P.1 Algebraic Expressions, Mathematical Models, and Real...

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P.1 Algebraic Expressions, Mathematical Models, and Real Numbers

P.2 Exponents and Scientific Notation Objectives: Evaluate algebraic expressions, find intersection and unions of sets, simplify algebraic expressions, use properties of exponents and scientific notation.

Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2

Algebraic Expressions

•  Variable: A letter is used to represent various numbers.

•  Algebraic Expression: A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots.

•  Evaluating an algebraic expression means to find the value of the expression for a given value of the variable.

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The Order of Operations Agreement

1.  Perform operations within the innnermost parentheses and work outward. If the algebraic expression involves and fraction, treat the numerator and the denominator as if they were each enclosed in parentheses.

2.  Evaluate all exponential expressions

3.  Perform multiplication and divisions as they occur, working from left to right.

4.  Perform additions and subtractions as they occur, working from left to right.

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Example: Evaluating an Algebraic Expression

Evaluate 8 + 6(x – 3)2 for x = 13.


Formulas and Mathematical Models

•  Equation: Formed when an equal sign is placed between two algebraic expressions.

•  Formula: Is an equation that uses variables to express a relationship between two or more quantities.

•  Mathematical Modeling: The process of finding formulas to describe real-world phenomena.


Example: Using a Mathematical Model

The formula models the average cost of tuition and fees, T, for public U.S. colleges for the school year ending x years after 2000. Use the formula to project the average cost of tuition and fees at public U.S. colleges for the school year ending in 2015.

24 341 3194T x x= + +

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Sets

•  Set is a collection of objects whose contents can be clearly determined.

•  Elements: The objects in a set. We use braces, { }, to indicate that we are representing a set.

•  {1, 2, 3, 4, 5, ...} is an example of the roster method of representing a set. The three dots after the 5, called an ellipsis, indicates that there is no final element and that the listing goes on forever.

•  Empty Set: If a set has no elements, or the null set, and is represented by the Greek letter phi, .∅


Set-Builder Notation

In set-builder notation, the elements of the set are described but not listed. is read, “The set of all x such that x is a counting number less than 6”. The same set written using the roster method is {1, 2, 3, 4, 5}.

{ } is a counting number less than 6x x


Definition of the Intersection of Sets

The intersection of sets A and B, written is the set of elements common to both set A and set B. This definition can be expressed in set-builder notation as follows:

A B,

A B = x x is an element of A and x is an element of B{ }.

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Example: Finding the Intersection of Two Sets

Find the intersection:

3,4,5,6,7{ } 3,7,8,9{ }.


Definition of the Union of Sets

The union of sets A and B, written , is the set of elements that are members of set A or of set B or of both sets. This definition can be expressed in set-builder notation as follows:

A BU

{ } is an element of OR is an element of .=UA B x x A x B


Example: Finding the Union of Two Sets

Find the union:

{ } { }3,4,5,6,7 3,7,8,9 .U

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The Set of Real Numbers

The sets that make up the Real Numbers, , are: the set of Natural Numbers, the set of Whole Numbers, the set of Integers, the set of Rational Numbers, and the set of Irrational Numbers. Irrational numbers cannot be expressed as a quotient of integers.

{1,2,3,4,5,...}{0,1,2,3,4,5,...}

{..., 5, 4, 3, 2, 1,0,1,2,3,4,5,...}− − − − −

= aba and b are integers and b ≠ 0⎧

⎨⎩

⎫⎬⎭


The Set of Real Numbers (continued)

The set of real numbers is the set of numbers that are either rational or irrational:

{ } is rational or is irrational .x x x


Example: Recognizing Subsets of the Real Numbers

Consider the following set of numbers: List the numbers in the set that are natural numbers.

{ }9, 1.3,0,0.3, , 9, 10 .2π− −

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Absolute Value

The absolute value of a real number a, denoted , is the distance from 0 to a on the number line. The distance is always taken to be nonnegative. Definition of absolute value

x


Example: Evaluating Absolute Value

Rewrite the expression without absolute value bars: answer:

Rewrite the expression without absolute value bars: answer:

2 1−

3π −


Properties of Absolute Value

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Distance between Points on a Real Number Line

If a and b are any two points on a real number line, then the distance between a and b is given by or a b− .−b a


Properties of the Real Numbers

The Commutative Property of Addition

The Commutative Property of Multiplication

The Associative Property of Addition

The Associative Property of Multiplication


Properties of the Real Numbers (continued)

The Distributive Property of Multiplication over Addition

The Identity Property of Addition

The Identity Property of Multiplication

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Properties of the Real Numbers (continued)

The Inverse Property of Addition

The Inverse Property of Multiplication


Example: Simplifying an Algebraic Expression

Simplify:

2 27(4 3 ) 2(5 ).+ + +x x x x


Properties of Negatives

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Properties of Exponents


Properties of Exponents (continued)



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Simplifying Exponential Expressions

An exponential expression is simplified when: • No parentheses appear. • No powers are raised to powers. • Each base occurs only once. • No negative or zero exponents appear.


Example: Simplifying Exponential Expressions

Simplify: Simplify:

3 6 4(2 ) .x y

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45 .

−⎛ ⎞⎜ ⎟⎝ ⎠xy

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Scientific Notation

A number is written in scientific notation when it is expressed in the form where the absolute value of a is greater than or equal to 1 and less than 10, and n is an integer.

10na×

(1 10),a≤ <


Example: Converting from Decimal Notation to Scientific Notation

Write in scientific notation: 5,210,000,000 Write in scientific notation: – 0.00000006893


Computations with Scientific Notation

Properties of exponents are used to perform computations with numbers that are expressed in scientific notation.

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Example: Computations with Scientific Notation

Perform the indicated computation, writing the answer in scientific notation:

( )( )5 77.1 10 5 10−× ×


Example: Computations with Scientific Notation

Perform the indicated computation, writing the answer in scientific notation:

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31.2 103 10−

××

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