6: Inequalities, Compound Inequalities, and Problem Solving
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128
Chapter 3 • Equations, Inequalities, and Problem Solving
Classroom Example
Solve
21x Ϫ 52 ϩ 71x ϩ 12 Յ 31x Ϫ 22 .
Solve 41x ϩ 32 ϩ 31x Ϫ 42 Ն 21x Ϫ 12 .
EXAMPLE 2
Solution
4(x ϩ 3) ϩ 3(x Ϫ 4) Ն 2(x Ϫ 1)
4x ϩ 12 ϩ 3x Ϫ 12 Ն 2x Ϫ 2
7x Ն 2x Ϫ 2
7x Ϫ 2x Ն 2x Ϫ 2 Ϫ 2x
5x Ն Ϫ2
5x
Ϫ2
Ն
5
5
2
xՆϪ
5
Distributive property
Combine similar terms
Subtract 2x from both sides
Divide both sides by 5
2
2
The solution set is ex0 x Ն Ϫ f or c Ϫ , qb in interval notation.
5
5
Classroom Example
3
1
2
Solve n Ϫ n Ͼ .
8
2
3
3
1
3
Solve Ϫ n ϩ n Ͻ .
2
6
4
EXAMPLE 3
Solution
1
3
3
Ϫ nϩ nϽ
2
6
4
1
3
3
12aϪ n ϩ nb Ͻ 12a b
2
6
4
3
1
3
12aϪ nb ϩ 12a nb Ͻ 12a b
2
6
4
Multiply both sides by 12, the LCD
of all denominators
Distributive property
Ϫ18n ϩ 2n Ͻ 9
Ϫ16n Ͻ 9
Ϫ16n
9
Ͼ
Ϫ16
Ϫ16
nϾϪ
The solution set is e n 0 n Ͼ Ϫ
Divide both sides by Ϫ16, which
reverses the inequality
9
16
9
9
f or aϪ , qb in interval notation.
16
16
9
In Example 3 we are claiming that all numbers greater than Ϫ will satisfy the original
16
inequality. Let’s check one number; we will check 0.
3
1
3
Ϫ nϩ nϽ
2
6
4
? 3
3
1
Ϫ (0) ϩ (0) Ͻ
2
6
4
0Ͻ
3
4
The check resulted in a true statement, which means that 0 is in the solution
set. Had we forgotten to reverse the inequality sign when we divided both sides by Ϫ16, then
3.6 • Inequalities, Compound Inequalities, and Problem Solving
129
the solution set would have been e n|n Ͻ Ϫ
9
f . Zero would not have been a member of that
16
solution set, and we would have detected the error by the check.
Compound Inequalities
The words “and” and “or” are used in mathematics to form compound statements. We use
“and” and “or” to join two inequalities to form a compound inequality.
Consider the compound inequality
x Ͼ 2 and x Ͻ 5
For the solution set, we must find values of x that make both inequalities true statements.
The solution set of a compound inequality formed by the word “and” is the intersection of
the solution sets of the two inequalities. The intersection of two sets, denoted by ʝ , contains
the elements that are common to both sets. For example, if A ϭ 51, 2, 3, 4, 5, 66 and
B ϭ 50, 2, 4, 6, 8, 106 , then A ʝ B ϭ 52, 4, 66 . So to find the solution set of the compound inequality x Ͼ 2 and x Ͻ 5, we find the solution set for each inequality and then
determine the solutions that are common to both solution sets.
Classroom Example
Graph the solution set for the compound inequality x Ͼ 3 and x Ͻ 7,
and write the solution set in interval
notation.
EXAMPLE 4
Graph the solution set for the compound inequality x Ͼ 2 and x Ͻ 5, and write the solution
set in interval notation.
