4: Equations Involving Parentheses and Fractional Forms
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112
Chapter 3 • Equations, Inequalities, and Problem Solving
Classroom Example
Solve 3(m Ϫ 1) ϭ 6 (m ϩ 2).
EXAMPLE 2
Solve 41x ϩ 32 ϭ 21x Ϫ 62 .
Solution
41x ϩ 32 ϭ 21x Ϫ 62
4x ϩ 12 ϭ 2x Ϫ 12
2x ϩ 12 ϭ Ϫ12
2x ϭ Ϫ24
x ϭ Ϫ12
Applied distributive property on each side
Subtracted 2x from both sides
Subtracted 12 from both sides
Divided both sides by 2
The solution set is 5Ϫ126.
It may be necessary to remove more than one set of parentheses and then to use the
distributive property again to combine similar terms. Consider the following two examples.
Classroom Example
Solve 2(t ϩ 4) ϩ 6(t ϩ 5) ϭ 32.
EXAMPLE 3
Solve 51w ϩ 32 ϩ 31w ϩ 12 ϭ 14.
Solution
51w ϩ 32 ϩ 31w ϩ 12 ϭ 14
5w ϩ 15 ϩ 3w ϩ 3 ϭ 14
8w ϩ 18 ϭ 14
8w ϭ Ϫ4
4
8
1
wϭϪ
2
wϭϪ
Applied distributive property
Combined similar terms
Subtracted 18 from both sides
Divided both sides by 8
Reduced
1
The solution set is eϪ f .
2
Classroom Example
Solve 4(w Ϫ 3) Ϫ 8(w Ϫ 2) ϭ 9.
EXAMPLE 4
Solve 61x Ϫ 72 Ϫ 21x Ϫ 42 ϭ 13.
Solution
61x Ϫ 72 Ϫ 21x Ϫ 42 ϭ 13
6x Ϫ 42 Ϫ 2x ϩ 8 ϭ 13
4x Ϫ 34 ϭ 13
4x ϭ 47
47
xϭ
4
The solution set is e
47
f.
4
Be careful with this sign!
Distributive property
Combined similar terms
Added 34 to both sides
Divided both sides by 4
3.4 • Equations Involving Parentheses and Fractional Forms
113
2
3
2
ϭ by adding to both sides.
3
4
3
If an equation contains several fractions, then it is usually easier to clear the equation of all
fractions by multiplying both sides by the least common denominator of all the denominators.
Perhaps several examples will clarify this idea.
In a previous section, we solved equations like x Ϫ
Classroom Example
3
7
1
Solve y ϩ ϭ .
3
4
12
EXAMPLE 5
1
2
5
Solve x ϩ ϭ .
2
3
6
Solution
1
2
5
xϩ ϭ
2
3
6
2
5
1
6a x ϩ b ϭ 6a b
2
3
6
6 is the LCD of 2, 3, and 6
1
2
5
6 a xb ϩ 6 a b ϭ 6 a b
2
3
6
Distributive property
3x ϩ 4 ϭ 5
Note how the equation has been cleared of all fractions
3x ϭ 1
1
xϭ
3
1
The solution set is e f .
3
Classroom Example
5x
2
4
Solve
Ϫ ϭ .
12
3
9
EXAMPLE 6
Solve
5n
1
3
Ϫ ϭ .
6
4
8
Solution
5n
1
3
Ϫ ϭ
6
4
8
24 a
24 a
1
3
5n
Ϫ b ϭ 24 a b
6
4
8
5n
1
3
b Ϫ 24 a b ϭ 24 a b
6
4
8
24 is the LCD of 6, 4, and 8
Distributive property
20n Ϫ 6 ϭ 9
20n ϭ 15
nϭ
15
3
ϭ
20
4
3
The solution set is e f .
4
We use many of the ideas presented in this section to help solve the equations in the next
examples. Study the solutions carefully and be sure that you can supply reasons for each step.
It might be helpful to cover up the solutions and try to solve the equations on your own.
114
Chapter 3 • Equations, Inequalities, and Problem Solving
Classroom Example
sϩ5
sϩ2
11
Solve
ϩ
ϭ .
