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2 Tables, Pictographs, and Bar Graphs

# 2 Tables, Pictographs, and Bar Graphs

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SECTION 11.6

755

The variable may appear in any position in an equation. Just apply the rules

carefully as you try to write an equivalent equation, and you will find the solution.

Example 2 illustrates this property.

c

Example 2

Solving Equations

Solve

3 Ϫ 2x ϭ 9

3 Ϫ 3 Ϫ 2x ϭ 9 Ϫ 3

Ϫ2x ϭ 6

First subtract 3 from both sides.

Now divide both sides by Ϫ2. This leaves x alone on the left.

NOTE

Ϫ2

ϭ 1, so we divide by Ϫ2

Ϫ2

to isolate x.

Ϫ2x

6

ϭ

Ϫ2

Ϫ2

x ϭ Ϫ3

The solution is Ϫ3. We leave it to you to check this result.

The Streeter/Hutchison Series in Mathematics

Basic Mathematical Skills with Geometry

Check Yourself 2

Solve and check.

10 ؊ 3x ‫ ؍‬1

Apago PDF Enhancer

You may also have to combine multiplication with addition or subtraction to solve

an equation. Consider Example 3.

c

Example 3

Solving Equations

(a) Solve

x

Ϫ3ϭ4

5

To get the x term alone, we first add 3 to both sides.

x

Ϫ3ϩ3ϭ4ϩ3

5

x

ϭ7

5

To undo the division, multiply both sides of the equation by 5.

5

΂5΃ ϭ 5 # 7

x

x ϭ 35

The solution is 35. Just return to the original equation to check the result.

(35)

Ϫ3՘4

5

7Ϫ3՘4

4ϭ4

(True)

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An Introduction to Algebra

(b) Solve

2

x ϩ 5 ϭ 13

3

2

x ϩ 5 Ϫ 5 ϭ 13 Ϫ 5

3

2

xϭ8

3

First subtract 5 from both sides.

3

2

Now multiply both sides by , the reciprocal of .

2

3

3 2

3

x ϭ

8

2 3

2

΂ ΃΂ ΃ ΂ ΃

or

x ϭ 12

The solution is 12. We leave it to you to check this result.

x

؉5‫؍‬3

6

(b)

3

x ؊ 8 ‫ ؍‬10

4

Apago PDF Enhancer

In Section 11.4, you learned how to solve certain equations when the variable

appeared on both sides. Example 4 shows you how to extend that work by using the

multiplication property of equality.

c

Example 4

Combining Properties to Solve an Equation

Solve

6x Ϫ 4 ϭ 3x Ϫ 2

We begin by bringing all the variable terms to one side. To do this, we subtract 3x

from both sides. This eliminates the variable term from the right side.

6x Ϫ 4 ϭ 3x Ϫ 2

6x Ϫ 4 Ϫ 3x ϭ 3x Ϫ 2 Ϫ 3x

3x Ϫ 4 ϭ Ϫ2

We now isolate the variable term by adding 4 to both sides.

3x Ϫ 4 ϭ Ϫ2

3x Ϫ 4 ϩ 4 ϭ Ϫ2 ϩ 4

3x ϭ 2

Finally, divide by 3.

3x

2

ϭ

3

3

2

3

The Streeter/Hutchison Series in Mathematics

(a)

Solve and check.

Basic Mathematical Skills with Geometry

Check Yourself 3

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Combining the Properties to Solve Equations

SECTION 11.6

757

Check:

6

΂3΃ Ϫ 4 ՘ 3΂3΃ Ϫ 2

2

2

4 Ϫ 4՘2Ϫ2

0ϭ0

(True)

The basic idea is to use our two properties to form an equivalent equation with the

x isolated. Here we subtracted 3x and then added 4. You can do these steps in either

order. Try it for yourself the other way. In either case, the multiplication property is

then used as the last step in finding the solution.

Check Yourself 4

Solve and check.

7x ؊ 5 ‫ ؍‬3x ؉ 5

c

Example 5

Combining Properties to Solve an Equation (Two Methods)

Solve 4x Ϫ 8 ϭ 7x ϩ 7.

