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6: Inequalities, Compound Inequalities, and Problem Solving

# 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

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

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.

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

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

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

19

The solution set is e

Solve equations that are

(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

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

4

3

4 3

4 2

a xb Ͻ a b

3 4

3 3

8

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:

2. Sketch any figure or diagram that might

3. Choose a meaningful variable.

4. Look for a guideline.

5. Form an equation.

6. Solve the equation.

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)

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6: Inequalities, Compound Inequalities, and Problem Solving

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