Addition subtraction-multi-divison(6)

Copyright © 2015, 2011, 2007 Pearson Education, Inc. Chapter 1-1
Presented by
Professional Aided Supplemental Instruction
(PASI)
Ivy Tech Community College
Indianapolis
Basic Mathematics
Addition
Subtraction
Multiplication
Division

Real
Numbers
Chapter 1

Addition of Real
Numbers

Number Lines
Evaluate 3 + (- 4) using a number line
1. Always begin with 0.
2. Since the first number is positive, the first arrow
starts at 0 and is drawn 3 units to the right.
3. The second arrow starts at 3 and is drawn 4 units
to the left , since the second addend is negative.
3 + (– 4) = -1
-5 -4 -3 -2 -1 0 1 2 3 4 5
3
-5 -4 -3 -2 -1 0 1 2 3 4 5
-4
3

Add Fractions
The LCD is 48. Rewriting the first fraction with the LCD
gives the following.















3
2
16
7Add
48
11
48
21
16
16
3
3
16
7






























48
32
3
2

Identify Opposites
Any two numbers whose sum is 0 are said to
be opposites, or additive inverses, of each
other.
a + (– a) = 0
The opposite of a is –a.
The opposite of –a is a.
Example:
The opposite of –5 is 5, since –5 + 5 = 0

Add Using Absolute Values
To add real numbers with the same sign,
add their absolute values. The sum has the
same sign as the numbers being added.
Example:
–6 + (–9) = –15 4 + 8 = 12
The sum of two positive numbers will
always be positive and the sum of two
negative numbers will always be negative.

Add Using Absolute Values
To add two signed numbers with different
signs, subtract the smaller absolute value from the
larger absolute value. The answer has the sign of
the number with the larger absolute value.
Example:
13 + (–4) = 9 –35 + 15 = -20
The sum of two numbers with different
signs may be positive or negative. The sign
of the sum will be the same as the sign of the
number with the larger absolute value.

Subtraction of
Real Numbers

Subtracting
In general, if a and b represent any two
real numbers, then
a – b = a + (-b)
Examples 1 and 2:
1.) 9 – (+4) =9 + (– 4) = 5
2.) 5 – 3 = 5 + (– 3) = 2

More Examples
Example: 3
1.) 3 – 10 =3 + (– 10) = -7
2.) -6 – 4 = -6 + (– 4) = -10

Kuta Software - Infinite Algebra 1
Adding and Subtracting Positive and Negative Numbers
1 (-2) + 3 9 (-14) + (-7)
2 3 - (-8) 10 (-9) + 14
3 (-8) - (-2) 11 5 + (-8)
4 (-27) - 24 12 (-41) + (-40)
5 38 - (-17) 13 (-44) + (-9)
6 (-16) - (-36) 14 (-6) - 24
7 (-16) - 6 + (-5) 15 15 - 13 + 2
8 16 - (-13) - (-5) 16 (-7) - (-2) - 9
1
11
-6
-51
55
20
-27
34
-21
5
-3
-81
-53
-30
4
-14

Multiplication and
Division of Real
Numbers

Sign of the Product Rule
The product of two numbers with like signs
is a positive number.
The product of two numbers with unlike
signs is a negative number.
Example:
a.) 4(– 5) = – 20
b.) (– 6)(7) = -42
c.) (– 9)(-3) = 27

Helpful Hint
At this point some students begin confusing
problems like -2 – 3 with (-2)(-3) and problems
like 2-3 and (-2)(-3). Make sure you understand
the difference between these problems.
Subtraction Problems
– 2 – 3 = – 5
2- 3 = – 1
Multiplication Problems
(-2) (– 3) = 6
(2)(-3)= – 6

Divide Numbers
1. The quotient of two numbers with like signs
is a positive number.
2. The quotient of two numbers with unlike
signs is a negative number.
Example:
2
5
10 
a.)
9-
5
45
b.)
6
6
36


c.)
The Sign of the Quotient of Two Real Numbers

Helpful Hint
(+)(+) = + (+)(+) = +
(–)(–) = + (–)(–) = +
(+)(–) = – (+)(–) = –
(–)(+) = – (–)(+) = –
Like signs give
positive products and
quotients.
Unlike signs give
negative products and
quotients.
For multiplication and division of two real numbers:

Remove Negative Signs from Denominators
If a and b represent any real numbers, b 0, then
b
a
b
a
b
a 

We generally do not write fractions with a negative
sign in the denominator. When a negative sign
appears in a denominator, we can move it to the
numerator or place it in front of the fraction.
The fraction would be written as or .
7
5
 7
5

7
5

Real Number Operations

Evaluate Divisions Involving Zero
If a represents any real number except 0, then
0  a = = 0a
0
Division by 0 is undefined. ?a 
0

