College Textbook Sample 4a

6e

Elementary Algebra

Alan S. Tussy Citrus College

Diane R. Koenig Rock Valley College

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States

CONTENTS

CHAPTER

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1 An Introduction to Algebra 1 1.1 1.2 1.3 1.4 1.5

Introducing the Language of Algebra 2 Fractions 11 The Real Numbers 25 Adding Real Numbers; Properties of Addition 35 Subtracting Real Numbers 44

1.6 Multiplying and Dividing Real Numbers; Multiplication 1.7 1.8 1.9

CHAPTER

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and Division Properties 51 Exponents and Order of Operations 62 Algebraic Expressions 74 Simplifying Algebraic Expressions Using Properties of Real Numbers 85

CHAPTER SUMMARY AND REVIEW 97 CHAPTER TEST 99 GROUP PROJECT 102 CUMULATIVE REVIEW 104

2 Equations, Inequalities, and Problem Solving 107 2.1 2.2 2.3 2.4 2.5 2.6 2.7

Solving Equations Using Properties of Equality 108 More about Solving Equations 119 Applications of Percent 129 Formulas 138 Problem Solving 152 More about Problem Solving 162 Solving Inequalities 173

CHAPTER SUMMARY AND REVIEW 175

CHAPTER TEST 177

GROUP PROJECT 178

CUMULATIVE REVIEW 179

v

2

Linear Equations and Ineqalities in Two Variables

from Campus to Careers

CHAPTER OUTLINE

Automotive Service Technician Anyone whose car has ever broken down appreciates the talents of automotive service technicians. To work on today’s high-tech cars and trucks require a person with diagnostic and problem-solving skills.

Courses in automotive repair, electronics, physics, chemistry, English, computers, and mathematics provide a good educational background for a career as a service technician.

One basic formula that automotive students study is d = rt, where d is the distance traveled by a vehicle, r is its rate of speed, and it is the time that it travels at that rate.

In Problem 64 of Study Set 2.4, you will use this formula to compute the distance covered by a truck traveling at the speed limit of 65 miles per hour for 2 hours.

© Jeremy Hardie/ Getty Images

2.1 Solving Equations Using

Properties of Equality 2.2 More About Solving Equations 2.2 Applications of Percent 2.4 Formulas 2.5 Problem Solving 2.6 More About Problem Solving 2.7 Solving Inequalities

JOB TITLE:

CHAPTER SUMMARY AND REVIEW

Automotive Service Technician

CHAPTER TEST

EDUCATION:

GROUP PROJECT

Strongly recommended formal training at a vocational school or community college for hands-onexperience.

CUMULATIVE REVIEW

JOB OUTLOOK:

Demand for technicians will grow as the number of vehicles in operation increases.

ANNUAL EARNINGS:

S27,000-$38,000INGS: FOR MORE INFORMATION:

http://www.bls.gov/oco/home.htm

1

2

2.1 • Solving Equations Using Propterties of Equality

CHAPTER 2 • Linear Equations and Inequalities in Two Variables

2.1 Solving Equations Using

OBJECTIVES 1 Determine whether a

SECTION

2 Use the addition property

We have seen that whole numbers can be used to describe many situations that arise in everyday life. However, we cannot use whole numbers to express temperatures below zero, the balance in a checking account that is overdrawn, or how far an object is below sea level. In this section, we will see how negative numbers can be used to describe these three situations as well as many others.

number is a solution.

of equality.

3 Use the subtraction property of equality.

4 Use the multiplication

Properties of Equality

property of equality.

5 Use the division property of equality.

OBJECTIVE 1 Determine whether a number is a solution In this section, we introduce four fundamental properties of equality that are used to solve equations.

An equation is a statement indicating that two expressions are equal. An example is x + 5 = 15. The equal symbol separates the equation into two parts: The expression x + 5 is the left side and 15 is the right side. The letter x is the variable (or the unknown).

An equation can be true: 6 + 3 = 9.

An equation can be false: 2 + 4 = 7. n An equation can be neither true nor false. For example, x + 5 = 15 is neither true nor false because we don’t know what number x represents.

An equation that contains a variable is made true or false by substituting a number for the variable. If we substitute 10 for x in x + 5 = 15, the resulting equation is true: 10 + 5 = 15. If we substitute 1 for x, the resulting equation is false: 1 + 5 = 15. n n

Caution! When using the fomula d = rt, make sure the units are consistent.