Solution
xϾ2
xϽ5
x Ͼ 2 and
xϽ5
Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
(a)
Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
(b)
Ϫ2 Ϫ1 0 1 2 3 4 5 6 7
(c)
Figure 3.9
Thus all numbers greater than 2 and less than 5 are included in the solution set, and the graph
is shown in Figure 3.9(c). In interval notation the solution set is (2, 5).
Classroom Example
Graph the solution set for the compound inequality x Ն Ϫ 1 and
x Ն Ϫ2, and write the solution set
in interval notation.
EXAMPLE 5
Graph the solution set for the compound inequality x Յ 1 and x Յ 4, and write the solution
set in interval notation.
Solution
xՅ1
xՅ4
xՅ1
and
xՅ4
Ϫ2 Ϫ1
0
1
2
3
4
5
(a)
Ϫ2 Ϫ1
0
1
2
3
4
5
(b)
Ϫ2 Ϫ1
0
1
2
3
4
5
(c)
Figure 3.10
The intersection of the two solution sets is x Յ 1. The solution set {x 0 x Յ 1} contains all the
numbers that are less than or equal to 1, and the graph is shown in Figure 3.10(c). In interval
notation the solution set is (Ϫq, 1΅.
130
Chapter 3 • Equations, Inequalities, and Problem Solving
The solution set of a compound inequality formed by the word “or” is the
union of the solution sets of the two inequalities. The union of two sets, denoted
by ഫ, contains all the elements in both sets. For example, if A ϭ 50, 1, 26 and
B ϭ 51, 2, 3, 46 , then A ഫ B ϭ 50, 1, 2, 3, 46 . Note that even though 1 and 2 are in both set
A and set B, there is no need to write them twice in A ഫ B.
To find the solution set of the compound inequality
x Ͼ 1 or x Ͼ 3
we find the solution set for each inequality and then take all the values that satisfy either
inequality or both.
Classroom Example
Graph the solution set for the compound inequality x Ͼ 0 or x Ͼ 2,
and write the solution set in interval
notation.
EXAMPLE 6
Graph the solution set for x Ͼ 1 or x Ͼ 3 and write the solution in interval notation.
Solution
xϾ1
−2 −1
0
1
2
3
4
5
(a)
xϾ3
−2 −1
0
1
2
3
4
5
(b)
−2 −1
0
1
2
3
4
5
(c)
xϾ1
or
xϾ3
Figure 3.11
Thus all numbers greater than 1 are included in the solution set, and the graph is shown in
Figure 3.11(c). The solution set is written as 11, q 2 in interval notation.
Classroom Example
Graph the solution set for the compound inequality x Յ Ϫ 1 or x Ն 3,
and write the solution set in interval
notation.
EXAMPLE 7
Graph the solution set for x Յ 0 or x Ն 2 and write the solution in interval notation.
Solution
xՅ0
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
(a)
xՆ2
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
(b)
Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5
(c)
xՅ0
or
xՆ2
Figure 3.12
Thus all numbers less than or equal to 0 and all numbers greater than or equal
to 2 are included in the solution set, and the graph is shown in Figure 3.12(c). Since the solution set contains two intervals that are not continuous, a ഫ symbol is used in the interval notation. The solution set is written as (Ϫq, 0] ഫ [2, q) in interval notation.
Back to Problem Solving
Let’s consider some word problems that translate into inequality statements. We gave suggestions for solving word problems in Section 3.3; these suggestions still apply, except that
3.6 • Inequalities, Compound Inequalities, and Problem Solving
131
here the situations described in the problems/examples will translate into inequalities instead
of equations.
Classroom Example
Sam had scores of 82, 93, and 75 on
his first three exams of the semester.
What score must he get on the fourth
exam to have an average of 85 or
better?
EXAMPLE 8
Ashley had scores of 95, 82, 93, and 84 on her first four exams of the semester. What score
must she get on the fifth exam to have an average of 90 or higher for the five exams?
