Download b - El Camino College

Chapter 1. Review of Pre-Algebra
1.1
1.2
1.3
1.4
1.5
1.6
Review of Integers
Review of Fractions
Review of Decimals and Square Roots
Review of Percents
Real Number System
Translations: Statements to Mathematical
Expressions
1
1.1 Review of Integers
1. Important terms and symbols
2. Operation on integers
2
1.1 Review of Integers
1. Important terms and symbols
•
•
Integers: set of integers = {…, –3, –2, –1, 0, 1, 2, 3}
Opposite of an integer:
– the opposite of 3 is –3, the opposite of –3 is 3
– Evaluate –(–5)
•
•
•
Additive inverse. Evaluate:
8 + (–8) = 0
Absolute value. Evaluate: | 3 | = 3, | –6 | = 6, | 0 | = 0
Number line.
−6 −5 −4
−3
−2
−1
0
1
2
3
4
5
6
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1.1 Review of Integers
2. Operation on integers (including order of operations)
(1)
–10 – 5 + 7
(2)
(–3)(–2)(–5)
(3)
6( −5)( −3)
9
(4)
5 + 3( −2)( −5)
( −2 ) 3 + 1
(5)
(6)
12 – 2[15 – 3(20 – 22 × 3)]
–10 + 5 – |–5|
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1.2 Review of Fractions
1. Important terms and symbols
2. Operation on fractions
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1.2 Review of Fractions
1. Important terms and Symbols
•
Divisor and factor
– In 48 ÷ 6 = 8, 6 is a divisor, we say 48 is divisible by 6.
– In 48 = 6 × 8, 6 is a factor of 48
•
•
•
•
•
Prime numbers: 7 (only factor: 1 and itself)
Composite number: 6 (has factors other than 1 and
itself)
Prime factorization: 12 = 2·2·3 (product of prim #s)
Multiples and LCM: LCM of {6, 9} = 18
Common factor and GCF: GCF of {6, 9} = 3
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1.2 Review of Fractions
2. Operation on fractions
(1) Find x, so that equation is true:
9
x
=
14 112
36
(2) Reduce the fraction completely: −
50
(3) Multiply:
7 3 4
× ×
8 8 7
(4) Divide:
7 3
÷
5 10
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1.2 Review of Fractions
2. Operation on fractions
(1) evaluate:
1 ⎛
1⎞
7 − ⎜− 2 ⎟
3 ⎝
3⎠
(2) evaluate :
2 2 ⎛ 5⎞
+ + ⎜− ⎟
3 5 ⎝ 6⎠
(3) evaluate :
1⎞
1
⎛ 1
⎜2 +1 ⎟ ÷ 2
4⎠
2
⎝ 2
(4) evaluate :
1 1 5 2 1
1 ÷ + × −
3 2 12 5 6
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1.3 Review of Decimals and Square Roots
1. Decimals
2. Decimals and Fractions
3. Radical
4. Right triangle
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1.3 Review of Decimals and Square Roots
1. Decimals
(1) Add:
(–29.78) + (–18.47)
(2) Subtract 3.4 from –7.4
(3) Find the product: –79.4 × (–0.01)
(4) Find the quotient: –2.75 ÷ 0.1
(5) Find the product: –7.483 × 102
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1.3 Review of Decimals and Square Roots
2. Decimals and Fractions
(1) Change the fraction to an exact decimal:
2
5
0.4
11
(2) Change the fraction to repeating decimal:
1 .8 3
6
3
(2) Change the decimal to a simplest fraction: 0.12 25
(3) Evaluate:
(1.5)2 + 2.8 ÷ 0.2
16.25
(4) Evaluate:
(2.5)2 – 5.2 ÷ 13
5.85
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1.3 Review of Decimals and Square Roots
3. Radicals
(1) Find the square roots of 4
2, –2
2, − 2
(2) Find the square roots of 2
(3) Evaluate:
64
8
7
(4) In the expression,
(a) what is the radicand?
(b) what is the radical sign?
7
12
1.3 Review of Decimals and Square Roots
4. Right triangle
B
c
a
90°
A
b
C
To sides of the angle C are called legs, and the side
opposite the angle C is called the hypotenuse.
Pythagorean Theorem:
a2 + b2 = c2
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1.4 Review of Percents
1. Equivalence of Percent
2. Basic Percent Problems
3. Business Related Terms and Concepts
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1.4 Review of Percents
1. Equivalence of Percent
(1) Convert the decimal to percent: 0.075
7.5%
(2) Convert the percent to fraction and simplify:
5
125%
4
(3) Convert the percent to decimal: 12.3%
(4) Convert the fraction the percent: 8
5
0.123
160%
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1.4 Review of Percents
2. Basic Percent Problems
(1) If $100 are made by investing $400, find the
25%
percent of profit.