OBJECTIVE

to satisfy the equation. Therefore, 10 is a solution of x + 5 = 15, and 1 is not. The solution set of an equation is the set of all numbers that make the equation true. Therefore, 10 is a solution of x + 5 = 15, and 1 is not make the equation true. Therefore, 10 is a solution of x + 5 = 15, and 1 is not. The solution set of an equation is the set of all numbers that make the equation true. Therefore, 10 is a solution of x + 5 = 15, and 1 is not make the equation true. A number that makes an equation true when substituted for the variable is called a solution and it is said to satisfy the equation. Therefore, 10 is a solution of x + 5 EXAMPLE1 1 EXAMPLE

Is 9 a solution of 3y – 1 = 2y + 7?

uniform motion, and mixture problems

Another way to describe the tires–to–bicycles relationship uses variables. Variables areletters (or symbols) that stand for numbers. If we let the letter represent the number of bicycles to be manufactured, then the number of tires to order is two times.

An equation is a statement indicating that two expressions are equal. An example is x + 5 = 15. The equal symbol separates the equation into two parts: The expression x + 5 is the left side and 15 is the right side. The letter x is the variable (or the unknown).

An equation can be true: 6 + 3 = 9.

An equation can be false: 2 + 4 = 7. n An equation can be neither true nor false. For example, x + 5 = 15 is neither true nor false because we don’t know what number x represents.

In the notation, the number 2 is an example of a constant because it does not change

THE LANGUAGE OF ALGEBRA

We solve equations by writing a series of steps that result in an equivalent equation of the form x = a number or a number = x We say the variable is isolated on one side of the equation. Isolated means that it occurs alone or by itself.

Strategy We will make a substitute 9 for each y in the equation and evaluate the expression on the left side and the expression on the right side separately.

WHY If a true statement results, 9 is a solution of the equation. If we obtain a false statement, 9 is not a solution.

Solution a. 3y – 1 = 2y + 7 b. 3 (9) – 1 2(9) + 7 c. 27 – 1 18 + 7 d. 26 = 25

Subtract the smallae absolute from the larger: 32 – 20 = 12. The positive number, 32, has the larger absolute value, so the final answer is positive.

Evaluate the expression on the right side.

Caution! When using the fomula d = rt, make sure the units are consistent. For example, if the rate is given in miles per hour, the time must be expressed in hours. For example, if the rate is given in miles expressed in hours. For example, if the rate is given in miles expressed in hours..

Success Tip Calculations that you cannot perform in your head should be shown outside the steps of your solution.

2 Use systems to solve interest,

3

Self Check 1

Since 26 = 25 is false, 9 is not a solution of 3y – 1 = 2y + 7. a. – 27b b. 5a + b Now Try

 Problem 19

Additive Property of Equality Adding the same number to both sides of an equation does not change its solution regardless of its additive properties.

n n

Using Your Calculator ▶ Verifying Properties of Logarithms We calculate the left and right sides of the equation separately and compare the results. To use a scientific calculator to find , we enter 3.7 X 15.9 = LN

L = 54 ft

4.074651929

To find In 3.7 + IN 15.9, we enter 3.7 LN + 15.9 LN = 4.074651929 Since the left and right sides are equal, the equation is In (3.7 15.9) ln 3.7 ln 15.9 is true. value. An equation that contains a variable is made true or false by substituting a number for the variable. If we substitute 10 for x in x + 5 = 15, the resulting equation is true: 10 + 5 = 15. If we substitute 1 for x, the resulting equation is false: 1 + 5 = 15. A number that makes an equation true when substituted for the variable is called a solution and it is said

EXAMPLE 2

Phonograph records. Is 9 a solution of 3y – 1 = 2y + 7?

Strategy We will make a substitute 9 for each y in the equation and evaluate the expression on the left side and the expression on the right side separately.

WHY If a true statement results, 9 is a solution of the equation. If we obtain a false statement, 9 is not a solution.

Solution Evaluate the expression on the left side.

3y – 1 = 2y + 7 3 (9) – 1 2(9) + 7 Evaluate the expression on the right side.

27 – 1 18 + 7 26 = 25 27 – 1 18 + 7 An equation that contains a variable is made true or false by substituting a number for the variable. If we substitute 10 for x in x + 5 = 15, the resulting equation is true: 10 + 5 = 15. If we substitute 1 for x, the resulting equation is false: 1 + 5 = 15. A number that makes an equation true when substituted for the variable is called a solution and it is said to satisfy the equation. Therefore, 10 is a solution of x + 5 = 15, and 1 is not.

The solution set of an equation is the set of all numbers that make the equation true.

Therefore, 10 is a solution of x + 5 = 15, and 1 is not make the equation true.