Method 1

Apago PDF Enhancer

4x Ϫ 8 Ϫ 7x ϭ 7x ϩ 7 Ϫ 7x

Ϫ3x Ϫ 8 ϭ 7

Ϫ3x Ϫ 8 ϩ 8 ϭ 7 ϩ 8

Ϫ3x ϭ 15

Ϫ3x

15

ϭ

Ϫ3

Ϫ3

x ϭ Ϫ5

The Streeter/Hutchison Series in Mathematics

Basic Mathematical Skills with Geometry

Here are two approaches to solving equations in which the coefficient on the right

side is greater than the coefficient on the left side.

Bring the variable terms to the same

(left) side.

Isolate the variable term.

Isolate the variable.

We let you check this result.

To avoid a negative coefficient (Ϫ3, in this example), some students prefer a different approach.

This time we work toward having the number on the left and the x term on the right, or

ϭ x.

Method 2

NOTE

It is usually easier to isolate

the variable term on the side

that results in a positive

coefficient.

4x Ϫ 8 ϭ 7x ϩ 7

4x Ϫ 8 Ϫ 4x ϭ 7x ϩ 7 Ϫ 4x

Ϫ8 ϭ 3x ϩ 7

Ϫ8 Ϫ 7 ϭ 3x ϩ 7 Ϫ 7

Ϫ15 ϭ 3x

Ϫ15

3x

ϭ

3

3

Ϫ5 ϭ x

Bring the variable terms to the same (right) side.

Isolate the variable term.

Isolate the variable.

Because Ϫ5 ϭ x and x ϭ Ϫ5 are equivalent equations, it really makes no difference;

the solution is still Ϫ5! You may use whichever approach you prefer.

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An Introduction to Algebra

Check Yourself 5

Solve 5x ؉ 3 ‫ ؍‬9x ؊ 21 by finding equivalent equations of the form

x‫؍‬

and

‫ ؍‬x to compare the two methods of finding the

solution.

When possible, we start by combining like terms on each side of the equation.

Solve.

7x Ϫ 3 ϩ 5x ϩ 4

12x ϩ 1

12x ϩ 1 Ϫ 6x

6x ϩ 1

6x ϩ 1 Ϫ 1

6x

6x

6

x

ϭ

ϭ

ϭ

ϭ

ϭ

ϭ

6x ϩ 25

6x ϩ 25

6x ϩ 25 Ϫ 6x

25

25 Ϫ 1

24

24

ϭ

6

ϭ4

Start by combining like terms.

Bring the variables to one side.

Isolate the variable term.

Isolate the variable.

The solution is 4. We leave the check to you.

Check Yourself 6

Apago PDF Enhancer

Solve and check.

9x ؊ 6 ؊ 3x ؉ 1 ‫ ؍‬2x ؉ 15

It may also be necessary to remove grouping symbols to solve an equation.

Example 7 illustrates this property.

c

Example 7

Solving Equations That Contain Parentheses

Solve and check.

NOTE

5(x Ϫ 3)

ϭ 5[x ϩ (Ϫ3)]

ϭ 5x ϩ 5(Ϫ3)

ϭ 5x ϩ (Ϫ15)

ϭ 5x Ϫ 15

5(x Ϫ 3) Ϫ 2x ϭ x ϩ 7

5x Ϫ 15 Ϫ 2x ϭ x ϩ 7

3x Ϫ 15 ϭ x ϩ 7

Apply the distributive property.

Combine like terms.

We now have an equation that we can solve by the usual methods. First, bring the variable terms to one side, then isolate the variable term, and finally, isolate the variable.

3x Ϫ 15 Ϫ x

2x Ϫ 15

2x Ϫ 15 ϩ 15

2x

2x

2

x

ϭ

ϭ

ϭ

ϭ

xϩ7Ϫx

7

7 ϩ 15

22

22

ϭ

2

ϭ 11

Subtract x to bring the variable

terms to the same side.

Add 15 to isolate the variable term.

Divide by 2 to isolate the variable.

