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Chapter 1. Solving Equations and Inequalities 1.1 – Expressions and Formulas.

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Presentation on theme: "Chapter 1. Solving Equations and Inequalities 1.1 – Expressions and Formulas."— Presentation transcript:

1 Chapter 1

2 Solving Equations and Inequalities

3 1.1 – Expressions and Formulas

4 Order of Operations

5 1.1 – Expressions and Formulas Order of Operations Parentheses

6 1.1 – Expressions and Formulas Order of Operations Parentheses Exponents

7 1.1 – Expressions and Formulas Order of Operations Parentheses Exponents Multiplication

8 1.1 – Expressions and Formulas Order of Operations Parentheses Exponents Multiplication Division

9 1.1 – Expressions and Formulas Order of Operations Parentheses Exponents Multiplication Division Addition

10 1.1 – Expressions and Formulas Order of Operations Parentheses Exponents Multiplication Division Addition Subtraction

11 1.1 – Expressions and Formulas Order of Operations ParenthesesPlease Exponents Multiplication Division Addition Subtraction

12 1.1 – Expressions and Formulas Order of Operations ParenthesesPlease ExponentsExcuse Multiplication Division Addition Subtraction

13 1.1 – Expressions and Formulas Order of Operations ParenthesesPlease ExponentsExcuse MultiplicationMy Division Addition Subtraction

14 1.1 – Expressions and Formulas Order of Operations ParenthesesPlease ExponentsExcuse MultiplicationMy DivisionDear Addition Subtraction

15 1.1 – Expressions and Formulas Order of Operations ParenthesesPlease ExponentsExcuse MultiplicationMy DivisionDear AdditionAunt Subtraction

16 1.1 – Expressions and Formulas Order of Operations ParenthesesPlease ExponentsExcuse MultiplicationMy DivisionDear AdditionAunt SubtractionSally

17 Example 1

18 Find the value of [2(10 - 4) 2 + 3] ÷ 5.

19 Example 1 Find the value of [2(10 - 4) 2 + 3] ÷ 5. [2(10 - 4) 2 + 3] ÷ 5 =

20 Example 1 Find the value of [2(10 - 4) 2 + 3] ÷ 5. [2(10 - 4) 2 + 3] ÷ 5 = [2(6) 2 + 3] ÷ 5

21 Example 1 Find the value of [2(10 - 4) 2 + 3] ÷ 5. [2(10 - 4) 2 + 3] ÷ 5 = [2(6) 2 + 3] ÷ 5 [2(36) + 3] ÷ 5

22 Example 1 Find the value of [2(10 - 4) 2 + 3] ÷ 5. [2(10 - 4) 2 + 3] ÷ 5 = [2(6) 2 + 3] ÷ 5 [2(36) + 3] ÷ 5 [72 + 3] ÷ 5

23 Example 1 Find the value of [2(10 - 4) 2 + 3] ÷ 5. [2(10 - 4) 2 + 3] ÷ 5 = [2(6) 2 + 3] ÷ 5 [2(36) + 3] ÷ 5 [72 + 3] ÷ 5 75 ÷ 5

24 Example 1 Find the value of [2(10 - 4) 2 + 3] ÷ 5. [2(10 - 4) 2 + 3] ÷ 5 = [2(6) 2 + 3] ÷ 5 [2(36) + 3] ÷ 5 [72 + 3] ÷ 5 75 ÷ 5 15

25 Example 2

26 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5.

27 Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) =

28 Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) = 8 2 – 1.5( )

29 Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) = 8 2 – 1.5( ) 8 2 – 1.5( )

30 Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) = 8 2 – 1.5( ) 8 2 – 1.5( ) 8 2 – 1.5(9.5)

31 Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) = 8 2 – 1.5( ) 8 2 – 1.5( ) 8 2 – 1.5(9.5) 64 – 1.5(9.5)

32 Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) = 8 2 – 1.5( ) 8 2 – 1.5( ) 8 2 – 1.5(9.5) 64 – 1.5(9.5) 64 – 14.25

33 Example 2 Evaluate x 2 – y(x + y) if x = 8 and y = 1.5. x 2 – y(x + y) = 8 2 – 1.5( ) 8 2 – 1.5( ) 8 2 – 1.5(9.5) 64 – 1.5(9.5) 64 –

34 Example 3

35 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5

36 Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = c 2 – 5

37 Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = (-4)(-3) c 2 – 5

38 Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = (-4)(-3) c 2 – 5 (-3) 2 – 5

39 Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = (-4)(-3) c 2 – 5 (-3) 2 – 5 = 8 + 2(-4)(-3)

40 Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = (-4)(-3) c 2 – 5 (-3) 2 – 5 = 8 + 2(-4)(-3) 9 – 5

41 Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = (-4)(-3) c 2 – 5 (-3) 2 – 5 = 8 + 2(-4)(-3) 9 – 5 = – 5

42 Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = (-4)(-3) c 2 – 5 (-3) 2 – 5 = 8 + 2(-4)(-3) 9 – 5 = – 5 = 32 4

43 Example 3 Evaluate a 3 + 2bc if a = 2, b = -4, and c = -3. c 2 – 5 a 3 + 2bc = (-4)(-3) c 2 – 5 (-3) 2 – 5 = 8 + 2(-4)(-3) 9 – 5 = – 5 = 32 = 8 4

44 Example 4

45 Find the area of the following trapezoid. 16 in. 10 in. 52 in.

46 Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b 2 ) 10 in. 52 in.