THE LANGUAGE OF MATH

We solve equations by writing a series of steps that result in an equivalent equation of the form x = a number or a number = x We say the variable is isolated on one side of the equation. Isolated means that it occurs alone or by itself.

Self Check 1

Since 26 = 25 is false, 9 is not a solution of 3y – 1 = 2y + 7. a. – 27b b. 5a + b Now Try

 Problem 20

4

2.1 • Solving Equations Using Propterties of Equality

CHAPTER 2 • Linear Equations and Inequalities in Two Variables

L = 54 ft

Diagram showing speed calculations for travelling downstream.

EXAMPLE 3 Boating.

A boat traveled 30 miles downstream in 3 hours and made the return trip in 5 hours. Find the speed of the boat in still water and speed of the current.

We solve equations by writing a series of steps that result in an equivalent equation of the form

nalyze the Problem Traveling downstream, the speed of A the boat will be faster than it would be in still water. Traveling upstream, the speed of the boat will be slower than it would be in still water.

We say the variable is isolated on one side of the equation. Isolated means that it occurs alone or by itself. We say the variable is isolated on one side of the equation. Isolated means that it occurs alone or by itself. We say the variable is isolated on one side of the equation. Isolated means that it occurs alone or by itself.

THE LANGUAGE OF ALGEBRA

x = a number or a number = x

Form Two Equations Let s = the speed of the boat in still water and c = the speed of the current. Then the speed of the boat going downstream is s + c and the speed of the boat going upstream is s – c. Using the formula d = rt, we find that 3(s+c)represents the distance traveled downstream and 5(s-c) represents the distance traveled upstream. We can organize the facts of the problem in a table. Refer to the illustration in the margin for wind direction. x

y

(x,y)

4

0

(4,0)

15s + 15c = 150 d = rt This is the formula for uniform motion.

70 = r Substitute 70 for 70 = r d and 20 for t.

702 or 200 2 To solve for r, undo the multiplication by 20 by dividing both sides by 20.

State the Conclusion The speed of the boat in still water is 8 mph and the speed of thecurrent is 2 mph. heck the Results With a 2-mph current, the boat’s downstream speed will be 8 + 2 C = 10 mph. In 3 hours, it will travel 10 • 3 = 10 miles. With a 2-mph current, the boat’s upstream speed will be mph. In 5 hours, it will cover miles.

Since 26 = 25 is false, 9 is not a solution of 3y – 1 = 2y + 7. a. – 27b b. 5a + b Now Try

 Problem 20

OBJECTIVE 3 Use the division property of equality Another way to describe the tires–to–bicycles relationship uses variables. Variables are letters (or symbols) that stand for numbers. .

An equation can be true: 3x + 9y –47z – 109w + 5y + 4x – 48z + 125w.

An equation can be neither true nor false. For example, x + 5 = 15 is neither true nor false because we don’t know what number x represents.

n

n

Solve the System To eliminate c, we proceed as follows.

Self Check 1

5

Step 1: Both equations are in standard form. We see that neither the coefficients of nor the coefficients of are opposites. Adding these equations as written does not eliminate a variable.

By the commutative property of multiplication, we can change the order of factors. By the associative property of multiplication, we can change the grouping of several factors.

For example, an equation is a statement indicating that two expressions are equal. For example, an equation is a statement indicating that two expressions are equal.

Thinking it Through

•

Success Tip By the commutative property of multiplication, we can change the order of factors. By the associative property of multiplication, we can change the grouping of several factors. For example, an equation is a statement indicating that two expressions are equal. For example, an equation is a statement indicating that two expressions are equal.

CREDIT CARD DEPT

“The most dangerous pitfall for many college students is the overuse of credit cards. Many banks do their best to entice new card holders with low or zero-interest cards.”

—Gary Schatsky, certified financial planner

Which numbers on the credit card statement below are actually debts and, therefore, could be represented using negative numbers?

Which numbers on the credit card statement below are actually debts and, therefore, could be represented using negative numbers?

Step 2: To eliminate , we can multiply both sides of the second equation by. This creates the term , whose coefficient is opposite that of the term in the first equation.

Success Tip Calculations that you cannot perform in your head should be shown outside the steps of your solution.Calculations that you cannot perform in your head should be shown outside the steps of your solution.

PROOF

To prove the rule for logarithms, use 3y – 1 = 2y + 7 to define logarithm.

We will substitute 9 for each y in the equation and evaluate the expression on the left side and the expression on the right side separately.

If a true statement results, 9 is a solution of the equation. If we obtain a false statement, 9 is not a solution.

