Download Some Basics of Algebra

Sect. 1.1 Some Basics of Algebra
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Numbers, Variables, and Constants
Operations and Exponents
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English phrases for operations
Algebraic Expressions vs. Equations
Evaluating Algebraic Expressions
Sets and Set Notation
Important Sets of Numbers
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Numbers, Variables, and Constants
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Numbers: 127, 4.39, 0, -11¾, square root of 3
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Integers, Decimals, Fractions, Mixed Numbers
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Variables: x, a, b, y, Q, B2 etc
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Constants: π, e, C=speed of light in vacuum
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Operations and Exponents
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Operations combine two numbers
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Addition
Subtraction
Multiplication
Division
Exponents
3 + 6.2
⅔–5
356 · 0.03 or 356(0.03)
19 / 3 or 19 ÷ 3
74
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Short for 7·7·7·7
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Class Exercise: Op’s + – •
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6 + 4 + 3 + 7 + 9 + 1 = 30
9 + 2 + 1 + 3 + 8 = 23
(-6) + (-2) + (-5) = -13
-6 – 2 – 5 = -13
8 + (-2) + (-9) + 6 + (-4) = 14 + (-15) = -1
6 • 2 • 5 = 60
-3 • 7 • (-2) = 42
2 • (-5) • (-3) • (-4) = -120
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Class Exercise: Op ÷, fractions
4 3
7
  
13 13
13
1
 3  5
    
4
 10   6 
4  2  1 4  6  2 10  1 5
    
 
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5  3  6 5  6  3 10  6  5
24  20  5
1
lcd  30


30
30
3 5 3 2 3
   
8 2 8 5 20
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Algebraic Expressions vs. Equations
Algebraic expressions have one or more terms
 Sometimes expressions can be simplified
 If each variable is replaced with a number, we can evaluate an
expression (reduce it to a single number)
 Today we will review how to evaluate expressions
 Tomorrow we’ll look at equations
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 An equation is two expressions separated by an equal sign – equations are
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not evaluated, they are solved1.1
Evaluating Algebraic Expressions
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Substitution is replacing a variable with a number
When every variable in an expression is substituted
with a number, we can evaluate that expression
Evaluate 3xz + y for x = 2, y = 5, and z = 7
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3xz + y
(write original problem)
3(2)(7) + (5) (put parentheses for each variable)
(insert the corresponding numbers)
42 + 5
(simplify according to “order of operations”)
47
(final answer)
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Class Exercise: mixed + • – ÷
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3+2•6= ?
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-3 – 3 = ?
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-6 or 0
3 • 22 = ?
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5 • 6 = 30 or
3 + 12 = 15
62 = 36 or
3 • 4 = 12
6+4÷2=?
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10 ÷ 2 = 5
6+2=8
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Rules for Order of Operations
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To make sure an expression is always evaluated in
the same way by different people, the Order of
Operations convention was defined
Mnemonic: “Please Excuse My Dear Aunt Sally”
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Parentheses
Exponents
Multiply/Divide
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Add/Subtract
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Always: Evaluate & Eliminate the innermost grouping first
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Order of Ops Example
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2 { 9 – 3 [ -2x – 4 ] }
2 { 9 + 6x + 12 }
2 { 6x + 21}
12x + 42
Remember: It’s an INSIDE job
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Class Exercise – Evaluate expressions
7x + 3
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7(5) + 3
35 + 3
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3z – 2y
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for x = 5
for y = 1 and z = 6
3(6) – 2(1)
18 – 2
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[17 – (a – b)]
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[17 – (-3 – 7)]
[17 – (-10)]
17 + 10
27
for a = -3 and b = 7
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Sets and Set Notation
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Finite sets and Infinite sets
Roster notation: {1, 2, 3, … } with ellipsis
Set-Builder notation: { x | x is an integer > 0}
Set of all real numbers: 
Empty Set (no members): 
Element of a set: 5  {1, 2, 3, 4, 5, 6}
Union of sets: {1, 2, 3}{3, 4, 5} = {1, 2, 3, 4, 5}
Intersection of sets: {1, 2, 3}{3, 4, 5} = { 3 }
Subset of a set: {1, 2, 3}  {1, 2, 3, 4, 5}
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Different Sets of Numbers
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Next time:
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1.2 Operations and Properties
of Real Numbers
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