:
9
4
9
4
)
11234678 5 2 3 4 5 60
x
:
9
4
9
4
)
11234678
x
1:
5 2 3 4 5 60
x 6
9
4
x 7 1
x 6
9
4
x 7 1
4 x 6 9 3x 6 3
4 x 3 6 6 3x 1 6 4
4x 3 6 6.3x 1 6 4
30 CHAPTER R A Review of Basic Concepts and Skills R–30
College Algebra Graphs & Models—
WORTHY OF NOTE
The graphs from Example 7 clearly
show the solution consists of two
disjoint (disconnected) intervals.
This is reflected in the “or”
statement: or and in
the interval notation. Also, note the
solution or is not
equivalent to as there
is no single number that is both
greater than 1 and less than at
the same time.
9
4
9
4
7 x 7 1,
x 7 1x 6
9
4
x 7 1,x 6
9
4
WORTHY OF NOTE
The inequality (a is less than
b) can equivalently be written as
(b is greater than a). In
Example 8, the solution is read,
“ and but if we
rewrite the first inequality as
(with the “arrowhead” still
pointing at we have
and and can clearly see
that x must be in the single interval
between and 2.6
x 6 2
6 6 x62,
6 6 x
x 6 2,”x 7 6
b 7 a
a 6 b
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The solution from Example 8 consists of the single interval indicat-
ing the original inequality could actually be joined and written as
called a joint inequality. We solve joint inequalities in much the same way as lin-
ear inequalities, but must remember they have three parts (left, middle, and right).
This means operations must be applied to all three parts in each step of the solution
process, to obtain a solution form such as smaller number larger number.
The same ideas apply when other inequality symbols are used.
EXAMPLE 9
䊳
Solving a Joint Inequality
Solve the joint inequality, then graph the solution set and write it in interval
notation:
Solution
䊳
original inequality
multiply all parts by 3; reverse the inequality symbols
subtract 5 from all parts
divide all parts by 2
• Graph:
• Interval notation:
Now try Exercises 73 through 78
䊳
E. Solving Basic Applications of Linear
Equations and Inequalities
Applications of linear equations and inequalities come in many forms. In most cases,
you are asked to translate written relationships or information given verbally into an
equation using words or phrases that indicate mathematical operations or relationships.
Here, we’ll practice this skill using ideas that were introduced in Section R.1, where
we translated English phrases into mathematical expressions. Very soon these skills
will be applied in much more significant ways.
EXAMPLE 10
䊳
Translating Written Information into an Equation
Translate the following relationships into equations, then solve:
In an effort to lower the outstanding balance on her credit card, Laura paid $10 less
than triple her normal payment. If she sent the credit card company $350.75, how
much was her normal payment? (See Section R.1, Examples 2 and 3.)
Solution
䊳
Let p represent her normal payment. Then “triple her normal payment” would be
3p, and “ten less than triple” would be 3p 10. Since “she sent the company
$350.75,” we have
x 僆 14,
13
2
4
[
)
2 15 4 3 23571 4 860
2
13
4 6 x
13
2
8 6 2x 13
3 6 2x 5 18
1 7
2x 5
3
6
1 7
2x 5
3
6.
6 x 6
6 6 x 6 2,
16, 22,
R–31 Section R.3 Solving Linear Equations and Inequalities 31
College Algebra Graphs & Models—
D. You’ve just seen how
we can solve compound
inequalities
cob19545_chR_025-039.qxd 7/28/10 2:35 PM Page 31
equation form
add 10
divide by 3
Laura’s normal payment is $120.25 per month.
A calculator check is shown in the figure.
Now try Exercises 83 through 90
䊳
Inequalities are widely used to help gather information, and to make comparisons
that will lead to informed decisions.
EXAMPLE 11
䊳
Using an Inequality to Compute Desired Test Scores
Justin earned scores of 78, 72, and 86 on the first three out of four exams. What
score must he earn on the fourth exam to have an average of at least 80?
Solution
䊳
The current scores are 78, 72, and 86. An average of at least 80 means
In organized form:
A 80.
p 120.25
3 p 360.75
3 p 10 350.75
32 CHAPTER R A Review of Basic Concepts and Skills R–32
College Algebra Graphs & Models—
Test 1 Test 2 Test 3 Test 4 Computed Average Minimum
78 72 86 x 80
78 72 86 x
4
Let x represent Justin’s score on the fourth exam, then
represents his average score.
average must be greater than or equal to 80
multiply by 4
simplify
solve for
x
(subtract 236)
Justin must score at least an 84 on the last exam to earn an 80 average.