Solution
Let s represent the score needed on the fifth exam. Since the average is computed by adding all
five scores and dividing by 5 (the number of scores), we have the following inequality to solve:
95 ϩ 82 ϩ 93 ϩ 84 ϩ s
Ն 90
5
Solving this inequality, we obtain
354 ϩ s
Ն 90
5
354 ϩ s
b Ն 51902
5
354 ϩ s Ն 450
354 ϩ s Ϫ 354 Ն 450 Ϫ 354
s Ն 96
5a
Simplify numerator of the left side
Multiply both sides by 5
Subtract 354 from both sides
She must receive a score of 96 or higher on the fifth exam.
Classroom Example
The Cougars have won 15 games
and have lost 13 games. They will
play 12 more games. To win more
than 60% of all their games, how
many of the remaining games must
they win?
EXAMPLE 9
The Cubs have won 40 baseball games and have lost 62 games. They will play 60 more games.
To win more than 50% of all their games, how many of the 60 games remaining must they win?
Solution
Let w represent the number of games the Cubs must win out of the 60 games remaining. Since
they are playing a total of 40 ϩ 62 ϩ 60 ϭ 162 games, to win more than 50% of their games,
they need to win more than 81 games. Thus we have the inequality
w ϩ 40 Ͼ 81
Solving this yields
w Ͼ 41
The Cubs need to win at least 42 of the 60 games remaining.
Concept Quiz 3.6
For Problems 1–5, answer true or false.
1. The solution set of a compound inequality formed by the word “and” is an intersection
of the solution sets of the two inequalities.
2. The solution set of a compound inequality formed by the words “and” or “or” is a union
of the solution sets of the two inequalities.
3. The intersection of two sets contains the elements that are common to both sets.
132
Chapter 3 • Equations, Inequalities, and Problem Solving
4. The union of two sets contains all the elements in both sets.
5. The intersection of set A and set B is denoted by A ʝ B.
For Problems 6–10, match the compound statement with the graph of its solution set
(Figure 3.13).
6. x Ͼ 4 or x Ͻ Ϫ 1
A.
7. x Ͼ 4 and x Ͼ Ϫ 1
−2 − 1
0
1
2
3
4
5
B.
−2 − 1
0
1
2
3
4
5
8. x Ͼ 4 or x Ͼ Ϫ 1
C.
−2 − 1
0
1
2
3
4
5
9. x Յ 4 and x Ն Ϫ 1
D.
−2 − 1
0
1
2
3
4
5
−2 − 1
0
1
2
3
4
5
10. x Ͼ 4 or x Ն Ϫ 1
E.
Figure 3.13
Problem Set 3.6
For Problems 1–50, solve each inequality. (Objectives 1
and 2)
1. 3x ϩ 4 Ͼ x ϩ 8
23. 3(x Ϫ 2) Ͻ 2(x ϩ 1)
24. 5(x ϩ 3) Ͼ 4(x Ϫ 2)
2. 5x ϩ 3 Ͻ 3x ϩ 11
25. 4(x ϩ 3) Ͼ 6(x Ϫ 5)
3. 7x Ϫ 2 Ͻ 3x Ϫ 6
26. 6(x Ϫ 1) Ͻ 8(x ϩ 5)
4. 8x Ϫ 1 Ͼ 4x Ϫ 21
5. 6x ϩ 7 Ͼ 3x Ϫ 3
6. 7x ϩ 5 Ͻ 4x Ϫ 12
27. 3(x Ϫ 4) ϩ 2(x ϩ 3) Ͻ 24
28. 2(x ϩ 1) ϩ 3(x ϩ 2) Ͼ Ϫ12
7. 5n Ϫ 2 Յ 6n ϩ 9
29. 5(n ϩ 1) Ϫ 3(n Ϫ 1) Ͼ Ϫ9
8. 4n Ϫ 3 Ն 5n ϩ 6
30. 4(n Ϫ 5) Ϫ 2(n Ϫ 1) Ͻ 13
9. 2t ϩ 9 Ն 4t Ϫ 13
10. 6t ϩ 14 Յ 8t Ϫ 16
31.
