Chapter 1 Algebra, Mathematical Models, and Problem Solving.

Chapter 1

Algebra, Mathematical Models, and Problem

Solving

§ 1.1

Algebraic Expressions and Real Numbers

Algebra uses letters such as x and y to represent numbers. If a letter is used to represent various numbers, it is called a variable. For example, the variable x might represent the number of minutes you can lie in the sun without burning when you are not wearing sunscreen.

Variables in Algebra

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 1.1

Suppose you are wearing number 6 sunscreen. If you can normally lie in the sun x minutes without burning, with the number 6 sunscreen, you can lie in the sun 6 times as long without burning - that is, 6 times x or 6x would represent your exposure time without burning.

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 1.1

A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots, is called an algebraic expression.

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 1.1

Algebraic Expressions

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 1.1

English Phrase Mathematical Operation

sum

plus

increased by

more than

Addition

difference

minus

decreased by

less than

Subtraction

product

times

of (used with fractions)

twice

Multiplication

quotient

divide

per

ratio

Division

Translating Phrases into Expressions

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 1.1

Translating Phrases into Expressions

EXAMPLEEXAMPLE

Write the English phrase as an algebraic expression. Let x represent

Four more than five times a number

SOLUTIONSOLUTION

45 x

the number.

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 1.1

Evaluating an Algebraic Expression

EXAMPLEEXAMPLE

The formula expresses the relationship between Fahrenheit temperature, F, and Celsius temperature, C. Use the formula to convert the given Fahrenheit temperature to its equivalent temperature on the Celsius scale.

SOLUTIONSOLUTION

Replace F with 50

329

5 FC

32509

5C

F50

329

5 FC

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 1.1

Evaluating an Algebraic Expression

Multiply

Therefore

CONTINUECONTINUEDD

Subtract 189

5C

10C

C10F50 .

In evaluating expressions, what comes first?

• #1 Start with the parentheses. Parentheses say “Me First!”

• #2 Then evaluate the exponential expressions.

• #3 Multiplications and divisions are equal in the order of operations – Perform them next.

• #4 Additions and subtractions are also equal to each other in order – and they come last.

Remember by “PEMDAS” - parentheses, exponents, multiplication, division, addition, subtraction

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 1.1

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 1.1

Order of Operations - PEMDAS

Order of Operations

1) First, perform all operations within grouping symbols

2) Next, Evaluate all exponential expressions.

3) Next, do all multiplications and divisions in the order in which they occur working from left to right.

4) Finally, do all additions and subtractions in the order in which they occur, working from left to right.

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 1.1

Order of Operations - PEMDAS

Evaluate for .

EXAMPLEEXAMPLE

Replace R with 3

Evaluate inside parentheses first

Evaluate – first exponent

SOLUTIONSOLUTION

43 62 RR 3R

43 62 RR

43 3623

43 323

43227 33

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 1.1

Order of Operations - PEMDAS

-135 Subtract

Multiply

CONTINUECONTINUEDD

27-162

27-2(81) Evaluate – second exponent43

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 1.1

Number Sets

Sets of Numbers Definition

Natural Numbers All numbers in the set {1,2,3,4,…}

Whole Numbers All numbers in the set {0,1,2,3,4,…}

Integers All numbers in the set {…-3,-2,-1,0,1,2,3,…}

Rational Numbers All numbers a/b such that a and b are integers

Irrational Numbers All numbers whose decimal representation neither terminate nor repeat

Real Numbers All numbers that are rational or irrational

NOTE: “…” means continue without end

Blitzer, Intermediate Algebra, 5e – Slide #15 Section 1.1

Three Common Number Sets

The natural numbers are the numbers we use for counting.

The set of whole numbers includes the natural numbers and 0. Zero is a whole number, but is not a natural number.

The set of integers includes all the whole numbers and their negatives. Every whole number is an integer, and every natural number is an integer.

These sets are just getting bigger and bigger…

Note that…

Blitzer, Intermediate Algebra, 5e – Slide #16 Section 1.1

Set-Builder Notation

{x | x is a real number and greater than 10}

Express x > 10 using set-builder notation

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Blitzer, Intermediate Algebra, 5e – Slide #17 Section 1.1

Rational Numbers

Rational numbers can be expressed either in fraction or in decimal notation. Every integer is rational because it can be

written in terms of division by one.

The set of rational numbers is the set of all numbers that can be expressed as the quotient of two integers with the

denominator not zero.

That is, a rational number is any number that can be written in the form a/b where a and b are integers and b is not zero.

DefinitionDefinition

Blitzer, Intermediate Algebra, 5e – Slide #18 Section 1.1

Symbols and

The symbol is used to indicate that a number or object is ina particular set. Here is an example:

7 {1,2,5,7,9}

The symbol is used to indicate that a number or object is not in a particular set. For example:

3 {4,6}

Blitzer, Intermediate Algebra, 5e – Slide #19 Section 1.1

Inequalities

Inequalities Meanings Examples

< is less than

10 < 32

-5 < 3

-7 < -2

> is greater than

6 > -4

11 > 8

-6 > -12

is less than or is equal to

3.4 4.5

-2 -2

is greater than or is equal to

5 5

0 -3