Review of Algebra
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INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences
2011 Pearson Education, Inc.
Chapter 0 Chapter 0 Review of AlgebraReview of Algebra
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2011 Pearson Education, Inc.
• A set is a collection of objects.
• An object in a set is called an element of that set.
• Different type of integers:
• The real-number line is shown as
Chapter 0: Review of Algebra
0.1 Sets of Real Numbers0.1 Sets of Real Numbers
... ,3 ,2 ,1integers positive of Set
1 ,2 ,3 ..., integers negative of Set
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2011 Pearson Education, Inc.
• Important properties of real numbers
1. The Transitive Property of Equality
2. The Closure Properties of Addition and Multiplication
3. The Commutative Properties of Addition and Multiplication
Chapter 0: Review of Algebra
0.2 Some Properties of Real Numbers0.2 Some Properties of Real Numbers
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numbers real unique are there numbers, real all For
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baababba and
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2011 Pearson Education, Inc.
4. The Commutative Properties of Addition and Multiplication
5. The Identity Properties
6. The Inverse Properties
7. The Distributive Properties
Chapter 0: Review of Algebra
0.2 Some Properties of Real Numbers
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2011 Pearson Education, Inc.
Chapter 0: Review of Algebra0.2 Some Properties of Real Numbers
Example 1 – Applying Properties of Real Numbers
Example 3 – Applying Properties of Real Numbers
354543 b.
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a. Show that
Solution:
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2011 Pearson Education, Inc.
• Properties:
Chapter 0: Review of Algebra
0.3 Exponents and Radicals0.3 Exponents and Radicals
1 4.
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2011 Pearson Education, Inc.
Chapter 0: Review of Algebra0.3 Exponents and Radicals
Example 1 – Exponents
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2011 Pearson Education, Inc.
Chapter 0: Review of Algebra0.3 Exponents and Radicals
• The symbol is called a radical.
n is the index, x is the radicand, and is the radical sign.
n x
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2011 Pearson Education, Inc.
• If symbols are combined by any or all of the operations, the resulting expression is called an algebraic expression.
• A polynomial in x is an algebraic expression of the form:
where n = non-negative integer cn = constants
Chapter 0: Review of Algebra
0.4 Operations with Algebraic Expressions0.4 Operations with Algebraic Expressions
011
1 cxcxcxc nn
nn
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2011 Pearson Education, Inc.
Chapter 0: Review of Algebra0.4 Operations with Algebraic Expressions
Example 3 – Subtracting Algebraic Expressions
Simplify
Solution:
.364123 22 xyxxyx
48
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)364()123(
364123
2
2
22
22
xyx
xyx
xyxxyx
xyxxyx
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2011 Pearson Education, Inc.
• A list of products may be obtained from the distributive property:
Chapter 0: Review of Algebra0.4 Operations with Algebraic Expressions
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2011 Pearson Education, Inc.
• If two or more expressions are multiplied together, the expressions are called the factors of the product.
Chapter 0: Review of Algebra
0.5 Factoring0.5 Factoring
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2011 Pearson Education, Inc.
Chapter 0: Review of Algebra0.5 Factoring
Example 1 – Common Factors
a. Factor completely.
Solution:
b. Factor completely.
Solution:
xkxk 322 93
kxxkxkxk 3393 2322
224432325 268 zxybayzbayxa
24232232
224432325
342
268
xyzbazbyxaya
zxybayzbayxa
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2011 Pearson Education, Inc.
Simplifying Fractions
• Allows us to multiply/divide the numerator and denominator by the same nonzero quantity.
Multiplication and Division of Fractions
• The rule for multiplying and dividing is
Chapter 0: Review of Algebra
0.6 Fractions0.6 Fractions
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a
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2011 Pearson Education, Inc.
Rationalizing the Denominator
• For a denominator with square roots, it may be rationalized by multiplying an expression that makes the denominator a difference of two squares.
Addition and Subtraction of Fractions
• If we add two fractions having the same denominator, we get a fraction whose denominator is the common denominator.
Chapter 0: Review of Algebra
0.6 Fractions
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2011 Pearson Education, Inc.
Chapter 0: Review of Algebra0.6 Fractions
Example 1 – Simplifying Fractions
a. Simplify
Solution:
b. Simplify Solution:
.127
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2011 Pearson Education, Inc.
Chapter 0: Review of Algebra0.6 Fractions
Example 3 – Dividing Fractions
41
2
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1
1
4
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2011 Pearson Education, Inc.
Equations
• An equation is a statement that two expressions are equal.
• The two expressions that make up an equation are called its sides.
• They are separated by the equality sign, =.
Chapter 0: Review of Algebra
0.7 Equations, in Particular Linear Equations0.7 Equations, in Particular Linear Equations
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2011 Pearson Education, Inc.
Chapter 0: Review of Algebra0.7 Equations, in Particular Linear Equations
Example 1 – Examples of Equations
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• A variable (e.g. x, y) is a symbol that can be replaced by any one of a set of different numbers.