4
5
20
EXAMPLE 7
Solve
xϩ3
xϩ4
3
ϩ
ϭ .
2
5
10
Solution
xϩ3 xϩ4
3
ϩ
ϭ
2
5
10
10 a
10 a
xϩ3 xϩ4
3
ϩ
b ϭ 10 a b
2
5
10
xϩ3
xϩ4
3
b ϩ 10 a
b ϭ 10 a b
2
5
10
10 is the LCD of 2, 5, and 10
Distributive property
51x ϩ 32 ϩ 21x ϩ 42 ϭ 3
5x ϩ 15 ϩ 2x ϩ 8 ϭ 3
7x ϩ 23 ϭ 3
7x ϭ Ϫ20
xϭϪ
The solution set is eϪ
Classroom Example
vϪ3
1
vϪ5
Ϫ
ϭ .
Solve
4
5
2
20
7
20
f.
7
EXAMPLE 8
Solve
xϪ2
2
xϪ1
Ϫ
ϭ .
4
6
3
Solution
xϪ1
xϪ2
2
Ϫ
ϭ
4
6
3
12 a
12 a
xϪ1
xϪ2
2
Ϫ
b ϭ 12 a b
4
6
3
xϪ2
2
xϪ1
b Ϫ 12 a
b ϭ 12 a b
4
6
3
12 is the LCD of 4, 6, and 3
Distributive property
31x Ϫ 12 Ϫ 21x Ϫ 22 ϭ 8
3x Ϫ 3 Ϫ 2x ϩ 4 ϭ 8
Be careful with this sign!
xϩ1ϭ8
The solution set is 576 .
xϭ7
Word Problems
We are now ready to solve some word problems using equations of the different types presented in this section. Again, it might be helpful for you to attempt to solve the problems on
your own before looking at the book’s approach.
3.4 • Equations Involving Parentheses and Fractional Forms
Classroom Example
Ian has 23 coins (dimes and nickels)
that amount to $1.45. How many
coins of each kind does he have?
115
EXAMPLE 9
Loretta has 19 coins (quarters and nickels) that amount to $2.35. How many coins of each
kind does she have?
Solution
Let q represent the number of quarters. Then 19 Ϫ q represents the number of nickels. We
can use the following guideline to help set up an equation:
Value of quarters in cents ϩ Value of nickels in cents ϭ Total value in cents
ϩ
25q
5(19 Ϫ q)
ϭ
235
Solving the equation, we obtain
25q ϩ 95 Ϫ 5q ϭ 235
20q ϩ 95 ϭ 235
20q ϭ 140
qϭ7
If q ϭ 7, then 19 Ϫ q ϭ 12, so she has 7 quarters and 12 nickels.
Classroom Example
Find a number such that 6 more than
three-fourths the number is equal to
two-thirds the number.
EXAMPLE 10
Find a number such that 4 less than two-thirds of the number is equal to one-sixth of the number.
Solution
Let n represent the number. Then
2
n Ϫ 4 represents 4 less than two-thirds of the number,
3
1
and n represents one-sixth of the number.
6
1
2
nϪ4ϭ n
3
6
2
1
6 a n Ϫ 4b ϭ 6 a nb
3
6
4n Ϫ 24 ϭ n
3n Ϫ 24 ϭ 0
3n ϭ 24
nϭ8
The number is 8.
Classroom Example
1
John is paid 1 times his normal
2
hourly rate for each hour he works
over 40 hours in a week. Last week
he worked 48 hours and earned $962.
What is his normal hourly rate?
EXAMPLE 11
1
Lance is paid 1 times his normal hourly rate for each hour he works over 40 hours in a week.
2
Last week he worked 50 hours and earned $462. What is his normal hourly rate?
Solution
3
1
Let x represent Lance’s normal hourly rate. Then x represents 1 times his normal hourly
2
2
rate. We can use the following guideline to set up the equation:
116
Chapter 3 • Equations, Inequalities, and Problem Solving
Regular wages for first 40 hours ϩ Wages for 10 hours of overtime ϭ Total wages
40x
ϩ
3
10a xb
2
ϭ
462
Solving this equation, we obtain
40x ϩ 15x ϭ 462
55x ϭ 462
x ϭ 8.40
Lance’s normal hourly rate is $8.40.
