1. Basic Concepts of Algebra
Chapter 1
Basic Concepts of Algebra

2. LANGUAGE OF ALGEBRA

3. SET– a collection or group of, things, objects, numbers, etc.

4. INFINITE SET – a set whose members cannot be counted.
If A= {1, 2, 3, 4, 5,…} then A is infinite

5. FINITE SET – a set whose members can be counted.
If A= {e, f, g, h, i, j} then A is finite and contains six elements

6. SUBSET – all members of a set are members of another set
If A= {e, f, g, h, i, j} and B = {e, i} , then BA

7. EMPTY SET or NULL SET – a set having no elements.
A= { } or B = { } are empty sets or null sets written as 

8. 1-1 Real Numbers and Their Graphs

9. Real Numbers

10. set of counting numbers
NATURAL NUMBERS -
set of counting numbers
{1, 2, 3, 4, 5, 6, 7, 8…}

11. WHOLE NUMBERS - set of counting numbers plus zero
{0, 1, 2, 3, 4, 5, 6, 7, 8…}

12. INTEGERS - set of the whole numbers plus their opposites
{…, -3, -2, -1, 0, 1, 2, 3, …}

13. RATIONAL NUMBERS -
numbers that can be expressed as a ratio of two integers a and b and includes fractions, repeating decimals, and terminating decimals

14. EXAMPLES OF RATIONAL NUMBERS
½, ¾, ¼, - ½, -¾, -¼, .05
.76, .333…, .666…, etc.

15. IRRATIONAL NUMBERS -
numbers that cannot be expressed as a ratio of two integers a and b and can still be designated on a number line

16. EXAMPLES OF IRRATIONAL NUMBERS
, 6, -29, …, etc .

17. Each point on a number line is paired with exactly one real number, called the coordinate of the point.
Each real number is paired with exactly one point on the line, called the graph of the number

18. 1-2 Simplifying Expressions

19. Definitions

20. NUMERICAL EXPRESSION or NUMERAL
a symbol or group of symbols used to represent a number
3 x
24 ÷ x 6

21. VALUE of a Numerical Expression
The number represented by the expression
Twelve is the value of
3 x
24 ÷ x 6

22. EQUATION
a sentence formed by placing an equals sign = between two expressions, called the sides of the equation. The equation is a true statement if both sides have the same value.

23. EXAMPLES OF EQUATIONS
= 6 – 2 or
4x + 3 = 19

24. ≥ - greater than or equal to
INEQUALITY SYMBOL
One of the symbols
< - less than
> greater than
≠ - does not equal
≤ - less than or equal to
≥ - greater than or equal to

25. INEQUALITY
a sentence formed by placing an inequality symbol between two expressions, called the sides of the inequality
-3 > -5
-3 < - 0.3

26. the result of adding numbers, called the terms of the sum
= 21
= 12
terms
sum

27. the result of subtracting one number from another
DIFFERENCE
the result of subtracting one number from another
8 – 6 = 2
= 8
difference

28. the result of multiplying numbers, called the factors of the product
6 x 15 = 80
10 · 2 = 20
product
factors

29. the result of dividing one number by another
QUOTIENT
the result of dividing one number by another
35 ÷ 7 = 5
10 ÷ 2 = 5
quotient

30. POWER, BASE, and EXPONENT
A power is a product of equal factors. The repeated factor is the base. A positive exponent tells the number of times the base occurs as a factor.

31. EXAMPLES OF POWER, BASE, and EXPONENT
Let the base be 3.
First power: 3 = 31
Second power: 3 x 3 = 32
Third power 3 x 3 x 3 = 33
Exponent is 1,2,3

32. GROUPING SYMBOLS
Pairs of parentheses ( ), brackets [ ], braces { }, or a bar — used to enclose part of an expression that represents a single number.
{ 3 + 4[(2 x 6) -22] ÷ 2}

33. VARIABLE – a symbol, usually a letter, used to represent any member of a given set, called the domain or replacement set, of the variable a, x, or y

34. If the domain of x is {0,1,2,3}, we write
EXAMPLES OF VARIABLES
If the domain of x is {0,1,2,3}, we write
x  {0,1,2,3}

35. VALUE of a Variable - the members of the domain of the variable
VALUE of a Variable - the members of the domain of the variable. If the domain of a is the set of positive integers, then a can have these values: 1,2,3,4,…

36. Algebraic Expression – a numerical expression; a variable; or a sum, difference, product, or quotient that contains one or more variables

37. EXAMPLES OF ALGEBRAIC EXPRESSIONS
x y2 – 2y + 6
a + b c2d – 4
c d

38. SUBSTITUTION PRINCIPLE
An expression may be replaced by another expression that has the same value.

39. Perform multiplications and divisions in order from left to right. and
ORDER OF OPERATIONS
Grouping symbols
Simplify powers
Perform multiplications and divisions in order from left to right. and