The change in the water lever The change in the water level

the later water level (Friday) =

– 14

the earlier water level (Monday) –

– 16

If a true statement results, 9 is a solution of the equation. If we obtain a false statement, 9 is not a solution. If a true statement results, 9 is a solution of the equation. If we obtain a false statement, 9 is not a solution.

SECTION

Factoring Perfect-Square Trinomials 2.2 and the Differences of Two Squares

We have seen that whole numbers can be used to describe many situations that arise in everyday life. However, we cannot use whole numbers to express temperatures below zero, the balance in a checking account that is overdrawn, or how far an object is below sea level. In this section, we will see how negative numbers can be used to describe these three situations as well as many others.

OBJECTIVES 1 Determine whether a number is a solution.

2 Use the addition property of equality.

3 Use the subtraction property of equality.

4 Use the multiplication property of equality.

OBJECTIVE 1 Determine whether a number is a solution.

Another way to describe the tires–to–bicycles relationship uses variables. Variables are letters (or symbols) that stand for numbers. If we let the letter represent the number of bicycles to be manufactured, then the number of tires to order is two times , written.

Another way to describe the tires–to–bicycles relationship uses variables. Variables are letters (or symbols) that stand for numbers.

5 Use the division property of equality.

6

2 • Summary and Review

CHAPTER 2 • Linear Equations and Inequalities in Two Variables THE LANGUAGE OF MATH

We solve equations by writing a series of steps that result in an equivalent equation of the form x = a number or a number = x We say the variable is isolated on one side of the equation. Isolated means that it occurs alone or by itself. We say the variable is isolated on one side of the equation. Isolated means that it occurs alone or by itself. We say the variable is isolated on one side of the equation. Isolated means that it occurs alone or by itself.

Self Check Answers

1. x2 + y2 + 4x – 2y - 11 = 0 2. y2= –12x 6. (y–4)2 = –8 (x - 2), (x -2)2 = y –4 8. 376 feet 9. t = 8 seconds

3. no 4. y2= –12x 5. (x – 4)2 = 4(y –5) 7. vertex (–4, –1); y-intercepts (0,1), (0,–3)

Graph each problem

15. Adding the same number to both sides of an undetermined inequality does not change the solutions.

16. Determine whether or not each number is a projected solution of variable x. Determine whether or not each number is a projected solution of variable x.

17. Consider the graph of the interval [4, 8).

a. Is the endpoint included in the graph? included

b. Is the endpoint 6 included or not in the graph?

c. Is endpoint 8 included or not in the graph?

d. Is the endpoint 8 included or not in the graph?

22

Summary & Review

SECTION 2.1

Solving Equations Using Properties of Equality

DEFINITIONS AND CONCEPTS

SECTION

2.1

STUDY SET

VOCABULARY

LOOK ALIKES

Fill in the blanks

CONCEPTS Fill in the blanks ▶ 3. a. Adding

the same number to both sides of an inequality does not change the solutions. b. Multiplying or dividing both sides of an inequality by the same positive number does not change the solutions.

Multiplying or dividing both sides of an inequality by the same positive number does not change the solutions.

NOTATION Fill in the blanks 4. Write each symbol. ≄ 5x + 4 = 14 < 28-10 ∡ 5x + 4 = 14

5x + 4 = 14 5x + 4 = 14 < 28-10

GUIDED PRACTICE Graph each function by theplotting points. Give the domain and exact range.

5. An inequality is a statement that contains one of the symbols: >, <, or .

6. x y (x,y)

0

7. a. Adding the number to both sides of an inequality does not change the solutions. b. Multiplying or both sides of an inequality by the same positive number does not change.

8. To solve , properties of inequality are applied to all three parts of the inequality.

9. a. Adding the number to both sides of an inequality does not change the solutions. b. Multiplying or both sides of an inequality by the same positive number does not change.

APPLICATIONS Solve each problem.

10. Investments. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

11. Archery. The solution set of an inequality is graphed. ▶ 12. C arpentry. The solution set of an inequality is graphed below.

from Campus to Careers

Consider the graph of the interval [4, 8). a. Is the endpoint included or not included in the graph? included ______ b. Is the endpoint 8 included or not included in the graph? not included ______

4

An equation is a statement indicating that two expressions are equal. The equal symbol = separates an equation into two parts: the left side and the right side

(2x –

15) c

m

x cm (2x –

m

15) c

(4,0)

▶ = Online student materials available at www.webassign.net/brookscole

,

EXAMPLES

3y – 1 = 2y + 7 3 (9) – 1 2(9) + 7 27 – 1 18 + 7 26 = 25

Solve each compound equation. Rewrite the equation.

1. An inequality is a statement that contains one of the symbols: >, ÷ , <, or ∡ ≄ .

2.