Classroom Example
Find two consecutive whole numbers
such that the sum of the first plus
four times the second is 179.
EXAMPLE 12
Find three consecutive whole numbers such that the sum of the first plus twice the second
plus three times the third is 134.
Solution
Let n represent the first whole number. Then n ϩ 1 represents the second whole number and
n ϩ 2 represents the third whole number.
n ϩ 21n ϩ 12 ϩ 31n ϩ 22 ϭ 134
n ϩ 2n ϩ 2 ϩ 3n ϩ 6 ϭ 134
6n ϩ 8 ϭ 134
6n ϭ 126
n ϭ 21
The numbers are 21, 22, and 23.
Keep in mind that the problem-solving suggestions we offered in Section 3.3 simply outline a general algebraic approach to solving problems. You will add to this list throughout this
course and in any subsequent mathematics courses that you take. Furthermore, you will be able
to pick up additional problem-solving ideas from your instructor and from fellow classmates
as problems are discussed in class. Always be on the alert for any ideas that might help you
become a better problem solver.
Concept Quiz 3.4
For Problems 1–10, answer true or false.
1. To solve an equation of the form a (x ϩ b) ϭ 14, the associative property would be
applied to remove the parentheses.
2. Multiplying both sides of an equation by the common denominator of all fractions in the
equation clears the equation of all fractions.
3. If Jack has 15 coins (dimes and quarters), and x represents the number of dimes, then x
Ϫ 15 represents the number of quarters.
4. The equation 3(x ϩ 1) ϭ 3x ϩ 3 has an infinite number of solutions.
5. The equation 2x ϭ 0 has no solution.
3.4 • Equations Involving Parentheses and Fractional Forms
117
The equation 4x ϩ 5 ϭ 4x ϩ 3 has no solution.
The solution set for the equation 3(2x Ϫ 1) ϭ 2x Ϫ 3 is {0}.
The solution set for the equation 5(3x ϩ 2) ϭ 4 (2x Ϫ 1) is {2}.
The answer for a word problem must be checked back into the statement of the problem.
xϩ1
xϪ3
5
7
10. The solution set for the equation
Ϫ
ϭ is e f .
2
4
8
2
6.
7.
8.
9.
Problem Set 3.4
For Problems 1–60, solve each equation. (Objectives 1 and 2)
1. 71x ϩ 22 ϭ 21
2. 41x ϩ 42 ϭ 24
3. 51x Ϫ 32 ϭ 35
4. 61x Ϫ 22 ϭ 18
5. Ϫ31x ϩ 52 ϭ 12
6. Ϫ51x Ϫ 62 ϭ Ϫ15
7. 41n Ϫ 62 ϭ 5
8. 31n ϩ 42 ϭ 7
9. 61n ϩ 72 ϭ 8
10. 81n Ϫ 32 ϭ 12
11. Ϫ10 ϭ Ϫ51t Ϫ 82
12. Ϫ16 ϭ Ϫ41t ϩ 72
13. 51x Ϫ 42 ϭ 41x ϩ 62
14. 61x Ϫ 42 ϭ 312x ϩ 52
15. 81x ϩ 12 ϭ 91x Ϫ 22
16. 41x Ϫ 72 ϭ 51x ϩ 22
17. 81t ϩ 52 ϭ 412t ϩ 102