Now try Exercises 91 through 100
䊳
F. Solving Applications of Basic Geometry
As your translation skills grow, your ability to solve a wider range of more significant
applications will grow as well. In many cases, the applications will involve some basic
geometry and the most often used figures and formulas appear here. For a more com-
plete review of geometry, see Appendix II Geometry Review with Unit Conversions,
which is posted online at www.mhhe.com/coburn.
Perimeter and Area
Perimeter is a measure of the distance around a two dimensional figure. As this is a lin-
ear measure, results are stated in linear units as in centimeters (cm), feet (ft), kilome-
ters (km), miles (mi), and so on. If no unit is specified, simply write the result as x
units. Area is a measure of the surface of a two dimensional figure, with results stated
in square units as in x units
2
. Some of the most common formulas involving perimeter
and area are given in Table R.2A.
x 84
236 x 320
78 72 86 x 320
78 72 86 x
4
80
78 72 86 x
4
E. You’ve just seen how
we can solve applications
of linear equations and
inequalities
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If an exercise or application uses a formula, begin by stating the formula first.
Using the formula as a template for the values substituted will help to prevent many
careless errors.
EXAMPLE 12A
䊳
Computing the Area of a Trapezoidal Window
A basement window is shaped like an isosceles
trapezoid (base angles equal, nonparallel sides
equal in length), with a height of 10 in. and
bases of 1.5 ft and 2 ft. What is the area of the
glass in the window?
R–33 Section R.3 Solving Linear Equations and Inequalities 33
College Algebra Graphs & Models—
C
r
10 in.
1.5 ft
2 ft
Table R.2A
Perimeter Formula Area Formula
Definition and Diagram (linear units or units) (square units or units
2
)
Triangle
a three-sided polygon
Rectangle
a quadrilateral with four right angles
and opposite sides parallel
Square
a rectangle with four equal sides
a quadrilateral with one pair of parallel sides
Trapezoid
(called bases b
1
and b
2
)
sum of all sides
the set of all points lying in a plane
Circle that are an equal distance
(called the radius r) from a
given point (called the center C). or
C ⫽ d
A ⫽ r
2
C ⫽ 2r
A ⫽
h
2
1b
1
⫹ b
2
2P ⫽ s
1
⫹ s
2
⫹ s
3
⫹ s
4
b
2
b
1
h
s
1
s
2
s
4
s
3
A ⫽ S
2
P ⫽ 4S
S
A ⫽ LWP ⫽ 2L ⫹ 2W
W
L
A ⫽
1
2
bhP ⫽ s
1
⫹ s
2
⫹ s
3
h
b
s
1
s
2
s
3
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Solution
䊳
Before applying the area formula, all measures must use the same unit. In inches,
we have 1.5 ft in. and 2 ft in.
given formula
substitute 10 for
h
, 18 for
b
1
, and 24 for
b
2
simplify
result
The area of the glass in the window is 210 in
2
.
Now try Exercises 101 and 102
䊳
Volume
Volume is a measure of the amount of space occupied by a three dimensional object and
is measured in cubic units. Some of the more common formulas are given in Table R.2B.
A ⫽ 210 in
2
A ⫽ 15 in.2142 in.2
A ⫽
10 in.
2
118 in. ⫹ 24 in.2
A ⫽
h
2
1b
1
⫹ b
2
2
⫽ 24⫽ 18
34 CHAPTER R A Review of Basic Concepts and Skills R–34
College Algebra Graphs & Models—
Table R.2B
Volume Formula
Definition and Diagram (cubic units or units
3
)
a six-sided, solid
Rectangular figure with opposite
solid faces congruent and
adjacent faces meeting
at right angles
a rectangular solid
Cube with six congruent,
square faces
the set of all points in space,
Sphere
an equal distance (called the
radius) from a given point
(called the center)
union of all line segments
Right circular connecting two congruent
cylinder circles in parallel planes,
meeting each at a right angle
union of all line segments
connecting a given point
Right circular (vertex) to a given circle
cone (base) and whose altitude
meets the center of the base
at a right angle
union of all line segments
connecting a given point
Right (vertex) to a given square
pyramid (base) and whose altitude
meets the center of the base
at a right angle
V ⫽
1
3
s
2
h
V ⫽
1
3
r
2
h
V ⫽ r
2
h
V ⫽
4
3
r
3
V ⫽ S
3
V ⫽ LWH
H
L
W
s
h
h
r
h
r
C
r
S
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EXAMPLE 12B
䊳
Computing the Volume of a Composite Figure
Sand at a cement factory is being dumped from a conveyor
belt into a pile shaped like a right circular cone atop a
right circular cylinder (see figure). How many cubic
feet of sand are there at the moment the cone is 6 ft high
with a diameter of 10 ft?