40. ORDER OF OPERATIONS
Perform additions and subtractions in order from left to right
Simplify the expression within each grouping symbol, working outward from the innermost grouping

41. DEFINITION of ABSOLUTE VALUE
For each real number a,
l a l = a if a >0
0 if a = 0
- a if a < 0

42. 1-3 Basic Properties of Real Numbers

43. Properties of Equality

44. Reflexive Property - a = a
Symmetric Property - If a = b, then b = a
Transitive Property - If a = b, and b = c, then a = c

45. Addition Property - If a = b, then a + c = b + c and c + a = c + b
Multiplication Property -If a = b, then ac = bc and ca = cb

46. Properties of Real Numbers

47. CLOSURE PROPERTIES
a + b and ab are unique
7 + 5 = 12
7 x 5 = 35

48. COMMUTATIVE PROPERTIES
a + b = b + a
ab = ba
2 + 6 = 6 + 2
2 x 6 = 6 x 2

49. ASSOCIATIVE PROPERTIES
(a + b) + c = a + (b +c)
(ab)c = a(bc)
(5 + 15) + 20 = 5 + (15 +20)
(5·15)20 = 5(15·20)

50. There are unique real numbers 0 and 1 (1≠0) such that:
IDENTITY PROPERTIES
There are unique real numbers 0 and 1 (1≠0) such that:
a + 0 = 0 + a = a
a · 1 = 1 ·a
= = -3
3 x 1 = 1 x 3 = 3

51. INVERSE PROPERTIES
PROPERTY OF OPPOSITES
For each a, there is a unique real number – a such that:
a + (-a) = 0 and (-a)+ a = (-a is called the opposite or additive inverse of a

52. INVERSE PROPERTIES
PROPERTY OF RECIPROCALS
For each a except 0, there is a unique real number 1/a such that:
a · (1/a) = 1 and (1/a)· a = (1/a is called the reciprocal or multiplicative inverse of a

53. DISTRIBUTIVE PROPERTY
a(b + c) = ab + ac
(b +c)a = ba + ca
5(12 + 3) = 5• •3 = 75
(12 + 3)5 = 12• • 5 = 75

54. 1-4
Sums and Differences

55. Rules for Addition

56. For real numbers a and b
If a and b are negative numbers, then a + b is negative and a + b = -(lal + lbl)
-5 + (-9) = - (l-5l + l-9l) = -14

57. For real numbers a and b
If a is a positive number, b is a negative number, and lal is greater than lbl, then a + b is a positive number and a + b = lal – lbl (-5) = l9l – l-5l = 4

58. For real numbers a and b
If a is a positive number, b is a negative number, and lal is less than lbl, then a + b is a negative number and a + b = -lbl – lal (-9) = -l-9l – l5l = -4

59. DEFINITION of SUBTRACTION
For all real number a and b,
a – b = a + (-b)
To subtract any real number, add its opposite

60. DISTRIBUTIVE PROPERTY
For all real number a ,b, and c
a(b - c)= ab – ac
and
(b – c)a = ba - ca

61. 1-5
Products

62. MULTIPLICATIVE PROPERTY OF 0
For every real number a,
a · 0 = 0 and 0 · a = 0

63. MULTIPLICATIVE PROPERTY OF -1
For every real number a,
a(-1) = -a and (-1)a = -a

64. Rules for Multiplication

65. The product of two positive numbers or two negative numbers is a positive number.

66. The product of a positive number and a negative number is a negative number.
(-5)(9) = -45 or (5)(-9) = -45

67. The absolute value of the product of two or more numbers is the product of their absolute values
l(-5)(9)l = l-5l l9l = 45

68. PROPERTY of the OPPOSITE of a PRODUCT
For all real number a and b,
-ab = (-a)b
and
-ab = a(-b)

69. PROPERTY of the OPPOSITE of a SUM
For all real number a and b,
-(a + b) = (-a) + (-b)

70. 1-6
Quotients

71. DEFINITION OF DIVISION
The quotient a divided by b is written a/b or a÷b. For every real number a and nonzero real number b,
a/b = a·1/b, or a÷b = a·1/b

72. DEFINITION OF DIVISION
To divide by any nonzero number, multiply by its reciprocal. Since 0 has no reciprocal, division by 0 is not defined.