Basic Mathematical Skills with Geometry

< Objective 2 >

Combining Terms to Solve an Equation

The Streeter/Hutchison Series in Mathematics

Example 6

c

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SECTION 11.6

759

The solution is 11. To check, substitute 11 for x in the original equation. Again note the

use of our rules for the order of operations.

5[(11) Ϫ 3] Ϫ 2(11) ՘ (11) ϩ 7

5 ؒ 8 Ϫ 2 ؒ 11 ՘ 11 ϩ 7

40 Ϫ 22 ՘ 11 ϩ 7

18 ϭ 18

Simplify terms in parentheses.

Multiply.

A true statement.

Check Yourself 7

Solve and check.

7(x ؉ 5) ؊ 3x ‫ ؍‬x ؊ 7

We say that an equation is “solved” when we have an equivalent equation of the form

or

ϭx

in which

is some number

The steps of solving a linear equation are as follows:

The Streeter/Hutchison Series in Mathematics

Basic Mathematical Skills with Geometry

Step by Step

To Solve a Linear

Equation

Step 1

Use the distributive property to remove any grouping symbols.

Step 2

Combine like terms on each side of the equation.

Step 3

Add or subtract variable terms to bring the variable term to one side

of the equation.

Step 4

Add or subtract numbers to isolate the variable term.

Step 5

Multiply by the reciprocal of the coefficient to isolate the variable.

Step 6

Apago PDF Enhancer

There are a host of applications involving linear equations.

c

Example 8

Applying Algebra

In an election, the winning candidate had 160 more votes than the loser did. If the total

We first set up the problem. Let x represent the number of votes received by the

This must total 3,260.

x ϩ (x ϩ 160) ϭ 3,260

2x ϩ 160 ϭ 3,260

2x ϭ 3,100

x ϭ 1,550

Remove the parentheses and combine like terms.

Subtract 160 from both sides.

Divide both sides by 2.

x ϩ 160 ϭ 1,550 ϩ 160 ϭ 1,710 votes.

Check Yourself 8

The Randolphs used 12 more gallons (gal) of fuel oil in October than

in September and twice as much oil in November as in September. If

they used 132 gal for the 3 months, how much was used each month?

CHAPTER 11

Page 760

An Introduction to Algebra

1. (a) x ϭ Ϫ4; (b) x ϭ 3

2. x ϭ 3

3. (a) x ϭ Ϫ12; (b) x ϭ 24

5

4. x ϭ

5. x ϭ 6

6. x ϭ 5

7. x ϭ Ϫ14

2

8. 30 gal in September, 42 gal in October, 60 gal in November

b

The following fill-in-the-blank exercises are designed to ensure that you

understand some of the key vocabulary used in this section.

SECTION 11.6

(a) The first goal for solving an equation is to

term on one side of the equation.

(b) Apply the

property.

the variable

property before applying the multiplication

(d) An equation in the form x ϭ

Apago PDF Enhancer

or

ϭ x has been

.

Basic Mathematical Skills with Geometry

8:56 AM

The Streeter/Hutchison Series in Mathematics

760

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8:56 AM

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Page 761

Challenge Yourself

|

Calculator/Computer

|

Career Applications

|

Above and Beyond

< Objective 1 >

ALEKS.com!

Solve and check.

1. 2x ϩ 1 ϭ 9

11.6 exercises

2. 3x Ϫ 1 ϭ 17

> Videos

3. 3x Ϫ 2 ϭ 7

4. 5x ϩ 3 ϭ 23

• Practice Problems

• Self-Tests

• NetTutor

• e-Professors

• Videos

Name

5. 4x ϩ 7 ϭ 35

6. 7x Ϫ 8 ϭ 13

Section

7. 2x ϩ 9 ϭ 5

8. 6x ϩ 25 ϭ Ϫ5

9. 4 Ϫ 7x ϭ 18

The Streeter/Hutchison Series in Mathematics

Basic Mathematical Skills with Geometry

11. 3 Ϫ 4x ϭ Ϫ9

13.

15.

x

ϩ1ϭ5

2

10. 8 Ϫ 5x ϭ Ϫ7

> Videos

12. 5 Ϫ 4x ϭ 25

14.