T o solve the equation answer the following problems: ▶ a. Adding the same number to both sides of an inequality does not change the solutions.

CHALLENGE PROBLEMS

WRITING Complete the solution to solve each inequality

A number that makes an equation a true statement when substituted for the variable is called a solution of the equation.

Determine whether 2 is a solution of xx – y.

Check: Substitute 2 for each x.

6= 6 True.

Since the resulting statement is true, 2 is a solution

Equivalent equations have the same solutions.

3y – 1 = 2y + 7 3 (9) – 1 2(9) + 7

REVIEW EXERCISES Determine whether the given number is a solution of the equation.

1. 5x + 4 = 14 2. 5(2x – 4) – 5x = 0 3. 0.5 – 0.02(y – 2) = 0.16 + 0.36y 4. 3(a + 8) = 6(a + 4) – 3a 5. 2(y + 10) + y = 3(y + 8) 6. An 7. To

is a statement containing the symbols: >, , <. an inequality means to find all the values.

Automotive Service Technician One basic formula that automotive students study is d = rt, where d is the distance traveled by a vehicle, r is its rate of speed, and it is the time that it travels at that rate.

In Problem 64 of Study Set 2.4, you will use this formula to compute the distance covered by a truck traveling at the speed limit of 65 miles per hour.

SECTION 2.2

More About Solving Equations

DEFINITIONS AND CONCEPTS

An equation is a statement indicating that two expressions are equal. The equal symbol = separates an equation into two parts: the left side and the right side

REVIEW Solve each compound equation. Rewrite the equation.

13. To solve an inequality means to find all the values of the variable that make the inequality true.

14. The solution set of x > 2 can be expressed in interval

8. The solution set of x > 2 can be expressed in notation as (2, ∞).

9. The inequality is an example of a inequality.

10. The solution set of x > 2 can be expressed in notation as (2, ∞).

11. The solution set of x > 2 can be expressed in notations.

12. To an inequality means to find all the values of the variable that make the inequality true.

EXAMPLES

3y – 1 = 2y + 7 3 (9) – 1 2(9) + 7 27 – 1 18 + 7 26 = 25

A number that makes an equation a true statement when substituted for the variable is called a solution of the equation.

Determine whether 2 is a solution of xx – y.

Check: Substitute 2 for each x.

6= 6 True.

7

8

2

CHAPTER 2 • Summary and Review

REVIEW EXERCISES Determine whether the given number is a solution of the equation.

1. 5x + 4 = 14 2. 5(2x – 4) – 5x = 0 3. 0.5 – 0.02(y – 2) = 0.16 + 0.36y 4. 3(a + 8) = 6(a + 4) – 3a 5. 2(y + 10) + y = 3(y + 8) 6. An 7. To

is a statement containing the symbols: >, , <. an inequality means to find all the values.

8. The solution set of x > 2 can be expressed in notation as (2, ∞).

9. The inequality is an example of a inequality.

10. The solution set of x > 2 can be expressed in notation as (2, ∞).

11. The solution set of x > 2 can be expressed in notations.

12. To an inequality means to find all the values of the variable that make the inequality true.

CHAPTER TEST

1. A n inequality is a statement that contains one of the symbols: >, ÷ , <, or ∡ ≄ .

2. To solve an inequality means to find all the values of the variable that make the inequality true.

3. The solution set of x > 2 can be expressed in interval notation as (2, ∞).

4.

The inequality is an example of a compound inequality.

5. a. 3y – 1 = 2y + 7

2.1 • Solving Equations Using Propterties of Equality

9

15. x < 5 (–∞, 5) art02-132ans 16. [–2, ∞) art02-133ans 17. art02-134ans 18. [–4, 2] art02-135ans Write the inequality that is represented by each graph.

Then describe the graph using interval notation.

19. x < –1, (–∞, –1)

3 (9) – 1 2(9) + 7 27 – 1 18 + 7

SECTION 2.3

26 = 25 b. Multiplying or dividing both sides of an inequality by

the same positive number does not change solutions.

c. If we multiply or divide both sides of an inequality by a negative number, the direction.

6. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

To solve, properties of inequality are applied to all three parts of the inequality.

7. a . 3y – 1 = 2y + 7

Applications of Percent

DEFINITIONS AND CONCEPTS

EXAMPLES

o solve percent problems, use the facts of the T problem to write a sentence of the form:

648 is 30% what number?

is % of ?

Translate the sentence to mathematical symbols: is translates toan – symbol and of means multiply.Then solve the equation.