Worksheet by Kuta Software LLC Kuta Software - Infinite Pre-Algebra
1 6 × −4 13 4 × 2
2 3 × −4 14 −6 × 4
3 5 × −4 15 −3 × 4
4 −5 × 6 16 −2 × −1
5 −8 ÷ −2 17 11 × 12
6 35 ÷ -5 18 9 ÷ −3
7 10 ÷ 5 19 16 ÷ 2
8 −49 ÷ 7 20 8 × −12
9 9 × 10 × 6 21 −6 × −10 × −8
10 7 × 9 × 7 22 6 × 6 × −2
11 −5 × −4 × −10 23 9 × 9 × −5
12 8 × 3 × 8 24 7 × 5 × −5
−24
−12
−20
−30
4
−7
2
−7
540
441
−200
192
8
−24
−12
2
132
−3
8
−96
− 480
− 72
− 405
− 175

Exponents,
Parentheses and
Order of Operations

Learn the Meaning of Exponents
In the expression 42, the 4 is called the base,
and the 2 is called the exponent.
exponent
42
base
43 is read “4 to the third power” and means
4·4·4 = 43
3 factors of 4

–x2 vs. (-x)2
An exponent refers only to the number or
variable that directly precedes it unless
parentheses are used to indicate otherwise.
– x2 = -(x)(x)
(– x)2 = (–x)(–x) = x2
Example: – 32 = – (3)(3) = – 9
(– 3)2 = (–3)(–3) = 9

Learn the Order of Operations
To evaluate mathematical expressions, use the
following order:
1. First, evaluate the information within parentheses ( ),
brackets  , or braces  .These are grouping
symbols, for they group information together. A
fraction bar, —, also serves as a grouping symbol. If
the expression contains nested grouping symbols (one
pair of grouping symbols within another pair),
evaluate the information in the innermost groping
symbols first.

Learn the Order of Operations
2. Next, evaluate all exponents.
3. Next, evaluate all the multiplications and
divisions in the order in which they occur,
working from left to right.
4. Finally, evaluate all additions or subtractions in
the order in which they occur, working from left
to right.

Order of Operations
Evaluate:
6 + 3 • 52 – 4 = Exponent
Multiply
Add
6 + 3 • 25 – 4 =
6 + 75 - 4=
81 – 4 =
77
Steps Taken

Order of Operations
Evaluate:
-7 + 2 [-6 + (36 / 32 )] = Exponent
Divide
Add
-7 + 2 [-6 + (36 / 9 )] =
-7 + 2 [-6 + 4] =
-7 + 2 [-2] =
-11
Steps Taken
-7 - 4 =
Multiply

Properties of the
Real Number System

Commutative Property
Commutative Property of Addition
If a and b represent any two real numbers, then
a + b = b + a 4 + 3 = 3 + 4
Commutative Property of Multiplication
If a and b represent any real numbers, then
a · b = b · a 6 · 3 = 3 · 6
Commutative (commute) changes the order.
*Note that the commutative property does not hold for
subtraction and division
7 = 7
18 = 18

Associative Property
Associative Property of Addition
If a, b, and c represent three real numbers, then
(a + b) + c = a + (b + c) (3 + 4) + 5 = 3 + (4 + 5)
Associative Property of Multiplication
If a, b, and c represent any three real numbers, then
(a · b) · c = a ·(b · c) (6 · 2) · 4 = 6 · (2 · 4)
Associative (associate) changes the grouping.
*Note that the associative property does not hold for
subtraction and division
7 + 5 = 3 + 9
12 = 12
12 · 4 = 6 · 8
48 = 48

Distributive Property
If a, b, and c represent three real numbers, then
a(b + c) = ab + ac
Distributive involves two operations (usually
multiplication and division).
2(3 + 4) = 2(3) + 2(4)
2(7) = 6 + 8
14 = 14

Identity Properties
If a represents any real number, then
a + 0 = a and 0 + a = a
a · 1 = a and 1 · a = 1
Identity Property of Addition
Identity Property of Multiplication
4 + 0 = 4 0 + 4 = 4
13 · 1 = 13 1 · 13 = 13

Inverse Properties
If a represents any real number, then
a + (-a)= 0 and (-a) + a = 0
Inverse Property of Addition
Inverse Property of Multiplication
a · = 1 and · a = 1 (a  0)a
1
a
1
7 + (-7) = 0 (-7) + 7 = 0
12 · = 1 · 12 = 112
1
12
1

Kuta Software -Infinite Pre-Algebra
Order of Operations Evaluate each expression.
1 (30 - 3) ÷ 3 13 9+6÷ (8-2)
2 (21 - 5) ÷ 8 14 4(4÷2+4)
3 1 + 72 15 6+ (5+8) × 4
4 5×4-8 16 6×6- (7+5)
5 8+6 × 9 17 (9 × 2) ÷ (2 + 1)
6 3 + 17 × 5 18 2 - (4 + 3 - 6)
7 7 + 12 × 11 19 7 × 7 - (8 - 2)
8 15 + 40 ÷ 20 20 9 - 7 - 6 ÷ 6
9 20 + 16 - 15 21 (4 - 1 + 8 ÷ 8) × 5
10 19 - 15 - 3 22 (10 × 2) ÷ (1 + 1)
11 9 × (3+3) ÷6 23 7 × 9 - 7 - 3 × 5
12 (9 + 18 - 3) ÷8 24 8 - 1 - (18 - 2) ÷ 8
9
2
50
12
62
88
139
17
21
1
9
3
10
24
48
24
6
1
43
1
20
10
41
5

Addition subtraction-multi-divison(6)

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