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4: Equations Involving Parentheses and Fractional Forms

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



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



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





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



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



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