Foundations of Algebra

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Chapter 2

• Foundations of Algebra

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Section 2-1

• Real Numbers

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Integers

• Whole numbers and their opposites
• ,-3, -2, -1, 0, 1, 2, 3, are integers
• Every integer, except 0, has an opposite

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Rational Number

• A number that can be expressed as a ratio
• These are written as a/b
• All rational numbers can be written as
  terminating or repeating decimals

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Irrational Numbers

• Numbers that are non-terminating and
  non-repeating decimals
• They cannot be written as a ratio
• Such as v2, p, v6

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Real Numbers

• Together the set of rational and irrational
  numbers make up the set of real numbers
• All numbers on a number line are real numbers

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Coordinate of a Point

• The number that corresponds to a point on a
  number line is the coordinate of that point

?
0
What is the coordinate?
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Graph of a Number

• The point that corresponds to a number and is
  indicated by a dot
• Draw a number line with the graph of -3

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Example

• Graph the following sets of numbers on a number
  line
• The real numbers from -3 through 2
• The integers from -3 to 2
• Real numbers greater than -1
• Real numbers greater than or equal to -1

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Variable

• A symbol used to represent a number
• Letters such as n and x are often used as
  variables

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Absolute Value

• The distance a number is from zero on the number
  line
• The absolute value is written as x

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Section 2-2

• Order of Operations

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Numerical Expression

• Two or more numbers joined by operations
• 2 7 3 9
• 2x 45 328

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Value

• The number represented by the numerical
  expression
• 2 11 1 5 x 3 3

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Simplify

• When you find the value of an expression you
  simplify the expression
• 2 3 10 3 x 4 7

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Order of Operations

• Parentheses and Brackets
• Exponents
• Multiplication and Division
• Addition and Subtraction

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Exponent

• Tells how many times a number is used as a
  factor
• 7x2 3y4x3 5xy8

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Variable Expression

• An expression containing one or more variables
• 3x 5 4m 7n
• (2t)(-3s) 14 7x

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Evaluate

• To evaluate a variable expression, substitute a
  given number for each variable
• Then simplify the numerical expression

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Example

• Evaluate the expressions when n 1.8
• 7 n2 (1/3)n 2.7
• 10.24 3.3

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Section 2-3

• Writing Variable Expressions

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Addition

• More than a number
• The sum of a number
• A number increased

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Subtraction

• The difference of a number and
• Less than a number
• Decreased by a number

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Multiplication

• Times a number
• The product of a number and
• A number multiplied by

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Division

• The quotient of a number and
• Divided by a number

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Translate each phrase into a variable expression

• A number increased by 33
• Negative three times a number
• The difference of 15 and a number
• The quotient of five and twice a number

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Translate each variable expression into a phrase

• 2n 9
• 9 n
• 9 - n
• n - 9

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Section 2-4

• Adding and Subtracting Variable Expressions

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Terms

• The part of a variable expression separated by
  addition or subtraction signs
• 2x - one term
• 3x 6y - two terms

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Like Terms

• Terms that have identical variable parts
• 2x and 4.5x
• 3st and -6st

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Unlike Terms

• Terms that have different variable parts
• 2b and 2a
• 3xyz and 7xy

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Simplifying

• Performing as many of the operations as possible
• 3x 2x 5x
• -9n n 3n ?

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Combining like Terms

• Simplifying a variable expression by adding or
  subtracting like terms
• 12xy 8xy (-15xy) ?

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Section 2-5

• Multiplying and Dividing Variable Expressions

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Multiplying

• Multiply each term within the parentheses by the
  term that is outside the parentheses

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Opposite of a Sum

• For all real numbers a and b
• -(a b) -1(a b)
• -a (-b)
• -8(4rs 7r)
• -(14 2x)

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Dividing

• Divide each term in the numerator by the
  denominator
• 2x 8 3y - 9
• 6 3

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Section 2-6

• Simplify Variable Expressions

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Remember Order of Operations

• Parentheses
• Exponents
• Multiply or divide left to right
• Add or subtract left to right

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Examples

• 3x 2(5x - 1)
• 9 - 4(x - 4)
• 2(x 3) 3(x - 4)
• 3(ab - a) - 3(b a)

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Section 2-7

• Properties of Exponents

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Exponential Form

• A shortcut to writing a number that is repeatedly
  multiplied by itself
• a a a a a 4

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Base

• Tells what factor is being multiplied

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Exponent

• Tells how many equal factors there are

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Product Rule

• To multiply numbers with the same base, write the
  base raised to the sum of the exponents
• x2 x4 x 24 x6

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Power Rule

• To raise an exponential number to an exponent,
  multiply exponents
• (x2)5 x 25 x10

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Power of a Product Rule

• To find the power of a product, find the power of
  each factor and multiply
• (2y3)2 (2)2 (y3)2 4y6

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Quotient Rule

• To divide numbers with the same base, write the
  base with the difference of the exponents
• 34 / 32 3 4-2 32

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Power of a Quotient Rule

• To find the power of a quotient, find the power
  of each number and divide
• (2/5)3 23/53

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Section 2-8

• Zero and Negative Exponents

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Zero Property of Exponents

• For any real number a, a0 1
• x0 1
• 30 1

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Property of Negative Exponents

• For any real number a, if n is a positive
  integer,
• a-n 1/an
• x7 / x-3
• c5 c-2

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Examples

• Simplify the following expressions
• x2/x8
• y3 1/y4
• (z-3)2

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Standard Form

• Any number written in decimal form
• 26.376, 0.0067, and 200,000 are written in
  standard form

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Scientific Notation

• A number that has two factors
• The first factor is between 1 and 10
• The second factor is a power of 10

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Examples

• Standard Scientific
• 0.0000268 2.68 10-5
• 370,000,000 3.7 108
• 0.00000154 1.54 10-6
• 250,000 2.5 105

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Section 2-9

• Problem Solving Skills Find a Pattern

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Sequence

• A set of numbers that is arranged according to a
  pattern
• 2, 4, 6, 8,
• 3, 6, 9, 12, ...

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Term

• Each number of the sequence
• By figuring out the pattern, you can find the
  next term

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Example

• Sun Li invests 2,000 in a mutual fund. The
  value of the investment will double every 6
  years. How long will it take for the investment
  to be worth 16,000?

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Solution

• Sequence 2000, 4000, 8000, 16000
• How many years?

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Example Sequences

• Give the next 5 terms of the sequence
• 4 add 4
• 1 subtract 3