> Videos

x

Ϫ5ϭ3

4

16.

x

Ϫ2ϭ3

3

x

ϩ3ϭ8

5

Apago PDF Enhancer

2

17. x ϩ 5 ϭ 17

3

3

18. x Ϫ 5 ϭ 4

4

4

19. x Ϫ 3 ϭ 13

5

5

20. x ϩ 4 ϭ 14

7

21. 5x ϭ 2x ϩ 9

> Videos

22. 7x ϭ 18 Ϫ 2x

23. 3x ϭ 10 Ϫ 2x

24. 11x ϭ 7x ϩ 20

25. 9x ϩ 2 ϭ 3x ϩ 38

26. 8x Ϫ 3 ϭ 4x ϩ 17

27. 4x Ϫ 8 ϭ x Ϫ 14

29. 5x ϩ 7 ϭ 2x Ϫ 3

Date

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

28. 6x Ϫ 5 ϭ 3x Ϫ 29

> Videos

30. 9x ϩ 7 ϭ 5x Ϫ 3

31. 7x Ϫ 3 ϭ 9x ϩ 5

32. 5x Ϫ 2 ϭ 8x Ϫ 11

33. 5x ϩ 4 ϭ 7x Ϫ 8

34. 2x ϩ 23 ϭ 6x Ϫ 5

SECTION 11.6

761

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11.6 exercises

35.

36.

37.

38.

39.

40.

< Objective 2 >

35. 2x Ϫ 3 ϩ 5x ϭ 7 ϩ 4x ϩ 2

36. 8x Ϫ 7 Ϫ 2x ϭ 2 ϩ 4x Ϫ 5

37. 6x ϩ 7 Ϫ 4x ϭ 8 ϩ 7x Ϫ 26

38. 7x Ϫ 2 Ϫ 3x ϭ 5 ϩ 8x ϩ 13

39. 9x Ϫ 2 ϩ 7x ϩ 13 ϭ 10x Ϫ 13

40. 5x ϩ 3 ϩ 6x Ϫ 11 ϭ 8x ϩ 25

41. 8x Ϫ 7 ϩ 5x Ϫ 10 ϭ 10x Ϫ 12

42. 10x Ϫ 9 ϩ 2x Ϫ 3 ϭ 8x Ϫ 18

43. SOCIAL SCIENCE There were 55 more yes votes than no votes on an election

measure. If 735 votes were cast in all, how many yes votes were there?

44. BUSINESS AND FINANCE Juan worked twice as many hours as Jerry. Marcia

41.

worked 3 more hours than Jerry. If they worked a total of 31 hours, how

many hours did each employee work?

42.

45. BUSINESS AND FINANCE Francine earns \$120 per month more than Rob. If they

earn a total of \$2,680 per month, how much does Francine earn each month?

43.

Basic Skills

46.

|

Challenge Yourself

| Calculator/Computer | Career Applications

|

Above and Beyond

Apago PDF Enhancer

Solve each equation.

47.

48.

49.

50.

51.

47. 7(2x Ϫ1) Ϫ 5x ϭ x ϩ 25

48. 9(3x ϩ 2) Ϫ 10x ϭ 12x Ϫ 7

49. 3x ϩ 2(4x Ϫ 3) ϭ 6x Ϫ 9

50. 7x ϩ 3(2x ϩ 5) ϭ 10x ϩ 17

51.

8

2

x Ϫ 3 ϭ x ϩ 15

3

3

52.

3

12

x ϩ 7 ϭ 31 Ϫ x

5

5

53.

2x

12x

Ϫ5ϭ

ϩ8

5

5

54.

24x

3x

Ϫ5ϭ

ϩ7

7

7

55. 5.3x Ϫ 7 ϭ 2.3x ϩ 5

52.

Basic Skills | Challenge Yourself | Calculator/Computer |

53.

56. 9.8x ϩ 2 ϭ 3.8x ϩ 20

Career Applications

|

Above and Beyond

57. AGRICULTURAL TECHNOLOGY The estimated yield Y of a field of corn (in

54.

bushels per acre) can be found by multiplying the rainfall r, in inches, during

the growing season by 16 and then subtracting 15. This relationship can be

modeled by the formula

55.