18. 71t Ϫ 52 ϭ 51t ϩ 32
19. 216t ϩ 12 ϭ 413t Ϫ 12
28. 41x Ϫ 12 ϩ 51x ϩ 22 ϭ 31x Ϫ 82
29. Ϫ1x ϩ 22 ϩ 21x Ϫ 32 ϭ Ϫ21x Ϫ 72
30. Ϫ21x ϩ 62 ϩ 313x Ϫ 22 ϭ Ϫ31x Ϫ 42
31. 512x Ϫ 12 Ϫ 13x ϩ 42 ϭ 41x ϩ 32 Ϫ 27
32. 314x ϩ 12 Ϫ 212x ϩ 12 ϭ Ϫ21x Ϫ 12 Ϫ 1
33. Ϫ1a Ϫ 12 Ϫ 13a Ϫ 22 ϭ 6 ϩ 21a Ϫ 12
34. 312a Ϫ 12 Ϫ 215a ϩ 12 ϭ 413a ϩ 42
35. 3(x Ϫ 1) ϩ 2(x Ϫ 3) ϭ Ϫ4(x Ϫ 2) ϩ 10(x ϩ 4)
36. Ϫ21x Ϫ 42 Ϫ 13x Ϫ 22 ϭ Ϫ2 ϩ 1Ϫ6x ϩ 22
37. 3 Ϫ 71x Ϫ 12 ϭ 9 Ϫ 612x ϩ 12
38. 8 Ϫ 512x ϩ 12 ϭ 2 Ϫ 61x Ϫ 32
3
2 5
39. x Ϫ ϭ
4
3 6
1
4
5
40. x Ϫ ϭ Ϫ
2
3
6
5
1
9
41. x ϩ ϭ Ϫ
6
4
4
3
1
7
42. x ϩ ϭ Ϫ
8
6
12
1
3 3
43. x Ϫ ϭ
2
5 4
1
2 5
44. x Ϫ ϭ
4
5 6
20. 61t ϩ 52 ϭ 213t ϩ 152
21. Ϫ21x Ϫ 62 ϭ Ϫ1x Ϫ 92
22. Ϫ1x ϩ 72 ϭ Ϫ21x ϩ 102
45.
n 5n 1
ϩ
ϭ
3
6
8
46.
n 3n
5
ϩ
ϭ
6
8
12
47.
5y 3 2y
Ϫ ϭ
6
5
3
48.
3y 1
y
ϩ ϭ
7
2 4
49.
h h
ϩ ϭ1
6 8
50.
h h
ϩ ϭ1
4 3
51.
x ϩ 2 x ϩ 3 13
ϩ
ϭ
3
4
3
52.
x Ϫ 1 x ϩ 2 39
ϩ
ϭ
4
5
20
23. Ϫ31t Ϫ 42 Ϫ 21t ϩ 42 ϭ 9
24. 51t Ϫ 42 Ϫ 31t Ϫ 22 ϭ 12
25. 31n Ϫ 102 Ϫ 51n ϩ 122 ϭ Ϫ86
26. 41n ϩ 92 Ϫ 71n Ϫ 82 ϭ 83
27. 31x ϩ 12 ϩ 412x Ϫ 12 ϭ 512x ϩ 32
118
Chapter 3 • Equations, Inequalities, and Problem Solving
53.
xϪ1 xϩ4
13
Ϫ
ϭϪ
5
6
15
54.
xϩ1 xϪ3 4
Ϫ
ϭ
7
5
5
x ϩ 8 x ϩ 10 3
55.
Ϫ
ϭ
2
7
4
56.
xϩ7 xϩ9 5
Ϫ
ϭ
3
6
9
57.
xϪ2
xϩ1
Ϫ1ϭ
8
4
58.
xϪ4
xϪ2
ϩ3ϭ
2
4
59.
xϩ1 xϪ3
ϭ
ϩ2
4
6
60.
xϩ3 xϪ6
ϭ
ϩ1
5
2
Solve each word problem by setting up and solving an
appropriate algebraic equation. (Objective 3)
61. Find two consecutive whole numbers such that the smaller
number plus four times the larger number equals 39.
62. Find two consecutive whole numbers such that the
smaller number subtracted from five times the larger
number equals 57.
63. Find three consecutive whole numbers such that twice
the sum of the two smallest numbers is 10 more than
three times the largest number.
64. Find four consecutive whole numbers such that the sum
of the first three numbers equals the fourth number.
65. The sum of two numbers is 17. If twice the smaller number is 1 more than the larger number, find the numbers.
66. The sum of two numbers is 53. If three times the smaller
number is 1 less than the larger number, find the numbers.
67. Find a number such that 20 more than one-third of the
number equals three-fourths of the number.
71. Suppose that a board 20 feet long is cut into two pieces.
Four times the length of the shorter piece is 4 feet less
than three times the length of the longer piece. Find the
length of each piece.