(2) Compute 15% of $50.
(3) 22% of what number is 11?
(4) What percent of 600 is 72?
$7.5
50
12%
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1.4 Review of Percents
3. Business Related Terms and Concepts
(1) Cost of a TV set to a store is $250 and the
manager decides to make 40% profit. What
should be the list price?
$350
(2) Ellen invested $10,000 at 11% simple interest.
How much interest should she earn in 2 years?
$2,200
(3) An investment of $200 resulted a income return
of $20. Find the rate of returns.
10%
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1.5 Real Number System
1. Real number system
2. Identify different kinds of numbers
3. Compare real numbers
4. Opposite (additive inverse) of a real number
5. Properties of a real number
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1.5 Real Number System
1.5
Real Number System
−6 −5 −4
−3
−2
−1
0
1
2
3
4
5
6
fractions, mixed numbers
rational numbers decimals (repeating or terminal)
Real numbers
integers
irrational numbers (e.g. π, 2 )
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1.5 Real Number System
2. Identify different kinds of numbers
(1) In { 5 , 1, 0.2, − 1, 73 , 1.7, 3, π , 0}
(a) identify integers:
{1, –1, 3, 0}
(b) identify rational numbers {1, 0.2, − 1, 73 , 1.7, 3, 0}
(c) identify irrational numbers
{ 5, π }
(d) identify those which are rational but not natural
numbers. {0.2, − 1, 73 , 1.7, 0}
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1.5 Real Number System
3. Compare real numbers – determine true or false
false
(1)
5 < –5
(2)
–13 < –12
(3)
1 1
>
5 10
True
(4)
2 4
<
3 5
True
(5)
–2.5 > 2
True
false
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1.5 Real Number System
4. Opposite of a real number
Find the opposite of each number:
–8
(1)
8
(2)
–9.2
(3)
–2
(4)
7
9
9.2
2
7
−
9
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1.5 Real Number System
5. Properties of a real number
Commutative property of addition: a + b = b + a
Associative property of addition: (a + b) + c = a + (b + c)
Identity property of addition:
a+0=a
Inverse property of addition:
a + (–a) = 0
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1.5 Real Number System
5. Properties of a real number
Commutative property of multiplication:
a·b = b·a
associative property of multiplication: (a·b)·c = a·(b·c)
Identity property of multiplication:
a ·1= a
Inverse property of multiplication :
1
a· =1
a
Zero property of multiplication:
0·a=0
(a ≠ 0)
(a ≠ 0)
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1.5 Real Number System
5. Properties of a real number
Distributive property:
a(b + c) = ab + ac
or
a(b – c) = ab – ac
or
(b – c)a = ba – ca
or
a(b + c + d ) = ab + ac + ad
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1.5 Real Number System
5. Properties of a real number
Identify the property:
Identify property of addition
(1) 0 + (–1) = –1
(2)
–2·3 = 3·(–2)
(3)
2 + (3 + 5) = (2 + 3) + 5 Associative property of addition
(4)
6 + (–6) = 0
commutative property of multiplicaion
inverse property of addition
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1.6
Translations: Statements to Mathematical Expressions
1. Change the phrase to an algebraic expression
2. Equations
3. Using inequality and equality symbols
4. Evaluating algebraic expressions
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1.6
Translations: Statements to Mathematical Expressions
1. Change the phrase to an algebraic expression (let x
represent the unknown:
(1) Eight less than three times a number
3x – 8
(2) The difference between nine and a number
(3) The ratio of 5 and a number
9–x
5
x
(4) The quotient of twice a number and 11
2x
11
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1.6
Translations: Statements to Mathematical Expressions
2. Equations
(5) Translate into symbols: six divided by 2 is three.
6÷2=3
(6) True or false: x = 2 is a solution of x – 2 = 0.
yes
(7) Identify the solution(s) from the given set: x + 10 = 16;
{6, 8, 4}
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(8) Is 2x2 + 7x an expression or equation?
equation
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1.6
Translations: Statements to Mathematical Expressions
3. Using inequality and equality symbols
(9) Insert the inequality symbol to make the statement true:
43 > 25
(10) Rewrite the inequality: 5 < 11, by changing the
relational symbol.
11 > 5
(11) Determine the following statement by evaluating
true
72 + 8÷2 ≠ 49
(12) Translate the statement into symbols:
six multiplied by eight is greater than twelve.
6 × 8 > 12
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