Some Basics of Algebra• Algebraic Expressions and Their

Use

• Translating to Algebraic Expressions

• Evaluating Algebraic Expressions

• Sets of Numbers

1.1

Terminology

A letter that can be any one of various numbers is called a variable. If a letter always represents a particular number that never changes, it is called a constant.

Algebraic Expressions

An algebraic expression consists of variables, numbers, and operation signs.

Examples:

, 2 2 , .4

yt l w m x b

When an equal sign is placed between two expressions, an equation is formed.

Translating to Algebraic Expressions

per of less than more than

ratio twicedecreased byincreased by

quotient of times minus plus

divided byproduct ofdifference of sum of

divide multiply subtract add

DivisionMultiplicationSubtractionAddition

Key Words

Example

Translate to an algebraic expression:

Eight more than twice the product of 5 and a number.

Solution 8 2 5 n

Eight more than twice the product of 5 and a number.

Evaluating Algebraic Expressions

When we replace a variable with a number, we are substituting for the variable. The calculation that follows is called evaluating the expression.

Example

Evaluate the expression

8 for 2, 7, and 3.xz y x y z

Solution

8xz – y = 8·2·3 – 7

= 41

= 48 – 7

Substituting

Multiplying

Subtracting

Example

The base of a triangle is 10 feet and the height is 3.1 feet. Find the area of the triangle.

1 1

2 2b h

Solution

10·3.1

= 15.5 square feet

h

b

Exponential Notation The expression an, in which n is a counting number means

n factors

In an, a is called the base and n is called the exponent, or power. When no exponent appears, it is assumed to be 1. Thus a1 = a.

a a a a a

Rules for Order of Operations

1. Simplify within any grouping symbols.

2. Simplify all exponential expressions.

3. Perform all multiplication and division working from left to right.

4. Perform all addition and subtraction working from left to right.

Example

Evaluate the expression

Solution

2(x + 3)2 – 12 x2

Substituting

Simplifying 52 and 22

Multiplying and Dividing

Subtracting

2 22 3 12 for 2.x x x

= 2(2 + 3)2 – 12 22

2 22 5 12 2 2 25 12 4

Working within parentheses

50 3 47

Example

Evaluate the expression 24 2 for 3, 2, and 8.x xy z x y z

Solution

4x2 + 2xy – z = 4·32 + 2·3·2 – 8

= 36 + 12 – 8

= 40

= 4·9 + 2·3·2 – 8

Substituting

Simplifying 32

Multiplying

Adding and Subtracting

Part 2 of 1.1

Sets of Numbers

Sets of NumbersNatural Numbers (Counting Numbers)

Numbers used for counting: {1, 2, 3,…}

Whole Numbers

The set of natural numbers with 0 included: {0, 1, 2, 3,…}

Integers

The set of all whole numbers and their opposites: {…,-3, -2, -1, 0, 1, 2, 3,…}

Rational Numbers

Numbers that can be expressed as an integer divided by a nonzero integer are called rational numbers:

Sets of Numbers

is an integer, is an integer, and 0 .p

p q qq

Converting Fractions to DecimalsDivide the numerator by the denominator

6

5 .8 3 3 3

6 5.0 0 0

4 8

2 0

1 8

2 0

38.0

...8333.0

36.011

4

6.03

2

375.08

3

Any fraction can be converted to a repeating decimal or a terminating decimal..

All integers can be written as fractions. Insert a denominator of 1.

1

1414

1

33

Look at the following conclusion.

Rationals include the following.

• All integers (-2, 5, 17, 0)• All fractions (proper,

improper, or mixed)

• All terminating decimals

• All repeating decimals

2

11,

8

7,

11

91

-2.34, 0.0456

784.3 ,2.1

Sets of Numbers

Real Numbers

Numbers that are either rational or irrational are called real numbers:

is rational or irrational .x x

Numbers like are said to be irrational. Decimal notation for irrational numbers neither terminates nor repeats.

5 and

Identify as natural,whole, integers, rational, or

irrational.

5 -3 105

4

0.457 16 0 6.2

3

25

2

342.0

9

400

Answers

• Natural: 10

• Whole: 10 0

• Integers: 10 0 -3

• Rational:

• Irrational:

16

16

16

9

400 ,342.0 ,

3

25 ,2.6 ,0 ,16 ,457.0 ,

5

4 ,10, 3

2 ,5

Set Notation

Roster notation: {2, 4, 6, 8}

Set-builder notation: {x | x is an even number between 1 and 9}

“The set of all x such that x is an even number between 1 and 9”

Write with Roster Notation12} and 5between number even an is |{ xx

}10,8,6{

7}least at number natural a is |{ xx

}7,6,5,4,3,2,1{

Write with Set-Builder Notation

1} and 6-between integer an is |{ xx

}17,15,13,11,9{ )2

18} and 8between number oddan is |{ xx

1) The set of all integers between -6 and 1

5)

33} and 21between 4 of multiple a is |{ xx

}32,28,24{

Elements and SubsetsIf B = { 1, 3, 5, 7}, we can write 3 B to indicate that 3 is an element or member of set B. We can also write 4 B to indicate that 4 is not an element of set B.

When all the members of one set are members of a second set, the first is a subset of the second. If A = {1, 3} and B = { 1, 3, 5, 7}, we write A B to indicate that A is a subset of B.

True or False?Use the following sets: N= Naturals, W = Wholes,

Z = integers, Q = Rationals, H = Irrationals,

and R = Reals

NW

ZH

RQ

Q

H

W

6

5

4

3.2

Answers

• False

• True

• False

• True

• True

• False