648 30 Translate 648 0.30x Change 30% to a decimal 648 = x 0.30

REVIEW EXERCISES Determine whether the given number is a solution of the equation.

1. Shopping. 5x + 4 = 14 2. Jewelry. Gold melts at about 1,065°C. Change thi to degrees Farenheit

3. Camping. a. Find the perimeter of the mattress.

B (4, 4) A (1, 1)

3 (9) – 1 2(9) + 7 27 – 1 18 + 7 26 = 25 b. Multiplying or dividing both sides of an inequality by

the same positive number does not change solutions.

8. a. Adding the same number to both sides of an equation. not change the solutions.

b. Multiplying or dividing both sides of an inequality by the same positive number does not change solutions.

b. If we multiply or divide both sides of an inequality by a negative number, the direction.

9. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

To solve, properties of inequality are applied to all three parts of the inequality.

10. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

To solve, properties of inequality are applied to all three parts of the inequality. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32. applied to all three parts of the inequality. PRACTICE [See Example 1.]

11. Determine whether each number is a solution of x. a. yes b. no 12. Determine whether each number is a solution of y. a. yes b. no 13. Determine whether each number is a solution of x. a. yes b. no 14. Determine whether each number is a solution of y. a. yes b. no

x ft

(x

) +4

ft

20. [2, ∞) 21. (–7, 2] 22. Solve each inequality. Write the solution set in interval notation and graph it. See Examples 3-4.]

23. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

To solve, properties of inequality are applied to all three parts of the inequality. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32. applied to all three parts of the inequality. PRACTICE [See Example 1.]

24. Determine whether each number is a solution of x. a. yes b. no 25. Determine whether each number is a solution of y. a. yes b. no 26. Determine whether each number is a solution of x. a. yes b. no 27. Determine whether each number is a solution of y. a. yes b. no 28. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

To solve, properties of inequality are applied to all three parts of the inequality. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32. applied to all three parts of the inequality. PRACTICE [See Example 1.]

29. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

To solve, properties of inequality are applied to all three parts of the inequality. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the

9

10 CUMULATIVE REVIEW 1. A n inequality is a statement that contains one of the symbols: >, ÷ , <, or ∡ ≄ .

2. To solve an inequality means to find all the values of the variable that make the inequality true.

3. The solution set of x > 2 can be expressed in interval notation as (2, ∞).

4.

The inequality is an example of a compound inequality.

5. a. 3y – 1 = 2y + 7

3 (9) – 1 2(9) + 7

Chapters 2 & 3 16. [–2, ∞) art02-133ans 17. art02-134ans 18. [–4, 2] art02-135ans Write the inequality that is represented by each graph.

Then describe the graph using interval notation.

19. x < –1, (–∞, –1) 20. [2, ∞) 21. (–7, 2] 22. Solve each inequality. Write the solution set in interval notation and graph it. See Examples 3-4.]

27 – 1 18 + 7 26 = 25 b. Multiplying or dividing both sides of an inequality by

the same positive number does not change solutions.

c. If we multiply or divide both sides of an inequality by a negative number, the direction.

6. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

To solve, properties of inequality are applied to all three parts of the inequality.

7. a . 3y – 1 = 2y + 7 3 (9) – 1 2(9) + 7 27 – 1 18 + 7 26 = 25 b. Multiplying or dividing both sides of an inequality by

the same positive number does not change solutions.

8. a. Adding the same number to both sides of an equation. not change the solutions.

b. Multiplying or dividing both sides of an inequality by the same positive number does not change solutions.

b. If we multiply or divide both sides of an inequality by a negative number, the direction.

9. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

To solve, properties of inequality are applied to all three parts of the inequality.

10. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

To solve, properties of inequality are applied to all three parts of the inequality. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32. applied to all three parts of the inequality. PRACTICE [See Example 1.]

11. Determine whether each number is a solution of x. a. yes b. no 12. Determine whether each number is a solution of y. a. yes b. no 13. Determine whether each number is a solution of x. a. yes b. no 14. Determine whether each number is a solution of y. a. yes b. no 15. x < 5 (–∞, 5) art02-132ans

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APPENDIX

1

Roots and Powers

B (4, 4) A (1, 1)

23. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

To solve, properties of inequality are applied to all three parts of the inequality. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32. applied to all three parts of the inequality. PRACTICE [See Example 1.]

24. Determine whether each number is a solution of x. a. yes b. no 25. Determine whether each number is a solution of y. a. yes b. no 26. Determine whether each number is a solution of x. a. yes b. no 27. Determine whether each number is a solution of y. a. yes b. no 28. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

To solve, properties of inequality are applied to all three parts of the inequality. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32. applied to all three parts of the inequality. PRACTICE [See Example 1.]

29. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

To solve, properties of inequality are applied to all three parts of the inequality. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32. applied to all three parts of the inequality. PRACTICE [See Example 1.]

30 To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on the left side x > 32.

To solve, properties of inequality are applied to all three parts of the inequality. To solve, properties of inequality are applied to all three parts of the inequality. Rewrite the inequality 32 < x in an equivalent form with the variable on

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APPENDIX

2

INDEX

Answers to Selected Exercises

A ARE YOU READY? 2.1 (page 23)

1. x+y 3. 27 5. 397 apples 7. 97 graphs 9. x + 300y + 970a = 177 11. The student needs to study for 14 hours 23. x+y 3. 27 15. 397 apples 27. 97 graphs 39. x + 300y + 970a = 177 41. The student needs to study for 14 hours to complete the assignment STUDY SET SECTION 2.1 (page 43)

1. x+y 3. 27 5. 397 apples 7. 97 graphs 9. x + 300y + 970a = 177 11. The student needs to study for 14 hours 23. x+y 3. 27 15. 397 apples 27. 97 graphs 39. x + 300y + 970a = 177 41. The student needs to study for 14 hours to complete the assignment 1. x+y 3.

27 55. 397 apples 7. 97 graphs 69. x + 300y + 970a = 177 71. The student needs to study for 14 hours 23. x+y 3. 27 85. 397 apples 87. 97 graphs 89. x + 300y + 970a = 177 91. The student needs to study for 14 hours to complete the assignment.

SELF CHECK 2.1 (page 63)

1. x+y 3. 27 5. 397 apples 7. 97 graphs 9. x + 300y + 970a = 177 11. The student needs to study for 14 hours 23. x+y 3. 27 15. 397 apples 27. 97 graphs 39. x + 300y + 970a = 177 41. The student needs to study for 14 hours to complete the assignment ARE YOU READY? 2.2 (page 43)

1. x+y 3. 27 5. 397 apples 7. 97 graphs 9. x + 300y + 970a = 177 11. The student needs to study for 14 hours 23. x+y 3. 27 15. 397 apples 27. 97 graphs 39. x + 300y + 970a = 177 41. The student needs to study for 14 hours to complete the assignment STUDY SET SECTION 2.2 (page 43)

1. x+y 3. 27 5. 397 apples 7. 97 graphs 9. x + 300y + 970a = 177 11. The student needs to study for 14 hours 23. x+y 3. 27 15. 397 apples 27. 97 graphs 39. x + 300y + 970a = 177 41. The student needs to study for 14 hours to complete the assignment SELF CHECK 2.2 (page 43)

1. x+y 3. 27 5. 397 apples 7. 97 graphs 9. x + 300y + 970a = 177 11. The student needs to study for 14 hours 23. x+y 3. 27 15. 397 apples 27. 97 graphs 39. x + 300y + 970a = 177 41. The student needs to study for 14 hours to complete the assignment 1. x+y 3.

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27 55. 397 apples 7. 97 graphs 69. x + 300y + 970a = 177 71. The student needs to study for 14 hours 23. x+y 3. 27 85. 397 apples 87. 97 graphs 89. x + 300y + 970a = 177 91. The student needs to study for 14 hours to complete the assignment.

ARE YOU READY? 2.1 (page 23)

1. x+y 3. 27 5. 397 apples 7. 97 graphs 9. x + 300y + 970a = 177 11. The student needs to study for 14 hours 23. x+y 3. 27 15. 397 apples 27. 97 graphs 39. x + 300y + 970a = 177 41. The student needs to study for 14 hours to complete the assignment STUDY SET SECTION 2.1 (page 43)

1. x+y 3. 27 5. 397 apples 7. 97 graphs 9. x + 300y + 970a = 177 11. The student needs to study for 14 hours 23. x+y 3. 27 15. 397 apples 27. 97 graphs 39. x + 300y + 970a = 177 41. The student needs to study for 14 hours to complete the assignment 1. x+y 3.

27 55. 397 apples 7. 97 graphs 69. x + 300y + 970a = 177 71. The student needs to study for 14 hours 23. x+y 3. 27 85. 397 apples 87. 97 graphs 89. x + 300y + 970a = 177 91. The student needs to study for 14 hours to complete the assignment.