2
1
n Ϫ n Ն Ϫ7
2
3
32.
3
1
nϩ nՅ1
4
6
33.
3
5
3
nϪ nϽ
4
6
8
34.
2
1
1
nϪ nϾ
3
2
4
35.
2
x
3x
Ϫ Ͼ
5
3
10
36.
5x
3
7x
ϩ Ͻ
4
8
12
11. Ϫ3x Ϫ 4 Ͻ 2x ϩ 7
12. Ϫx Ϫ 2 Ͼ 3x Ϫ 7
13. Ϫ4x ϩ 6 Ͼ Ϫ2x ϩ 1
14. Ϫ6x ϩ 8 Ͻ Ϫ4x ϩ 5
15. 5(x Ϫ 2) Յ 30
16. 4(x ϩ 1) Ն 16
17. 2(n ϩ 3) Ͼ 9
18. 3(n Ϫ 2) Ͻ 7
19. Ϫ3(y Ϫ 1) Ͻ 12
20. Ϫ2(y ϩ 4) Ͼ 18
21. Ϫ2(x ϩ 6) Ͼ Ϫ17
37. n Ն 3.4 ϩ 0.15n
22. Ϫ3(x Ϫ 5) Ͻ Ϫ14
38. x Ն 2.1 ϩ 0.3x
3.6 • Inequalities, Compound Inequalities, and Problem Solving
39. 0.09t ϩ 0.1(t ϩ 200) Ͼ 77
40. 0.07t ϩ 0.08(t ϩ 100) Ͼ 38
41. 0.06x ϩ 0.08(250 Ϫ x) Ն 19
42. 0.08x ϩ 0.09(2x) Յ 130
43.
xϪ1
xϩ3
1
ϩ
Ͼ
2
5
10
xϩ3
xϪ5
1
44.
ϩ
Ͻ
4
7
28
45.
xϩ2
xϩ1
Ϫ
Ͻ Ϫ2
6
5
46.
xϪ6
xϩ2
Ϫ
Ͼ Ϫ1
8
7
47.
nϩ3
nϪ7
ϩ
Ͼ3
3
2
48.
nϪ4
nϪ2
ϩ
Ͻ4
4
3
49.
xϪ3
xϪ2
9
Ϫ
Յ
7
4
14
133
For Problems 67–78, solve each problem by setting up and
solving an appropriate inequality. (Objective 4)
67. Five more than three times a number is greater than 26.
Find all of the numbers that satisfy this relationship.
68. Fourteen increased by twice a number is less than or
equal to three times the number. Find the numbers that
satisfy this relationship.
69. Suppose that the perimeter of a rectangle is to be no
greater than 70 inches, and the length of the rectangle
must be 20 inches. Find the largest possible value for the
width of the rectangle.
70. One side of a triangle is three times as long as another
side. The third side is 15 centimeters long. If the perimeter of the triangle is to be no greater than 75 centimeters,
find the greatest lengths that the other two sides can be.
71. Sue bowled 132 and 160 in her first two games. What
must she bowl in the third game to have an average of at
least 150 for the three games?
72. Mike has scores of 87, 81, and 74 on his first three algebra tests. What score must he get on the fourth test to
have an average of 85 or higher for the four tests?
xϪ1
xϩ2
7
50.
Ϫ
Ն
5
6
15
73. This semester Lance has scores of 96, 90, and 94 on his
first three algebra exams. What must he average on the
last two exams to have an average higher than 92 for all
five exams?
For Problems 51– 66, graph the solution set for each compound inequality. (Objective 3)
74. The Mets have won 45 baseball games and lost 55
games. They have 62 more games to play. To win more
than 50% of all their games, how many of the 62 games
remaining must they win?
