Copyright 2013, 2009, 2005, 2002 Pearson, Education, Inc.

1.2

Algebraic Expressions and Sets of Numbers

A variable is a letter used to represent any number.

A constant is either a fixed number or a letter that represents a fixed number.

An algebraic expression is formed by numbers and variables connected by the operation of addition, subtraction, multiplication, division, raising to powers, and/or taking roots.

Algebraic Expressions

3 5x 25 6 40a a 4 2 15

4y y

y

Evaluating Algebraic Expressions

To evaluate an algebraic expression, substitute the numerical value for each variable into the expression and simplify the result.

Example

Evaluate each expression for the given value.

a) 5x – 2 for x = 8

= 5(8) – 2

= 40 – 2

= 38

b) 3a2 + 2a + 4 for a = – 4

= 3(– 4)2 + 2(– 4) + 4

= 3(16) + (– 8) + 4

= 44

Set of Numbers

• Natural numbers – {1, 2, 3, 4, 5, 6 . . .}• Whole numbers – {0, 1, 2, 3, 4 . . .}• Integers – {. . . –3, -2, -1, 0, 1, 2, 3 . . .}• Rational numbers – the set of all numbers that can be expressed as a quotient of integers, with denominator 0

• Irrational numbers – the set of all numbers that can NOT be expressed as a quotient of integers

• Real numbers – the set of all rational and irrational numbers combined

The Number Line

A number line is a line on which each point is associated with a number.

2– 2 0 1 3 4 5– 1– 3– 4– 5

Negative numbers

Positive numbers

– 4.8 1.5

Vocabulary

Variable is a symbol used to represent a number.

Algebraic expressions are a collection of numbers, variables, operations, grouping symbols, but NO = or inequalities.

In describing some of the previous sets, we used a . . . symbol, called an ellipsis. It means to continue in the same pattern.

The members of a set are called its elements.

When we list (or attempt to list with an ellipsis) the elements of a set, the set is written in roster form.

A set can also be written in set builder notation. This notation describes the members of a set, but does not list them.

{ x | x is an even natural number less than 10}

The set of all x

such that

x is an even natural number less than 10

Set Builder Notation

Example:

Example

Given the set of numbers

list the numbers in this set that belong to the set of:

a. Natural numbers:b. Whole numbers:c. Integers:d. Rational numbers:e. Irrational numbers:f. Real numbers:

1 3

1 , 2.5, , 3.75, 15, 2, 0, 54 2

15

0, 150, 15, 5

1 3

0, 15, 5, 1 , 3.75, 2.5, 4 2

2all the numbers in the set

Absolute Value

The absolute value of a real number a, denoted by |a|, is the distance between a and 0 on the number line.

2– 2 0 1 3 4 5– 1– 3– 4– 5

| – 4| = 4

Distance of 4

Symbol for absolute value

|5| = 5

Distance of 5

Example

Find each absolute value.

a.

b. c.

d.

9 9

0 0

6 6

4 4

5 5

Finding the Opposite of a Number

Opposite

The opposite of a number a is the number a.

Double Negative Property

For every real number a, (a) = a.

Example

Write the opposite of each number.

a. 12 The opposite of 12 is 12.

b. The opposite of 2/3 is 2/3.

c. 10.4 The opposite of 10.4 is 10.4.

2

3

Translating Phrases

Addition

(+)

Subtraction

(–)

Multiplication

(·)

Division

()

sum

plus

added to

more than

increased by

total

difference

minus

subtract

less than

decreased by

less

product

times

multiply

multiplied by

of

double/triple

quotient

divide

shared equally

among

divided by

divided into

Example

Write as an algebraic expression. Let x represent the unknown number.

a. 5 decreased by a number 5 – x

b. The quotient of a number and 12

12 or 12xx

Translate the following phrases into algebraic expressions.

a. The sum of 4 and twice y

4 + 2y

b. x less than the product of y and z

yz x

Example