Solution
䊳
verbal model
formula model (note
substitute 5 for
r
, 3 for
h
1
, and 6 for
h
2
simplify
result (exact form)
There are about 392.7 ft
3
of sand in the pile.
Now try Exercises 103 and 104
䊳
⫽ 125
⫽ 75 ⫹ 50
⫽ 152
2
132⫹
1
3
152
2
162
h
1
⫽ h
2
2 V ⫽ r
2
h
1
⫹
1
3
r
2
h
2
Total Volume ⫽ volume of cylinder ⫹ volume of cone
R–35 Section R.3 Solving Linear Equations and Inequalities 35
College Algebra Graphs & Models—
3 ft
10 ft
6 ft
4. The of sets A and B is written
The of sets A and B is written A ´ B.
A 傽 B.
5. Discuss/Explain the similarities and differences
between the properties of equality for equations
and those for inequalities.
6. Discuss/Explain the use of the words “and” and
“or” in the statement of compound inequalities.
Include a few examples to illustrate.
䊳
CONCEPTS AND VOCABULARY
Fill in each blank with the appropriate word or phrase. Carefully reread the section, if necessary.
R.3 EXERCISES
1. A(n) is an equation that is always true,
regardless of the value while a(n)
is an equation that is always false,
regardless of the value.
2. For inequalities, the three ways of writing a
solution set are notation, a number line
graph, and notation.
3. The mathematical sentence is a(n)
inequality, while is
a(n) inequality.
⫺2 6 3x ⫹ 5 6 7
3x ⫹ 5 6 7
䊳
DEVELOPING YOUR SKILLS
Solve each equation. Check your answer by substitution.
7.
8.
9.
10.
11.
12. 2a ⫹ 41a ⫺ 12⫽ 3 ⫺ 12a ⫹ 12
8 ⫺ 13b ⫹ 52⫽⫺5 ⫹ 21b ⫹ 12
⫺12 ⫺ 5w ⫽⫺9 ⫺ 16w ⫹ 72
21 ⫺ 12v ⫹ 172⫽⫺7 ⫺ 3v
15 ⫺ 2x ⫽⫺41x ⫹ 12⫹ 9
4x ⫹ 31x ⫺ 22⫽ 18 ⫺ x
Solve each equation.
13.
14.
15.
16.
17. 18.
19. 20.
z ⫺ 4
6
⫺ 2 ⫽
z
2
x ⫹ 3
5
⫹
x
3
⫽ 7
⫺4 ⫹
2
3
y ⫽
1
2
y ⫺ 5
1
2
x ⫹ 5 ⫽
1
3
x ⫹ 7
4
5
1n ⫺ 102⫽
⫺8
9
2
3
1m ⫹ 62⫽
⫺1
2
1
6
1n ⫺ 122⫽
1
4
1n ⫹ 82⫺ 2
1
5
1b ⫹ 102⫺ 7 ⫽
1
3
1b ⫺ 92
F. You’ve just seen how
we can solve applications
of basic geometry
cob19545_chR_025-039.qxd 11/23/10 6:17 PM Page 35
Solve each inequality and write the solution in set
notation.
47.
48.
49.
50.
51.
52.
Determine the intersection and union of sets A, B, C,
and D as indicated, given
and
.
53. 54. and
55. and 56. and
57. and 58. and
Express the compound inequalities graphically and in
interval notation.
59. or 60. or
61. and 62. and
63. and 64. and
Solve the compound inequalities and graph the
solution set.
65. or
66. or
67. and
68. and
69. and
70. and
71. or
72. or
73.
74.
75.
76.
77.