51. x Ͼ Ϫ1 and x Ͻ 2
52. x Ͼ 1 and x Ͻ 4
53. x Ͻ Ϫ2 or x Ͼ 1
54. x Ͻ 0 or x Ͼ 3
55. x Ͼ Ϫ2 and x Յ 2
56. x Ն Ϫ1 and x Ͻ 3
57. x Ͼ Ϫ1 and x Ͼ 2
58. x Ͻ Ϫ2 and x Ͻ 3
59. x Ͼ Ϫ4 or x Ͼ 0
60. x Ͻ 2 or x Ͻ 4
61. x Ͼ 3 and x Ͻ Ϫ1
62. x Ͻ Ϫ3 and x Ͼ 6
75. An Internet business has costs of $4000 plus $32 per
sale. The business receives revenue of $48 per sale.
What possible values for sales would ensure that the revenues exceed the costs?
76. The average height of the two forwards and the center of
a basketball team is 6 feet, 8 inches. What must the average height of the two guards be so that the team’s average height is at least 6 feet, 4 inches?
77. Scott shot rounds of 82, 84, 78, and 79 on the first four
days of the golf tournament. What must he shoot on the
fifth day of the tournament to average 80 or less for the
5 days?
78. Sydney earns $2300 a month. To qualify for a mortgage,
her monthly payments must be less than 35% of her
monthly income. Her monthly mortgage payments must
be less than what amount in order to qualify for the
mortgage?
63. x Յ 0 or x Ն 2
64. x Յ Ϫ2 or x Ն 1
65. x Ͼ Ϫ4 or x Ͻ 3
66. x Ͼ Ϫ1 or x Ͻ 2
Additional word problems can be found in Appendix B.
All of the problems in the Appendix marked as (3.6) are
appropriate for this section.
134
Chapter 3 • Equations, Inequalities, and Problem Solving
Thoughts Into Words
79. Give an example of a compound statement using the
word “and” outside the field of mathematics.
81. Give a step-by-step description of how you would solve
the inequality 3x Ϫ 2 Ͼ 41x ϩ 62 .
80. Give an example of a compound statement using the
word “or” outside the field of mathematics.
Answers to the Concept Quiz
1. True
2. False
3. True
4. True
5. True
6. B
7. E
8. A
9. D
10. C
Chapter 3 Summary
OBJECTIVE
SUMMARY
Solve first-degree equations.
Numerical equations can be true or false.
Algebraic equations (open sentences) contain one or more variables. Solving an
equation refers to the process of finding the
number (or numbers) that makes an algebraic equation a true statement. A firstdegree equation of one variable is an equation that contains only one variable, and
this variable has an exponent of one. The
properties found in this chapter provide the
basis for solving equations. Be sure you are
able to use these properties to solve the
variety of equations presented.
Solve equations using the
addition-subtraction property
of equality.
Any number can be added to or subtracted
from both sides of an equation.
EXAMPLE
Solve 0.8 ϭ x Ϫ 0.3.
Solution
0.8 ϭ x Ϫ 0.3
(Section 3.1/Objective 1)
0.8 ϩ 0.3 ϭ x Ϫ 0.3 ϩ 0.3
1.1 ϭ x
The solution set is {1.1}.
Solve equations using the
multiplication-division
property of equality.
An equivalent equation is obtained whenever both sides of an equation are multiplied or divided by a nonzero real number.
2
Solve Ϫ x ϭ 8.
3
Solution
(Section 3.1/Objective 2)
2
Ϫ xϭ8
3
3
2
3
aϪ baϪ xb ϭ 8aϪ b
2
3
2
x ϭ Ϫ12
The solution set is {Ϫ12}
Solve equations using both
the addition-subtraction
property of equality and the
multiplication-division
property of equality.
(Section 3.2/Objective 1)
To solve most equations, both properties
must be applied.
Solve 5n Ϫ 2 ϭ 8.