47 Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b 2 ) 10 in. 52 in. A = ½h(b 1 + b 2 )

48 Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b 2 ) 10 in. = h 52 in. A = ½h(b 1 + b 2 )

49 Example 4 Find the area of the following trapezoid. 16 in. = b 1 A = ½h(b 1 + b 2 ) 10 in. = h 52 in. A = ½h(b 1 + b 2 )

50 Example 4 Find the area of the following trapezoid. 16 in. = b 1 A = ½h(b 1 + b 2 ) 10 in. = h 52 in. = b 2 A = ½h(b 1 + b 2 )

51 Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b 2 ) 10 in. 52 in. A = ½h(b 1 + b 2 ) = ½10( )

52 Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b 2 ) 10 in. 52 in. A = ½h(b 1 + b 2 ) = ½10( ) = ½10(68)

53 Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b 2 ) 10 in. 52 in. A = ½h(b 1 + b 2 ) = ½10( ) = ½10(68) = 5(68)

54 Example 4 Find the area of the following trapezoid. 16 in. A = ½h(b 1 + b2) 10 in. 52 in. A = ½h(b 1 + b 2 ) = ½10( ) = ½10(68) = 5(68) = 340

55 1.2 – Properties of Real Numbers

56 Real Numbers

57 1.2 – Properties of Real Numbers Real Numbers (R)

58 1.2 – Properties of Real Numbers Real Numbers (R)

59 1.2 – Properties of Real Numbers Real Numbers (R) Rational

60 1.2 – Properties of Real Numbers Real Numbers (R) Rational (⅓)

61 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓)

62 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓)

63 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Integers

64 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Integers (-6)

65 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6)

66 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6)

67 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) Whole #’s

68 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) Whole #’s (0)

69 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0)

70 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0)

71 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) Natural #’s

72 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) Natural #’s (7)

73 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (7)

74 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

75 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Irrational (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

76 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) Irrational √ 5 (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

77 1.2 – Properties of Real Numbers Real Numbers (R) (Q) Rational (⅓) (I) Irrational √ 5 (Z) Integers (-6) (W) Whole #’s (0) (N) Natural #’s (1)

78 Real Rational Irrational Integers Whole Natural

79 Example 1

80 Name the sets of numbers to which each apply.

81 Example 1 Name the sets of numbers to which each apply.

82 Example 1 Name the sets of numbers to which each apply.

83 Example 1 Name the sets of numbers to which each apply. (a) √ 16

84 Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4

85 Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N

86 Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W

87 Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W, Z

88 Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W, Z, Q

89 Example 1 Name the sets of numbers to which each apply. (a) √ 16 = 4 - N, W, Z, Q, R

90 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b)-185

91 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z

92 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q

93 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R

94 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20

95 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R

96 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R (d) -⅞

97 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q

98 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R

99 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R __ (e) 0.45

100 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R __ (e) Q

101 Example 1 Name the sets of numbers to which each apply. (a)√ 16 = 4 - N, W, Z, Q, R (b) Z, Q, R (c)√ 20 - I, R (d) -⅞ - Q, R __ (e) Q, R

102 Properties of Real Numbers PropertyAdditionMultiplication Commutativea + b = b + aa·b = b·a Associative (a+b)+c = a+(b+c) (a · b) · c = a · (b · c) Identitya+0 = a = 0+aa·1 = a = 1·a Inversea+(-a) =0= -a+aa·1 =1= 1·a a a Distributivea(b+c)=ab+ac and (b+c)a=ba+ca

103 Example 2

104 Name the property used in each equation.

105 Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7)

106 Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7) Commutative Addition

107 Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7) Commutative Addition (b) 3(4x) = (3·4)x

108 Example 2 Name the property used in each equation. (a) (5 + 7) + 8 = 8 + (5 + 7) Commutative Addition (b) 3(4x) = (3·4)x Associative Multiplication

109 Example 3 What is the additive and multiplicative inverse for -1¾?

110 Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾

111 Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + = 0

112 Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0

113 Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0 Multiplicative: -1¾

114 Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0 Multiplicative: -1¾ · = 1

115 Example 3 What is the additive and multiplicative inverse for -1¾? Additive: -1¾ + 1¾ = 0 Multiplicative: (-1¾)(- 4 / 7 ) = 1

116 Example 4

117 Simplify 2(5m+n)+3(2m–4n).

118 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n)

119 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n)

120 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)

121 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+

122 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+

123 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)

124 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+

125 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+

126 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)

127 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-

128 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-

129 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n)

130 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m

131 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m +

132 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n

133 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n +

134 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m

135 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m –

136 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n

137 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n

138 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n

139 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n 10m + 6m + 2n – 12n

140 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n 10m + 6m + 2n – 12n 16m

141 Example 4 Simplify 2(5m+n)+3(2m–4n). 2 (5m+n) + 3 (2m–4n) 2(5m)+2(n)+3(2m)-3(4n) 10m + 2n + 6m – 12n 10m + 6m + 2n – 12n 16m – 10n


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