Y ϭ 16r Ϫ 15

56.

If a farmer wants a yield of 159 bushels per acre, then we can write the

equation shown to determine the amount of rainfall required.

159 ϭ 16r Ϫ 15

How much rainfall is necessary to achieve a yield of 159 bushels of corn

per acre?

57.

762

SECTION 11.6

The Streeter/Hutchison Series in Mathematics

45.

during aerobic training, subtract the person’s age from 220, and then

9

multiply the result by . Determine the age of a person if the person’s upper

10

limit heart rate is 153.

44.

Basic Mathematical Skills with Geometry

46. SCIENCE AND MEDICINE To determine the upper limit for a person’s heart rate

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11.6 exercises

58. CONSTRUCTION TECHNOLOGY The number of studs s required to build a wall

(with studs spaced 16 inches on center) is equal to the one more than

the length of the wall w, in feet. We model this with the formula

3

times

4

3

sϭ wϩ1

4

58.

If a contractor uses 22 studs to build a wall, how long is the wall?

59. ALLIED HEALTH The internal diameter D [in millimeters (mm)] of an endotra-

cheal tube for a child is calculated using the formula

t ϩ 16

4

59.

60.

61.

in which t is the child’s age (in years).

How old is a child who requires an endotracheal tube with an internal

diameter of 7 mm?

62.

63.

The Streeter/Hutchison Series in Mathematics

Basic Mathematical Skills with Geometry

60. MECHANICAL ENGINEERING The number of BTUs required to heat a house is

3

2 times the volume of the air in the house (in cubic feet). What is the maxi4

mum air volume that can be heated with a 90,000-BTU furnace?

Basic Skills

|

Challenge Yourself

|

Calculator/Computer

|

Career Applications

|

64.

Above and Beyond

Apago PDF Enhancer

61. Create an equation of the form ax ϩ b ϭ c that has 2 as a solution.

62. Create an equation of the form ax ϩ b ϭ c that has Ϫ6 as a solution.

63. The equation 3x ϭ 3x ϩ 5 has no solution, whereas the equation 7x ϩ 8 ϭ 8

has zero as a solution. Explain the difference between an equation that has

zero as a solution and an equation that has no solution.

64. Construct an equation for which every real number is a solution.

1. x ϭ 4

13. x ϭ 8

3. x ϭ 3

5. x ϭ 7

7. x ϭ –2

9. x ϭ Ϫ2

11. x ϭ 3

15. x ϭ 32

17. x ϭ 18

19. x ϭ 20

21. x ϭ 3

23. x ϭ 2

25. x ϭ 6

27. x ϭ Ϫ2

33. x ϭ 6

35. x ϭ 4

37. x ϭ 5

45. \$1,400

29. x ϭ Ϫ

10

3

39. x ϭ Ϫ4

47. x ϭ 4

49. x ϭ Ϫ

7

13

55. x ϭ 4

57. 10 in.

59. 12 yr

2

8

61. Above and Beyond

63. Above and Beyond

31. x ϭ Ϫ4

5

3

51. x ϭ 9

41. x ϭ

3

5

53. x ϭ Ϫ

SECTION 11.6

763

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summary :: chapter 11

Example

Section 11.1

The sum of x and 5 is x ϩ 5.

7 more than a is a ϩ 7.

b increased by 3 is b ϩ 3.

Subtraction x Ϫ y means the difference of x and y, or x

minus y. Some other words indicating subtraction are less

than and decreased by.

The difference of x and 3 is x Ϫ 3.

5 less than p is p Ϫ 5.

a decreased by 4 is a Ϫ 4.

Multiplication

The product of m and n is mn.

The product of 2 and the sum of a and

b is 2(a ϩ b).

x#y

(x)(y) s These all mean the product of x and y, or x times y.

xy

x

means x divided by y, or the quotient when x is

y

divided by y.

Division

n

n divided by 5 is . The sum of a

5

aϩb

and b, divided by 3, is

.

3

Evaluating Algebraic Expressions

Step 1

Step 2

Replace each variable with the given number value.