Solution
5n Ϫ 2 ϭ 8
5n Ϫ 2 ϩ 2 ϭ 8 ϩ 2
5n ϭ 10
1
1
15n2 ϭ 1102
5
5
nϭ2
The solution set is {2}.
(continued)
Chapter 3 • Summary
135
136
Chapter 3 • Equations, Inequalities, and Problem Solving
OBJECTIVE
SUMMARY
EXAMPLE
Solve equations that involve
the use of the distributive
property.
To solve equations in which the variable is
part of an expression enclosed in parentheses, the distributive property is used. The
distributive property removes the parentheses, and the resulting equation is solved in
the usual way.
Solve 31x Ϫ 42 ϭ 21x ϩ 12.
(Section 3.4/Objective 1)
Solution
31x Ϫ 42 ϭ 21x ϩ 12
3x Ϫ 12 ϭ 2x ϩ 2
x Ϫ 12 ϭ 2
x ϭ 14
The solution set is {14}.
Solve equations that involve
fractional forms.
(Section 3.4/Objective 2)
When an equation contains several fractions, it is usually best to start by clearing
the equation of all fractions. The fractions
can be cleared by multiplying both sides of
the equation by the least common denominator of all the denominators.
Solve
3n
n
7
ϩ ϭ .
4
5
10
Solution
3n
n
7
ϩ ϭ
4
5
10
3n
n
7
20 a
ϩ b ϭ 20 a b
4
5
10
20 a
3n
n
b ϩ 20 a b ϭ 14
4
5
15n ϩ 4n ϭ 14
19n ϭ 14
14
nϭ
19
The solution set is e
Solve equations that are
contradictions or identities.
(Section 3.3/Objectives 2
and 3)
When an equation is not true for any value
of x, then the equation is called “a contradiction.”
When an equation is true for any permissible value of x, then the equation is called
“an identity.”
14
f.
19
Solve the following equations:
(a) 2(x ϩ 4) ϭ 2x ϩ 5
(b) 4x Ϫ 8 ϭ 2(2x Ϫ 4)
Solution
(a) 2(x ϩ 4) ϭ 2x ϩ 5
2x ϩ 8 ϭ 2x ϩ 5
2x Ϫ 2x ϩ 8 ϭ 2x Ϫ 2x ϩ 5
8ϭ5
This is a false statement so there is no solution. The solution is л.
(b) 4x Ϫ 8 ϭ 2(2x Ϫ 4)
4x Ϫ 8 ϭ 4x Ϫ 8
4x Ϫ 4x Ϫ 8 ϭ 4x Ϫ 4x Ϫ 8
Ϫ8 ϭ Ϫ8
This is a true statement so any value of x is
a solution. The solution set is {All reals}.
(continued)
Chapter 3 • Summary
137
OBJECTIVE
SUMMARY
EXAMPLE
Show the solution set of an
inequality in set-builder
notation and by graphing.
The solution set of x ϩ 2 Ͼ 5 is all
numbers greater than 3. The solution set
in set-builder notation is 5x0 x Ͼ 36 and is
read “the set of all x such that x is greater
than 3.” A number line is used to graph
the solution. A parenthesis on the number
line means that number is not included in
the solution set. A bracket on the number
line indicates that the number is included
in the solution set.
Write the solution set of the inequalities in
set-builder notation and graph the solution.
(a) x Յ 2
(b) x Ͼ Ϫ1
(Section 3.5/Objective 2)
Solution
(a) {x0 x Յ 2}
Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4
(b) {x0 x Ͼ Ϫ1}
−3 −2 −1 0 1 2 3 4
Solve first-degree
inequalities.
(Section 3.5/Objective 1)
Properties for solving inequalities are similar to the properties for solving equations—
except for properties that involve multiplying or dividing by a negative number. When
multiplying or dividing both sides of an
inequality by a negative number, you must
reverse the inequality symbol.
Solve Ϫ 4n Ϫ 3 Ͼ 7.