73. Rules for Division

74. The quotient of two positive numbers or two negative numbers is a positive number
-24/-3 = 8 and 24/3 = 8

75. The quotient of two numbers when one is positive and the other negative is a negative number.

76. PROPERTY For all real numbers a and b and nonzero real number c,
(a + b)/c = a/c + b/c
and
(a-b)/c = a/c – b/c

77. 1-7 Solving Equations in One Variable

78. DEFINITION
Open sentences – an equation or inequality containing a variable.
Examples: y + 1= 1 + y
5x -1 = 9

79. 3 is a solution or root because
DEFINITION
Solution – any value of the variable that makes an open sentence a true statement.
Examples: 2t – 1 = 5
3 is a solution or root because
2·3 -1= 5 is true

80. DEFINITION
Solution Set – the set of all solutions of an open sentence. Finding the solution set is called solving the sentence.
Examples: y(4 - y) = 3
when y{0,1,2,3}
y  {1,3}

81. DEFINITION
Domain – the given set of numbers that a variable may represent
Example:
5x – 1 = 9
The domain of x is {1,2,3}

82. DEFINITION
Equivalent equations – equations having the same solution set over a given domain.
Examples: y(4 - y) = 3
when y{0,1,2,3} and
y2 – 4y = -3 y  {1,3}

83. DEFINITION
Empty set – the set with no members and is denoted by 

84. DEFINITION
Identity – the solution set is the set of all real numbers.

85. DEFINITION
Formula – is an equation that states a relationship between two or more variables usually representing physical or geometric quantities.
Examples: d = rt
A = lw

86. Transformations that Produce Equivalent Equations
Simplifying either side of an equation.

87. Adding to (or subtracting from) each side of an equation the same number or the same expression.

88. Multiplying (or dividing) each side of an equation by the same nonzero number.

89. 1-8
Words into Symbols

90. CONSECUTIVE NUMBERS Integers – {n-1, n, n+1}
{… -3, -2, -1, 0, 1, 2, 3,….}
Even Integers – {n-2, n, n+2}
{…-4,-2, 0, 2, 4,….}
Odd Integers – {n-2, n, n+2}
{…-5,-3, -1, 1, 3, 5,….}

91. The sum of 8 and x A number increased by 7 5 more than a number
Addition - Phrases
The sum of 8 and x
A number increased by 7
5 more than a number

92. Addition - Translation
8 + x
n + 7
n + 5

93. The difference between a number and 4 A number decreased by 8
Subtraction - Phrases
The difference between a number and 4
A number decreased by 8
5 less than a number
6 minus a number

94. Subtraction - Translation
x - 4
x- 8
n – 5
6 - n

95. Multiplication - Phrases
The product of 4 and a number
Seven times a number
One third of a number

96. Multiplication - Translation
1/3x

97. The quotient of a number and 8 A number divided by 10
Division - Phrases
The quotient of a number and 8
A number divided by 10

98. Division - Translation

99. 1-9 Problem Solving with Equations

100. Plan for Solving Word Problems
Read the problem carefully. Decide what numbers are asked for and what information is given. Making a sketch may be helpful.

101. Plan for Solving Word Problems
Choose a variable and use it with the given facts to represent the number(s) described in the problem. Labeling your sketch or arranging the given information in a chart may help.

102. Plan for Solving Word Problems
Reread the problem. Then write an equation that represents relationships among the numbers in the problem.

103. Plan for Solving Word Problems
Solve the equation and find the required numbers.
Check your results with the original statement of the problem. Give the answer

104. Solve using the five-step plan.
EXAMPLES
Solve using the five-step plan.
Two numbers have a sum of 44. The larger number is 8 more than the smaller. Find the numbers.

105. Solution
n + (n + 8) = 44
2n + 8 = 44
2n = 36
n = 18

106. EXAMPLES Translate the problem into an equation.
Marta has twice as much money as Heidi.
Together they have $36.
How much money does each have?

107. Let h = Heidi’s amount Then 2h = Marta’s amount h + 2h = 36
Translation
Let h = Heidi’s amount
Then 2h = Marta’s amount
h + 2h = 36

108. EXAMPLES Translate the problem into an equation.
A wooden rod 60 in. long is sawed into two pieces.
One piece is 4 in. longer than the other.
What are the lengths of the pieces?

109. Let x = the shorter length Then x + 4 = longer length x + (x + 4) = 60
Translation
Let x = the shorter length
Then x + 4 = longer length
x + (x + 4) = 60

110. EXAMPLES Translate the problem into an equation.
The area of a rectangle is 102 cm2.
The length of the rectangle is 6 cm.
Find the width of the rectangle?

111. Let w = width of rectangle Then 6 = length of rectangle 6w = 102
Translation
Let w = width of rectangle
Then 6 = length of rectangle
6w = 102

112. EXAMPLES Solve using the five-step plan.
Jason has one and a half times as many books as Ramon. Together they have 45 books. How many books does each boy have?

113. Let b = number of Ramon’s books Then 1.5b = number of Jason’s books
Translation
Let b = number of Ramon’s books
Then 1.5b = number of Jason’s books
b + 1.5b = 45

114. Solution
b + 1.5b = 45
2.5b = 45
b = 18

115. The End