Do the necessary arithmetic operations, following

the rules for the order of operations.

p. 703

Section 11.2

Apago D

P Evaluate

F Enhancer

4a Ϫ b

p. 711

2c

if a ϭ Ϫ6, b ϭ 8, and c ϭ Ϫ4.

4(Ϫ6) Ϫ (8)

4a Ϫ b

ϭ

2c

2(Ϫ4)

Ϫ24 Ϫ 8

ϭ

Ϫ8

Ϫ32

ϭ4

ϭ

Ϫ8

Simplifying Algebraic Expressions

Section 11.3

Term A number, or the product of a number and one or

more variables, raised to a power.

4a2 and 3a2 are like terms.

Like terms Terms that contain exactly the same variables

raised to the same powers.

5x2 and 2xy2 are not like terms.

p. 721

Combining Like Terms

Step 1

Add or subtract the numerical coefficients.

Step 2

Attach the common variables.

764

5a ϩ 3a ϭ 8a

7xy Ϫ 3xy ϭ 4xy

p. 723

Basic Mathematical Skills with Geometry

Addition x ϩ y means the sum of x and y, or x plus y.

Some other words indicating addition are more than and

increased by.

The Streeter/Hutchison Series in Mathematics

From Arithmetic to Algebra

Reference

Definition/Procedure

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summary :: chapter 11

Definition/Procedure

Example

Reference

Using the Addition Property to Solve an Equation

Section 11.4

Equation A statement that two expressions are equal.

3x Ϫ 5 ϭ 7 is an equation.

Solution Any value for the variable that makes an equation

a true statement.

4 is a solution to the equation

because

The Streeter/Hutchison Series in Mathematics

Basic Mathematical Skills with Geometry

3(4) Ϫ 5 ՘ 7

12 Ϫ 5 ՘ 7

7ϭ7

(True)

Equivalent equations Equations that have exactly the

same set of solutions.

3x Ϫ 5 ϭ 7 and x ϭ 4 are equivalent

equations.

p. 732

The addition property If a ϭ b, then a ϩ c ϭ b ϩ c.

Adding (or subtracting) the same quantity to both sides of

an equation yields an equivalent equation.

xϪ5ϭ 7

ϩ 5 ϩ5

x

ϭ 12

p. 732

Using the Multiplication Property to Solve an Equation

Section 11.5

The multiplication property If a ϭ b and c 0, then

ac ϭ bc. Multiplying (or dividing) both sides of an equation

by the same nonzero number yields an equivalent equation.

5x ϭ 20

20

5x

ϭ

5

5

xϭ4

p. 743

To solve a percent problem algebraically, translate the

problem into algebra (writing the rate as a decimal) and use

the multiplication rule to solve.

30% of what number is 45?

p. 747

Apago PDF Enhancer

0.3x ϭ 45

0.3x

45

ϭ

0.3

0.3

x ϭ 150

Combining the Properties to Solve Equations

p. 730

Solving linear equations We say that an equation is solved

when we have an equivalent equation of the form x ϭ or

ϭ x in which is some number.

The steps for solving a linear equation follow.

Step 1

Use the distributive property to remove any

grouping symbols.

Step 2

Combine like terms on each side of the equation.

Step 3

Add or subtract variable terms to bring the variable

term to one side of the equation.

Step 4

Add or subtract numbers to isolate the variable term.

Step 5

Multiply by the reciprocal of the coefficient to

isolate the variable.

Step 6

Section 11.6

Solve:

3x Ϫ 6 ϩ 4x

7x Ϫ 6

7x Ϫ 6 Ϫ 3x

4x Ϫ 6

4x Ϫ 6 ϩ 6

4x

4x

4

x

p. 759

ϭ

ϭ

ϭ

ϭ

ϭ

ϭ

3x ϩ 14

3x ϩ 14

3x ϩ 14 Ϫ 3x

14

14 ϩ 6

20

20

ϭ

4

ϭ5

Check:

3(5) Ϫ 6 ϩ 4(5) ՘ 3(5) ϩ 14

29 ϭ 29 True

765

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