Solution
Ϫ 4n Ϫ 3 Ͼ 7
Ϫ 4n Ͼ 10
Ϫ 4n
10
Ͻ
4
Ϫ4
Ϫ5
nϽ
2
The solution set is
5
5
en 0 n Ͻ Ϫ f or aϪq,Ϫ b
2
2
Solve inequalities that involve
the use of the distributive
property.
(Section 3.6/Objective 1)
To solve inequalities when the variable is
part of an expression enclosed in parentheses, use the distributive property. The distributive property removes the parentheses,
and the resulting inequality is solved in the
usual way.
Solve 15 Ͻ Ϫ 2(x Ϫ1) Ϫ5.
Solution
15 Ͻ Ϫ 2x ϩ 2 Ϫ5
15 Ͻ Ϫ 2x Ϫ3
15 ϩ 3 Ͻ Ϫ2x Ϫ3 ϩ 3
18 Ͻ Ϫ 2x
18
Ϫ2x
Ͼ
Ϫ2
Ϫ2
Ϫ9 Ͼ x
The solution set is 5x0 x Ͻ Ϫ 96
or (Ϫq,Ϫ9) .
(continued)
138
Chapter 3 • Equations, Inequalities, and Problem Solving
OBJECTIVE
SUMMARY
EXAMPLE
Solve inequalities that involve
fractional forms.
When an inequality contains several
fractions, it is usually best to clear the
inequality of all fractions. The fractions
can be cleared by multiplying both sides
of the equation by the LCD of all the
denominators.
3
2
Solve x Ͻ .
4
3
(Section 3.6/Objective 2)
Solution
3
2
xϽ
4
3
4 3
4 2
a xb Ͻ a b
3 4
3 3
8
xϽ
9
8
8
The solution set is ex 0 x Ͻ f or aϪq, b.
9
9
Solve compound inequalities
formed by the word “and.”
(Section 3.6/Objective 3)
Solve compound inequalities
formed by the word “or.”
(Section 3.6/Objective 3)
Solve word problems.
(Section 3.2/Objective 2;
Section 3.3/Objective 5;
Section 3.4/Objective 3)
The solution set of a compound inequality
formed by the word “and” is the
intersection of the solution sets of the two
inequalities. To solve inequalities
involving “and,” we must satisfy all of the
conditions. Thus the compound inequality
x Ͼ 1 and x Ͻ 3 is satisfied by all
numbers between 1 and 3.
Solve the compound inequality x Ͼ Ϫ 4
and x Ͼ 2.
The solution set of a compound inequality
formed by the word “or,” is the union of
the solution sets of the two inequalities. To
solve inequalities involving “or” we must
satisfy one or more of the conditions. Thus
the compound inequality x Ͻ 1 or x Ͻ 5 is
satisfied by all numbers less than 5.
Solve the compound inequality x Ͼ Ϫ 1
or x Ͻ 2.
Keep these suggestions in mind as you
solve word problems:
1. Read the problem carefully.
2. Sketch any figure or diagram that might
be helpful.
3. Choose a meaningful variable.
4. Look for a guideline.
5. Form an equation.
6. Solve the equation.
7. Check your answer.
The difference of two numbers is 14. If 35
is the larger number, find the smaller
number.
Solution
All of the conditions must be satisfied. Thus
the compound inequality x Ͼ Ϫ 4 and
x Ͼ 2 is satisfied by all numbers greater
than 2. The solution set is 5x0 x Ͼ 26.
Solution
One or more of the conditions must be
satisfied. Thus the compound inequality
x Ͼ Ϫ 1 or x Ͻ 2 is satisfied by all real
numbers. The solution set is {All reals}.
Solution
Let n represent the smaller number.
Guideline
Larger number Ϫ smaller number ϭ 14
35 Ϫ n ϭ 14
Ϫn ϭ Ϫ21
n ϭ 21
The smaller number is 21.
(continued)