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Prealgebra 2e

Key Concepts

Prealgebra 2eKey Concepts
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

3.1 Introduction to Integers

  • Opposite Notation
    • aa means the opposite of the number aa
    • The notation aa is read the opposite of a.a.
  • Absolute Value Notation
    • The absolute value of a number nn is written as |n||n|.
    • |n|0|n|0 for all numbers.

3.2 Add Integers

  • Addition of Positive and Negative Integers
    5+35+3 −5+(−3)−5+(−3)
    both positive, sum positive both negative, sum negative
    When the signs are the same, the counters would be all the same color, so add them.
    −5+3−5+3 5+(−3)5+(−3)
    different signs, more negatives different signs, more positives
    Sum negativesum positive
    When the signs are different, some counters would make neutral pairs; subtract to see how many are left.

3.3 Subtract Integers

  • Subtraction of Integers
    5353 –5(–3)–5(–3)
    22 –2–2
    2 positives 2 negatives
    When there would be enough counters of the color to take away, subtract.
    –53–53 5(–3)5(–3)
    –8–8 88
    5 negatives, want to subtract 3 positives 5 positives, want to subtract 3 negatives
    need neutral pairs need neutral pairs
    When there would not be enough of the counters to take away, add neutral pairs.
    Table 3.13
  • Subtraction Property
    • ab=a+(−b)ab=a+(−b)
    • a(−b)=a+ba(−b)=a+b
  • Solve Application Problems
    • Step 1. Identify what you are asked to find.
    • Step 2. Write a phrase that gives the information to find it.
    • Step 3. Translate the phrase to an expression.
    • Step 4. Simplify the expression.
    • Step 5. Answer the question with a complete sentence.

3.4 Multiply and Divide Integers

  • Multiplication of Signed Numbers
    • To determine the sign of the product of two signed numbers:
      Same Signs Product
      Two positives
      Two negatives
      Positive
      Positive

      Different Signs Product
      Positive • negative
      Negative • positive
      Negative
      Negative
  • Division of Signed Numbers
    • To determine the sign of the quotient of two signed numbers:
      Same Signs Quotient
      Two positives
      Two negatives
      Positive
      Positive

      Different Signs Quotient
      Positive • negative
      Negative • Positive
      Negative
      Negative
  • Multiplication by −1−1
    • Multiplying a number by −1−1 gives its opposite: −1a=a−1a=a
  • Division by −1−1
    • Dividing a number by −1−1 gives its opposite: a÷(−1)=−aa÷(−1)=−a

3.5 Solve Equations Using Integers; The Division Property of Equality

  • How to determine whether a number is a solution to an equation.
    • Step 1. Substitute the number for the variable in the equation.
    • Step 2. Simplify the expressions on both sides of the equation.
    • Step 3. Determine whether the resulting equation is true.
      • If it is true, the number is a solution.
      • If it is not true, the number is not a solution.
  • Properties of Equalities
    Subtraction Property of Equality Addition Property of Equality
    For any numbersa,b,c,For any numbersa,b,c,
    ifa=bthenac=bc.ifa=bthenac=bc.
    For any numbersa,b,c,For any numbersa,b,c,
    ifa=bthena+c=b+c.ifa=bthena+c=b+c.
  • Division Property of Equality
    • For any numbers a,b,c,a,b,c, and c0c0
      If a=ba=b, then ac=bcac=bc.
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