Chapter 1

Tools of Algebra

In Chapter 1, You Will…

Review and extend your knowledge of algebraic expressions and your skills in solving equations and inequalities.

Solve absolute value equations and inequalities by changing them to compound equations and inequalities.

1-1 Properties of Real Numbers

What You’ll Learn …

To graph and order real numbersTo identify and use properties of real

numbers

Number Classification

• Natural numbers are the counting numbers. 1,2,3,4,…..

• Whole numbers are natural numbers and zero. 0,1,2,3,4,….

• Integers are whole numbers and their opposites. ….-3,-2,-1,0,1,2,3,….

• Rational numbers can be written as a fraction. 7/5, -3/2, 0, 0.3, -1.2, 9

• Irrational numbers cannot be written as a fraction. 2 , 7 ,

• All of these numbers are real numbers.

Real Numbers

Rational Numbers Irrational Numbers

Integers

Whole Numbers

04/2

625

-4

-10 5

-3

58

-2 3 0.37

0.31

10

- 123

23

0.1011001000…

Example 1: Real World Connection

Many mathematical relationships involving variables are related to amusement parks. Which set of numbers best describes the values for each variable?

a. The cost C of admission for n people

b. The maximum speed s in meters per second on a roller coaster of height h in meters

c. The park’s profit (or loss) P in dollars for each week w of the year

Example 1: Check Your Understanding

The number is the ratio of the number of adult tickets sold to the number of children’s tickets sold. Which set of numbers best describes the values of ? Which set of numbers best describes the average cost per family for tickets?

Example 1: Check Your Understanding

A pilot uses the formula to find the distance in miles that can be seen when flying at an altitude of miles. Which set of numbers best describes the values for each variable?

Example 2: Graphing Numbers on a Number Line

- 3 1.7 5

2

Change toDecimals

Example 2: Check Your Understanding

Graph the numbers…

Change toDecimals

Example 2: Check Your Understanding

Graph the numbers…

Change toDecimals

−𝟑𝟒 √𝟕 𝟑 .𝟔

Example 3: Ordering Real Numbers

An inequality is a mathematical sentence that compares the value of two expressions using an inequality symbol.

< Less than

< Less than or equal to

> Greater than

> Greater than or equal to

Compare and . Use the symbols and .

Example 3: Check Your Understanding

Compare and . Use the symbols and .

Example 3: Check Your Understanding

Compare and . Use the symbols and .

The opposite or additive inverse of any number is .

* The sum of opposites is 0.

The reciprocal or multiplicative inverse of any nonzero number is .

* The product of reciprocals is 1.

Properties of Real Numbers

Find the opposite and the reciprocal of each number.

A.

B.

C.

D.

Example 4: Finding Inverses

Find the opposite and the reciprocal of each number.

A.

B.

C.

D.

Example 4: Check Your Understanding

Properties of Real Numbers

Property Addition Multiplication

Closure is a real number is a real number

Commutative

Associative

Identity or or

Inverse

𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐Distributive

Example 5: Identify Properties of Real Numbers

Which property is illustrated?

A.

B.

Example 5: Check Your Understanding

Which property is illustrated?

A.

B.

C.

D.

The absolute value of a real number is its distance from zero on the number line.

Absolute Value

Example 6: Finding Absolute Value

Find the absolute value of:

A.

B.

C.

Example 6: Finding Absolute Value

Find the absolute value of:

A.

B.

C.

Example 6: Finding Absolute Value

Find the absolute value of:

D.

E.

F.

Critical Thinking

For what values of does ?