72. Ellen is paid time and a half for each hour over
40 hours she works in a week. Last week she worked 44
hours and earned $391. What is her normal hourly rate?
73. Lucy has 35 coins consisting of nickels and quarters
amounting to $5.75. How many coins of each kind does
she have?
74. Suppose that Julian has 44 coins consisting of pennies
and nickels. If the number of nickels is two more than
twice the number of pennies, find the number of coins of
each kind.
75. Max has a collection of 210 coins consisting of nickels,
dimes, and quarters. He has twice as many dimes as
nickels, and 10 more quarters than dimes. How many
coins of each kind does he have?
76. Ginny has a collection of 425 coins consisting of
pennies, nickels, and dimes. She has 50 more nickels
than pennies and 25 more dimes than nickels. How
many coins of each kind does she have?
77. Maida has 18 coins consisting of dimes and quarters
amounting to $3.30. How many coins of each kind does
she have?
78. Ike has some nickels and dimes amounting to $2.90. The
number of dimes is one less than twice the number of
nickels. How many coins of each kind does
he have?
79. Mario has a collection of 22 specimens in his aquarium
consisting of crabs, fish, and plants. There are three
times as many fish as crabs. There are two more plants
than crabs. How many specimens of each kind are in the
collection?
80. Tickets for a concert were priced at $8 for students and
$10 for nonstudents. There were 1500 tickets sold for a
total of $12,500. How many student tickets were sold?
N
N
NO UDE
ST
C
at
L
Se
O
50
O
w
C
Ro
T
S
.00
E
0
$1
N
w
50
at
Se
25
Figure 3.2
T
Row 03 Seat 10
U
Ro
70. Raoul received a $30 tip for waiting on a large party.
This was $5 more than one-fourth of the tip the headwaiter received. How much did the headwaiter receive
for a tip?
OOL
25
69. Mrs. Nelson had to wait 4 minutes in line at her bank’s
automated teller machine. This was 3 minutes less than
one-half of the time she waited in line at the grocery
store. How long in minutes did she wait in line at the
grocery store?
STUDENT
T
68. The sum of three-eighths of a number and five-sixths of
the same number is 29. Find the number.
$8.00
UNES
Row 03 Seat 10
3.4 • Equations Involving Parentheses and Fractional Forms
119
81. The supplement of an angle is 30Њ larger than twice its
complement. Find the measure of the angle.
85. The supplement of an angle is 10Њ smaller than three
times its complement. Find the size of the angle.
82. The sum of the measure of an angle and three times its
complement is 202Њ. Find the measure of the angle.
86. In triangle ABC, the measure of angle C is eight times
the measure of angle A, and the measure of angle B is
10Њ more than the measure of angle C. Find the measure
of each angle of the triangle.
83. In triangle ABC, the measure of angle A is 2Њ less
than one-fifth of the measure of angle C. The measure
of angle B is 5Њ less than one-half of the measure of
angle C. Find the measures of the three angles of the
triangle.
84. If one-fourth of the complement of an angle plus onefifth of the supplement of the angle equals 36Њ, find the
measure of the angle.
Additional word problems can be found in Appendix B.
All of the problems in the Appendix marked as (3.4) are
appropriate for this section.
Thoughts Into Words
87. Discuss how you would solve the equation
89. Consider these two solutions:
3(x Ϫ 2) Ϫ 5(x ϩ 3) ϭ Ϫ4(x ϩ 9).
31x ϩ 22 ϭ 9
88. Why must potential answers to word problems be
checked back into the original statement of the
problem?
31x ϩ 22
3
ϭ
31x Ϫ 42 ϭ 7
3x Ϫ 12 ϭ 7
3x ϭ 19
19
xϭ
3
9
3
xϩ2ϭ3
xϭ1
Are both of these solutions correct? Comment on the
effectiveness of the two different approaches.
