1.1 Fractions

• Multiplying or dividing the numerator (top) and the denominator (bottom) of a fraction by the same number does not change the value of a fraction.

• Writing a fraction in lowest terms:1. Factor the top and bottom completely

2. Divide the top and bottom by the greatest common factor

1.1 Fractions

• Multiplying fractions:

• Dividing fractions:

db

ca

d

c

b

a

cb

da

c

d

b

a

d

c

b

a

1.1 Fractions

• Adding fractions with the same denominator:

• Subtracting fractions with the same denominator:

b

ca

b

c

b

a

b

ca

b

c

b

a

1.1 Fractions

• To add or subtract fractions with different denominators - get a common denominator.

• Using the least common denominator:1. Factor both denominators completely2. Multiply the largest number of repeats of each

prime factor together to get the LCD3. Multiply the top and bottom of each fraction

by the number that produces the LCD in the denominator

1.1 Fractions

• Adding fractions with different denominators:

• Subtracting fractions with different denominators:

db

cbda

d

c

b

a

db

cbda

d

c

b

a

1.1 Fractions

• Try these:

(simplify) 16

12?

14

3

9

7

?5

3

10

9 ?

9

5

9

1

?21

2

7

5 ?

4

1

9

5

1.2 Exponents, Order of Operations, and Inequality

• Exponents:

• Note:

81333334

4334

1.2 Exponents, Order of Operations, and Inequality

• PEMDAS (Please Excuse My Dear Aunt Sally)1. Parenthesis2. Exponentiation3. Multiplication / Division

(evaluate left to right)4. Addition / Subtraction

(evaluate left to right)• Note: the fraction bar implies parenthesis

1.2 Exponents, Order of Operations, and Inequality

• Symbols of equality / inequality

1. = is equal to

2. is not equal to

3. < is less than

4. is less than or equal to

5. > is greater than

6. is greater than or equal to

1.3 Variables, Expressions, and Equations

• Variable – usually a letter such as x, y, or z, used to represent an unknown number

• Evaluating expressions – replace the variable(s) with the given value(s) and evaluate using PEMDAS (order of operations)

1.3 Variables, Expressions, and Equations

• Changing word phrases to expressions:

The sum of a number and 9 x + 9

7 minus a number 7 - x

Subtract 7 from a number x – 7

The product of 11 and a number 11x

5 divided by a number

The product of 2 and the sum of a number and 8

2(x + 8)x5

1.3 Variables, Expressions, and Equations

• Equation: statement that two algebraic expressions are equal.

Expression Equation

x – 7 x – 7 = 3

No equal sign Contains equal sign

Can be evaluated or simplified

Can be solved

1.4 Real Numbers and the Number Line• Classifications of Numbers

Natural numbers {1,2,3,…}

Whole numbers {0,1,2,3,…}

Integers {…-2,-1,0,1,2,…}

Rational numbers – can be expressed as where p and q are integers

-1.3, 2, 5.3147,

Irrational numbers – not rational

qp

523

137 ,

,47,5

1.4 Real Numbers and the Number Line

• The real number line:

• Real numbers:{xx is a rational or an irrational number}

-3 -2 -1 0 1 2 3

1.4 Real Numbers and the Number Line• Ordering of Real Numbers:

a < b a is to the left of b on the number linea > b a is to the right of b on the number line

• Additive inverse of a number x: -x is a number that is the same distance from 0 but on the opposite side of 0 on the number line

1.4 Real Numbers and the Number Line• Double negative rule:

-(-x) = x• Absolute Value of a number x: the distance

from 0 on the number line or alternatively

How is this possible if the absolute value of a number is never negative?

x0 if

0 if

xx

xx

1.5 Adding and Subtracting Real Numbers

• Adding numbers on the number line (2 + 2):

-3 -2 -1 0 1 2 3-4 4

22

1.5 Adding and Subtracting Real Numbers

• Adding numbers on the number line (-2 + -2):

-3 -2 -1 0 1 2 3-4 4

-2-2

1.5 Adding and Subtracting Real Numbers

• Adding numbers with the same sign:Add the absolute values and use the sign of both numbers

• Adding numbers with different signs:Subtract the absolute values and use the sign of the number with the larger absolute value

1.5 Adding and Subtracting Real Numbers

• Subtraction:

• To subtract signed numbers:Change the subtraction to adding the number with the opposite sign

)( yxyx

12)7(5)7(5

1.6 Multiplying and Dividing Real Numbers

• Multiplication by zero:For any number x,

• Multiplying numbers with different signs:For any positive numbers x and y,

• Multiplying two negative numbers:For any positive numbers x and y,

00 x

)()()( xyyxyx

xyyx ))((

1.6 Multiplying and Dividing Real Numbers

• Reciprocal or multiplicative inverse:If xy = 1, then x and y are reciprocals of each other. (example: 2 and ½ )

• Division is the same as multiplying by the reciprocal:

yyx x 1

1.6 Multiplying and Dividing Real Numbers

• Division by zero:For any number x,

• Dividing numbers with different signs:For any positive numbers x and y,

• Dividing two negative numbers:For any positive numbers x and y,

undefined 0 x

)( yx

yx

yx

yx

yx

1.7 Properties of Real Numbers

• Commutative property (addition/multiplication)

• Associative property (addition/multiplication)

baab

abba

)()(

)()(

bcacab

cbacba

1.7 Properties of Real Numbers

• Identity property (addition/multiplication)

• Inverse property (addition/multiplication)

• Distributive property

aaaa 1 0

11

0)( a

aaa

cabaacb

acabcba

)(

)(

1.8 Simplifying Expressions -Terms

• Term: product or quotient of numbers, variables, and variables raised to powers

• Coefficient: number before the variablesIf none is present, the coefficient is 1

• Factors vs. terms:In “5x +y”, 5x is a term.In “5xy”, 5x is a factor.

xzyx

zyx ,

3 ,15 ,

2

52

32

1.8 Simplifying Expressions -Terms

• When you read a sentence, it split up into words. There is a space between each word.

• Likewise, an is split up into terms by the +/-/= sign:

• The only trick is that if the +/-/= sign is in parenthesis, it doesn’t count:

33 61

21

32 xxx

3612

21

32 yxx

1.8 Simplifying Expressions

• Like Terms: terms with exactly the same variables that have the same exponents

• Examples of like terms:

• Examples of unlike terms

yx

xx22 x5 andy 3

12 and 5

zx

xyxy22

2

x2 andy

7 and 2

1.8 Simplifying Expressions

• Combining Like Terms: the distributive property allows you to combine like terms

• Examples of combining like terms:

yxyxyxyx

xxxx2222 8)53(53

7)125()12 ( 5