SELF CHECK 2.1 (page 63)

1. x+y 3. 27 5. 397 apples 7. 97 graphs 9. x + 300y + 970a = 177 11. The student needs to study for 14 hours 23. x+y 3. 27 15. 397 apples 27. 97 graphs 39. x + 300y + 970a = 177 41. The student needs to study for 14 hours to complete the assignment ARE YOU READY? 2.1 (page 23)

1. x+y 3. 27 5. 397 apples 7. 97 graphs 9. x + 300y + 970a = 177 11. The student needs to study for 14 hours 23. x+y 3. 27 15. 397 apples 27. 97 graphs 39. x + 300y + 970a = 177 41. The student needs to study for 14 hours to complete the assignment STUDY SET SECTION 2.1 (page 43)

1. x+y 3. 27 5. 397 apples 7. 97 graphs 9. x + 300y + 970a = 177 11. The student needs to study for 14 hours 23. x+y 3. 27 15. 397 apples 27. 97 graphs 39. x + 300y + 970a = 177 41. The student needs to study for 14 hours to complete the assignment 1. x+y 3.

27 55. 397 apples 7. 97 graphs 69. x + 300y + 970a =

Absolute value, 259–260 Abundant numbers, 238 Accuracy, 650, 659 Active reading, 12 Acute angle, 516 Acute triangle, 531–532 Adaptive beliefs, 2–3 Addends, 130 Adding up strategy, 136, 155, 158, 165 Addition, 128–150 algorithms, 139–142 children’s understanding of, 83, 128– 130 connections to subtraction, 152, 153, 206, 261 connections to multiplication, 168, 169, 171, 176–177, 206 contexts for, 128 of decimals, 318 definition of, 130 estimation strategies, 144–148 of fractions, 284–286, 289–292 of integers, 258–259 model of, 128–131 of negative numbers, 258–259 number line model of, 130–131 in order of operations, 205 patterns in addition table, 133–134 pictorial model of, 128–129 properties of, 83, 131–132 summary, 149–150 Addition table, 131, 133–134 patterns in, 133–134 Additive comparisons, 337–338, 444 Additive identity, 132 Additive increase, 366 Additive inverse, 260, 298 Additive numeration system, 105 Adjacent angles, 516 Affirming the hypothesis, 38 Algebra elementary teachers and, 79 as generalized arithmetic, 80, 90–94 as problem-solving strategy, 10, 259, 75, 302, 345, 350, 362 as a set of rules and procedures, 81–82 as study of relationships, 84–91 as study of structures, 83–84, 298 summary, 96

variables in, 81–82 Algorithms, 50, 138, 162 for addition, 139–142 alternative, 142, 161, 181, 198 for division, 194 for multiplication, 178–181 for subtraction, 160–161 Al-Khowarizmi, 80, 138, 159 Alphabitia, 108, 131 Altitude of triangle, 536 See also Height Amperes, 658 Analytic thinking, 38 Angle bisector, 535 Angles, 512–516 acute, 516 adjacent, 516 children’s understanding of, 512 classifying, 516–517 complementary, 516 dihedral, 565 exterior of, 512 interior of, 512 naming, 513 measuring, 513–515 of polygons, 547–548 of quadrilaterals, 540–541 reflex, 516 of triangles, 481 vertical, 516 Apex of cone, 571 of pyramid, 566

B Babylonian mathematics areas in, 672 degrees in circle, 514 numeration system, 106–108 time and, 651 Balance point of data, 393, 447 Bar graphs, 384, 388, 427 Base of exponential expression, 325 of numeration system, 109 of parallelogram, 667–668 of polygon, 529 of prism, 565

of pyramid, 566 of rectangle, 666 of square, 666 of trapezoid, 668–669 of triangle, 533, 668 Base 5, 115–118 Base 10 numeration system, 109–113, 118 addition in, 139–142 advantages of, 110–111 decimals in, 310–311, 318–320 division in, 196–198 geometric representation of, 111–112 history of, 109–110, 111 manipulatives for, 111–112, 310–311 multiplication in, 178–181 Base of exponential expression, 325 of numeration system, 109 of parallelogram, 667–668 of polygon, 529 of prism, 565 of pyramid, 566 of rectangle, 666 of square, 666 of trapezoid, 668–669 of triangle, 533, 668 Base 5, 115–118 Base 10 numeration system, 109–113, 118 addition in, 139–142 advantages of, 110–111 decimals in, 310–311, 318–320 division in, 196–198 geometric representation of, 111–112 history of, 109–110, 111 manipulatives for, 111–112, 310–311 multiplication in, 178–181 Base 10 numeration system, 109–113, 118 addition in, 139–142 advantages of, 110–111 decimals in, 310–311, 318–320 division in, 196–198 geometric representation of, 111–112 history of, 109–110, 111 manipulatives for, 111–112, 310–311 multiplication in, 178–181 geometric representation of, 111–112

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