Further Investigations
90. Solve each equation.
(f)
(a) Ϫ21x Ϫ 12 ϭ Ϫ2x ϩ 2
(g) 41x Ϫ 22 Ϫ 21x ϩ 32 ϭ 21x ϩ 62
(b) 31x ϩ 42 ϭ 3x Ϫ 4
(h) 51x ϩ 32 Ϫ 31x Ϫ 52 ϭ 21x ϩ 152
(c) 51x Ϫ 12 ϭ Ϫ5x Ϫ 5
(d)
xϪ3
ϩ4ϭ3
3
(i) 71x Ϫ 12 ϩ 41x Ϫ 22 ϭ 151x Ϫ 12
91. Find three consecutive integers such that the sum of the
smallest integer and the largest integer is equal to twice
the middle integer.
xϩ2
xϪ2
(e)
ϩ1ϭ
3
3
Answers to the Concept Quiz
1. False
2. True
3. False
4. True
9. True
10. False
xϪ1
x Ϫ 11
Ϫ2ϭ
5
5
5. False
6. True
7. True
8. False
120
Chapter 3 • Equations, Inequalities, and Problem Solving
3.5
Inequalities
OBJECTIVES
1
Solve ﬁrst-degree inequalities
2
Write the solution set of an inequality in set-builder notation or interval notation
3
Graph the solution set of an inequality
Just as we use the symbol ϭ to represent is equal to, we use the symbols Ͻ and Ͼ to represent is less than and is greater than, respectively. Here are some examples of statements of
inequality. Notice that the first four are true statements and the last two are false.
6ϩ4Ͼ7
True
8 Ϫ 2 Ͻ 14
True
# 8Ͼ4 # 6
5 # 2Ͻ5 # 7
True
5 ϩ 8 Ͼ 19
False
9Ϫ2Ͻ3
False
4
True
Algebraic inequalities contain one or more variables. These are examples of algebraic
inequalities:
xϩ3Ͼ4
2x Ϫ 1 Ͻ 6
x ϩ 2x Ϫ 1 Ͼ 0
2
2x ϩ 3y Ͻ 7
7ab Ͻ 9
An algebraic inequality such as x ϩ 1 Ͼ 2 is neither true nor false as it stands; it is called
an open sentence. Each time a number is substituted for x, the algebraic inequality
x ϩ 1 Ͼ 2 becomes a numerical statement that is either true or false. For example, if x ϭ 0,
then x ϩ 1 Ͼ 2 becomes 0 ϩ 1 Ͼ 2, which is false. If x ϭ 2, then x ϩ 1 Ͼ 2 becomes
2 ϩ 1 Ͼ 2, which is true. Solving an inequality refers to the process of finding the numbers
that make an algebraic inequality a true numerical statement. We say that such numbers,
called the solutions of the inequality, satisfy the inequality. The set of all solutions of an
inequality is called its solution set. We often state solution sets for inequalities with set
builder notation. For example, the solution set for x ϩ 1 Ͼ 2 is the set of real numbers
greater than 1, expressed as 5x 0 x Ͼ 16. The set builder notation 5x 0 x Ͼ 16 is read as
“the set of all x such that x is greater than 1.” We sometimes graph solution sets
for inequalities on a number line; the solution set for 5x 0 x Ͼ 16 is pictured in
Figure 3.3.
Ϫ4 Ϫ3 Ϫ2 Ϫ1
0
1
2
3
4
Figure 3.3
The left-hand parenthesis at 1 indicates that 1 is not a solution, and the red part of the line to
the right of 1 indicates that all real numbers greater than 1 are solutions. We refer to the red
portion of the number line as the graph of the solution set 5x 0 x Ͼ 16.
The solution set for x ϩ 1 Յ 3 (Յ is read “less than or equal to”) is the set of real numbers less than or equal to 2, expressed as 5x 0 x Յ 26. The graph of the solution set for
3.5 • Inequalities
121
5x 0x Յ 26 is pictured in Figure 3.4. The right-hand bracket at 2 indicates that 2 is included in
the solution set.
Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1
0
1
2
3
4
5
Figure 3.4
It is convenient to express solution sets of inequalities using interval notation. The solution
set 5 x0 x Ͼ 6 6 is written as (6, q) using interval notation. In interval notation, parentheses are
used to indicate exclusion of the endpoint. The Ͼ and Ͻ symbols in inequalities also indicate
the exclusion of the endpoint. So when the inequality has a Ͼ or Ͻ symbol, the interval notation uses a parenthesis. This is consistent with the use of parentheses on the number line.