78. 21
2
3
x 9 6 7
7 6
3
4
x 1 11
8.2 6 1.4 x 6 0.9
0.5 0.3 x 1.7
2 6 3x 4 19
3 2x 5 6 7
x 3 7 2
2x
5
x
10
6 2
x 1 7 5
3x
8
x
4
6 3
3x 6 2
2
3
x
5
6
0
4x 7 1
3
5
x
1
2
7
3
10
5x 03x 5 17
2x 02x 7 3
x 3 131x 227 15
x 6 7 941x 12 20
x 7x 5x 1x 3
x 6 3x 4x 2x 6 5
x 7 5x 6 5x 7 1x 6 2
C ´ DC 傽 DB ´ DB 傽 D
B ´ CB 傽 CA ´ DA 傽 D
A ´ CA 傽 CA 傽 B and A ´ B
D ⴝ 54, 5, 6, 76
C ⴝ 5ⴚ 4, ⴚ2, 0, 2, 46,B ⴝ 52, 4, 6, 86,
A ⴝ 5ⴚ3, ⴚ2, ⴚ1, 0, 1, 2, 36,
91w 12 3w 215 3w2 1
61p 12 2p 212p 32
8 16 5m27 9m 13 4m2
413x 52 18 6 215x 12 2x
3 61x 52 217 3x2 1
7 21x 32 4x 61x 32
21. 22.
23.
24.
25.
26.
27.
28.
29.
30.
Identify the following equations as an identity, a
contradiction, or a conditional equation, then state the
solution.
31.
32.
33.
34.
35.
36.
Write the solution set illustrated on each graph in set
notation and interval notation.
37.
38.
39.
40.
Solve the inequality and write the solution in set
notation. Then graph the solution and write it in
interval notation.
41.
42.
43.
44.
45. 46.
2y
5
y
10
6 2
3x
8
x
4
6 4
51x 22 3 6 3x 11
21n 32 4 5n 1
8n 5 7 2n 12
5a 11 2a 5
[
)
31123240
[
[
112320 3
)
112302 3
[
1123230
15x 32 2x 11 41x 22
414x 526 218x 72
2a 41a 12 1 312a 12
8 813n 525 611 n2
5x 9 2 512 x2 1
314z 5215z 20 3z
q
6
1 3q 2 4q
q
8
3p
p
4
5
p
6
2p 6
m
3
2
5
m
4
n
2
n
5
2
3
7.9 2.6w 1.5w 19.1 2.1w2
6.2v 12.1v 52 1.1 3.7v
0.4117 4.25b2 3.15 4.16
0.2124 7.5a2 6.1 4.1
15
2q
9
2115 6
3p
8
36 CHAPTER R A Review of Basic Concepts and Skills R–36
College Algebra Graphs & Models—
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R–37 Section R.3 Solving Linear Equations and Inequalities 37
College Algebra Graphs & Models—
䊳
WORKING WITH FORMULAS
79. Euler’s Polyhedron Formula:
Discovered by Leonhard Euler
in 1752, this simple but
powerful formula states that in
any regular polyhedron, the
number of vertices V and faces
F is always two more than the
number of edges E. (a) Verify
the formula for a simple cube. (b) Verify the
formula for the octahedron shown in the figure.
(c) If a dodecahedron has 12 faces and 30 edges,
how many vertices does it have?
80. Area of a Regular Polygon:
The area of any regular
polygon can be found
using the formula
shown, where a is the
apothem of the polygon
(perpendicular distance
from center to any edge),
and p is the perimeter.
(a) Verify the formula using a square with sides of
length 6 cm. (b) If the hexagon shown has an area
of 259.8 cm
2
with sides 10 cm in length, what is
the length a of the apothem?
A ⴝ
1
2
ap
V ⴙ F ⴚ E ⴝ 2
81. Body mass index:
The U.S. government publishes a body mass index
formula to help people consider the risk of heart
disease. An index “B” of 27 or more means that a
person is at risk. Here W represents weight in
pounds and H represents height in inches. If your
height is what range of weights will help
ensure you remain safe from the risk of heart
disease?
Source:
www.surgeongeneral.gov/topics.
82. Lift capacity:
The capacity in pounds of the lift used by a roofing
company to place roofing shingles and buckets of
roofing nails on rooftops is modeled by the formula
shown, where S represents packs of shingles and B
represents buckets of nails. Use the formula to find
(a) the largest number of shingle packs that can be
lifted, (b) the largest number of nail buckets that
can be lifted, and (c) the largest number of shingle
packs that can be lifted along with three nail
buckets.