In this same example, 5x0 x Ͼ 66 , the solution set has no upper endpoint, so the infinity
symbol, q , is used to indicate that the interval continues indefinitely. The solution set for
5x0 x Ͻ 36 is written as (Ϫq, 3) in interval notation. Here the solution set has no lower endpoint, so a negative sign precedes the infinity symbol because the interval is extending
indefinitely in the opposite direction. The infinity symbol always has a parenthesis in interval
notation because there is no actual endpoint to include.
The solution set 5x0 x Ն 56 is written as [ 5, q) using interval notation. In interval notation square brackets are used to indicate inclusion of the endpoint. The Ն and Յ symbols
in inequalities also indicate the inclusion of the endpoint. So when the inequality has a
Ն or Յ symbol, the interval notation uses a square bracket. Again the use of a bracket in
interval notation is consistent with the use of a bracket on the number line.
The examples in the table below contain some simple algebraic inequalities, their solution sets, graphs of the solution sets, and the solution sets written in interval notation. Look
them over very carefully to be sure you understand the symbols.
Algebraic
inequality
Solution
set
xϽ2
5x 0x Ͻ 26
x Ͼ Ϫ1
5x 0x Ͼ Ϫ16
3Ͻx
5x 0x Ͼ 36
5x 0x Ն 16
xՆ1
(Ն is read “greater
than or equal to”)
5x 0x Յ 26
xՅ2
(Յ is read “less than
or equal to”)
5x 0x Յ 16
1Նx
Graph of solution set
Interval
notation
(Ϫq, 2)
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5
(Ϫ1, q)
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5
(3, q)
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5
΄1, q)
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5
Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1 0 1 2 3 4 5
(Ϫq, 2΅
(Ϫq, 1΅
Figure 3.5
The general process for solving inequalities closely parallels that for solving equations. We
continue to replace the given inequality with equivalent, but simpler inequalities. For example,
2x ϩ 1 Ͼ 9
(1)
2x Ͼ 8
(2)
xϾ4
(3)
122
Chapter 3 • Equations, Inequalities, and Problem Solving
are all equivalent inequalities; that is, they have the same solutions. Thus to solve inequality
(1), we can solve inequality (3), which is obviously all numbers greater than 4. The exact
procedure for simplifying inequalities is based primarily on two properties, and they
become our topics of discussion at this time. The first of these is the addition-subtraction
property of inequality.
Property 3.4 Addition-Subtraction Property of Inequality
For all real numbers a, b, and c,
1. a Ͼ b if and only if a ϩ c Ͼ b ϩ c.
2. a Ͼ b if and only if a Ϫ c Ͼ b Ϫ c.
Property 3.4 states that any number can be added to or subtracted from both sides of an
inequality, and an equivalent inequality is produced. The property is stated
in terms of Ͼ, but analogous properties exist for Ͻ, Ն, and Յ. Consider the use of this property in the next three examples.
Classroom Example
Solve x Ϫ 5 Ͻ Ϫ 3 and graph the
solutions.
Solve x Ϫ 3 Ͼ Ϫ1 and graph the solutions.
EXAMPLE 1
Solution
x Ϫ 3 Ͼ Ϫ1
x Ϫ 3 ϩ 3 Ͼ Ϫ1 ϩ 3
xϾ2
Add 3 to both sides
The solution set is 5x 0x Ͼ 26, and it can be graphed as shown in Figure 3.6. The
solution, written in interval notation, is (2, q) .
Ϫ4 Ϫ3 Ϫ2 Ϫ1
0
1
2
3
4
Figure 3.6
Classroom Example
Solve x ϩ 7 Ն 10 and graph the
solutions.
Solve x ϩ 4 Յ 5 and graph the solutions.
EXAMPLE 2
Solution
xϩ4Յ5
xϩ4Ϫ4Յ5Ϫ4
Subtract 4 from both sides
xՅ1
The solution set is 5x0 x Յ 16 , and it can be graphed as shown in Figure 3.7. The
solution, written in interval notation, is (Ϫq, 1΅.
Ϫ4 Ϫ3 Ϫ2 Ϫ1
Figure 3.7
0
1
2
3
4