75S ⴙ 125B ⱕ 750
5¿8–
B ⴝ
704W
H
2
a
䊳
APPLICATIONS
83. Celebrity Travel: To avoid paparazzi and
overzealous fans, the arrival gates of planes carrying
celebrities are often kept secret until the last possible
moment. While awaiting the arrival of Angelina
Jolie, a large crowd of fans and photographers had
gathered at Terminal A, Gate 18. However, the
number of fans waiting at Gate 32 was twice that
number increased by 5. If there were 73 fans at
Gate 32, how many were waiting at Gate 18? (See
Section R.1, Example 2a.)
84. Famous Architecture: The Hall of Mirrors is the
central gallery of the Palace of Versailles and is one
of the most famous rooms in the world. The length
of this hall is 11 m less than 8 times the width. If
the hall is 73 m long, what is its width? (See
Section R.1, Example 2b.)
85. Dietary Goals: At the picnic, Mike abandoned his
diet and consumed 13 calories more than twice the
number of calories he normally allots for lunch. If
he consumed 1467 calories, how many calories are
normally allotted for lunch?
86. Marathon Training: While training for the
Chicago marathon, Christina’s longest run of the
week was 5 mi less than double the shortest. If the
longest run was 11.2 mi, how long was the
shortest?
87. Actor’s Ages: At the time of this writing, actor
Will Smith (Enemy of the State, Seven Pounds,
others), was 1 yr older than two-thirds the age of
Samuel Jackson (The Negotiator, Die Hard III,
others). If Will Smith was 41 at this time, how old
was Samuel Jackson?
88. Football versus Fútbol: The area of a regulation
field for American football is about 410 square
meters (m
2
) less than three-fifths of an Olympic-
sized soccer field. If an American football field
covers 5350 m
2
, what is the area of an Olympic
soccer field?
Write an equation to model the given information and solve.
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38 CHAPTER R A Review of Basic Concepts and Skills R–38
College Algebra Graphs & Models—
96. Area of a triangle: Using the triangle shown, find
the height that will guarantee an area equal to or
greater than 48 in
2
.
97. Heating and cooling subsidies: As long as the
outside temperature is over and less than
the city does not issue heating or
cooling subsidies for low-income families. What is
the corresponding range of Celsius temperatures
C? Recall that
98. U.S. and European shoe sizes: To convert a
European male shoe size “E” to an American male
shoe size “A,” the formula can be
used. Lillian has five sons in the U.S. military, with
shoe sizes ranging from size 9 to size 14
What is the corresponding range of
European sizes? Round to the nearest half-size.
99. Power tool rentals: Sunshine Equipment Co. rents
its power tools for a $20 fee, plus $4.50/hr.
Kealoha’s Rentals offers the same tools for an $11
fee plus $6.00/hr. How many hours h must a tool
be rented to make the cost at Sunshine a better
deal?
100. Moving van rentals: Stringer Truck Rentals will
rent a moving van for $15.75/day plus $0.35 per
mile. Bertz Van Rentals will rent the same van for
$25/day plus $0.30 per mile. How many miles m
must the van be driven to make the cost at Bertz a
better deal?
19 A 142.
A 0.76E 23
F
9
5
C 32.
145 6 F 6 852,
85°F45°F
12 in.
h
91. Exam scores: Jacques is going to college on an
academic scholarship that requires him to maintain at
least a 75% average in all of his classes. So far he has
scored 82%, 76%, 65%, and 71% on four exams.
What scores are possible on his last exam that will
enable him to keep his scholarship?
92. Timed trials: In the first three trials of the 100-m
butterfly, Johann had times of 50.2, 49.8, and
50.9 sec. How fast must he swim the final timed trial
to have an average time of at most 50 sec?
93. Checking account balance: If the average daily
balance in a certain checking account drops below
$1000, the bank charges the customer a $7.50
service fee. The table
gives the daily balance
for one customer. What
must the daily balance be
for Friday to avoid a
service charge?
94. Average weight: In the
National Football
League, many consider
an offensive line to be
“small” if the average
weight of the five down
linemen is less than
325 lb. Using the table,
what must the weight of the right tackle be so that
the line will not be considered small?
95. Area of a rectangle: Given the rectangle shown,
what is the range of values for the width, in order
to keep the area less than 150m
2
?
w
20 m
Weekday Balance
Monday $1125
Tuesday $850
Wednesday $625
Thursday $400
Lineman Weight
Left tackle 318 lb
Left guard 322 lb
Center 326 lb
Right guard 315 lb
Right tackle ?
89. Forensic Studies: In forensic studies, skeletal
remains are analyzed to determine the height,
gender, race, age, and other characteristics of the
decedent. For instance, the height of a male
individual is approximated as 34 in. more than
three and one-third times the length of the radial
bone. If a live individual is 74 in. tall, how long is
his radial bone?
90. Famous Waterways: The Suez Canal and the
Panama Canal are two of the most important
waterways in the world, saving ships thousands of
miles as they journey from port to destination. The
length of the Suez Canal is 39 kilometers (km) less
than three times the length of the Panama Canal. If
the Egyptian canal is 192 km long, how long is the
Central American canal?
Write an inequality to model the given information and solve.
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R–39 Section R.4 Factoring Polynomials and Solving Polynomial Equations by Factoring 39
College Algebra Graphs & Models—
101. Cost of drywall:After
the studs are up, the
wall shown in the figure
must be covered in
drywall. (a) How many
square feet of drywall
are needed? (b) If drywall
is sold only in 4-ft by 8-ft sheets, approximately how
many sheets are required for this job?
102. Paving a walkway: Current plans
call for building a circular fountain
6 m in diameter with a circular
walkway around it that is 1.5 m
wide. (a) What is the approximate
area of the walkway? (b) If the
concrete for the walkway is to be 6 cm deep, what
volume of cement must be used ?
11 cm ⫽ 0.01 m2
103
. Trophy bases: The
base of a new trophy
has the form of a
cylinder sitting atop
a rectangular solid.
If the base is to be
cast in a special
aluminum, determine the volume of aluminum to
be used.
104. Grain storage: The dimensions of
a grain silo are shown in the figure.
If the maximum storage capacity of
the silo is 95% of the total volume
of the silo, how many cubic meters
of corn can be stored?
10 in.
10 in.
2 in.
7 in.
5 in.
6 m
1.5 m
16 m
6 m
䊳
EXTENDING THE CONCEPT
105. Solve for x:
106. Solve for n:
107. Use your local library, the Internet, or another
resource to find the highest and lowest point on each
of the seven continents. Express the range of altitudes
for each continent as a joint inequality. Which
continent has the greatest range?
108. The sum of two consecutive even integers is greater
than or equal to 12 and less than or equal to 22. List
all possible values for the two integers.
⫺655 ⫹ 23n ⫺ 1019 ⫹ n246
553 ⫺ 34 ⫺ 215 ⫺ 9n246 ⫹ 15 ⫽
614 ⫺ x ⫺ 2x
2
2
⫺
19
⫺314x
2
⫹ 5x ⫺ 22⫹ 7x ⫽
Place the correct inequality symbol in the blank to make
the statement true.
109. If and , then mn 0.
110. If and , then mp np.
111. If and , then mp np.
112. If and , then mp np.
113. If , then .
114. If , then .
115. If and , then m
2
n.
116. If , then m
3
0.m 6 0
n 6 0m 7 0
1
n
1
m
0 6 m 6 n
⫺n
⫺mm 7 n
p 6 0m ⱕ n
p 7 0m 6 n
p 7 0m 7 n
n 6 0m 7 0
R.4 Factoring Polynomials and Solving Polynomial
Equations by Factoring
It is often said that knowing which tool to use is just as important as knowing how to use
the tool. In this section, we review the tools needed to factor an expression, an important
part of solving polynomial equations. This section will also help us decide which factor-
ing tool is appropriate when many different factorable expressions are presented.
A. The Greatest Common Factor
To factor an expression means to rewrite the expression as an equivalent product. The
distributive property is an example of factoring in action. To factor we might
first rewrite each term using the common factor then
apply the distributive property to obtain We commonly say that we have
factored out 2x. The greatest common factor (or GCF) is the largest factor common
to all terms in the polynomial.
2x1x ⫹ 32.
2x: 2x
2
⫹ 6x ⫽ 2x
#
x ⫹ 2x
#
3,
2x
2
⫹ 6x,
LEARNING OBJECTIVES
In Section R.4 you will review:
A. Factoring out the greatest
common factor
B. Common binomial factors
and factoring by grouping
C. Factoring quadratic
polynomials
D. Factoring special forms
and quadratic forms
E. Solving Polynomial
Equations by Factoring
19 ft
10 ft
15 ft
3 ft
7 ft
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