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Prealgebra 2e

Key Concepts

Prealgebra 2eKey Concepts
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  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

8.1 Solve Equations Using the Subtraction and Addition Properties of Equality

  • Determine whether a number is a solution to an equation.
    1. Step 1. Substitute the number for the variable in the equation.
    2. Step 2. Simplify the expressions on both sides of the equation.
    3. 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.
  • Subtraction and Addition Properties of Equality
    • Subtraction Property of Equality
      For all real numbers a, b, and c,
      if a = b then a-c=b-ca-c=b-c.
    • Addition Property of Equality
      For all real numbers a, b, and c,
      if a = b then a+c=b+ca+c=b+c.
  • Translate a word sentence to an algebraic equation.
    1. Step 1. Locate the “equals” word(s). Translate to an equal sign.
    2. Step 2. Translate the words to the left of the “equals” word(s) into an algebraic expression.
    3. Step 3. Translate the words to the right of the “equals” word(s) into an algebraic expression.
  • Problem-solving strategy
    1. Step 1. Read the problem. Make sure you understand all the words and ideas.
    2. Step 2. Identify what you are looking for.
    3. Step 3. Name what you are looking for. Choose a variable to represent that quantity.
    4. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
    5. Step 5. Solve the equation using good algebra techniques.
    6. Step 6. Check the answer in the problem and make sure it makes sense.
    7. Step 7. Answer the question with a complete sentence.

8.2 Solve Equations Using the Division and Multiplication Properties of Equality

  • Division and Multiplication Properties of Equality
    • Division Property of Equality: For all real numbers a, b, c, and c0c0, if a=ba=b, then ac=bcac=bc.
    • Multiplication Property of Equality: For all real numbers a, b, c, if a=ba=b, then ac=bcac=bc.

8.3 Solve Equations with Variables and Constants on Both Sides

  • Solve an equation with variables and constants on both sides
    1. Step 1. Choose one side to be the variable side and then the other will be the constant side.
    2. Step 2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
    3. Step 3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
    4. Step 4. Make the coefficient of the variable 1, using the Multiplication or Division Property of Equality.
    5. Step 5. Check the solution by substituting into the original equation.
  • General strategy for solving linear equations
    1. Step 1. Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
    2. Step 2. Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
    3. Step 3. Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
    4. Step 4. Make the coefficient of the variable term to equal to 1. Use the Multiplication or Division Property of Equality. State the solution to the equation.
    5. Step 5. Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.

8.4 Solve Equations with Fraction or Decimal Coefficients

  • Solve equations with fraction coefficients by clearing the fractions.
    1. Step 1. Find the least common denominator of all the fractions in the equation.
    2. Step 2. Multiply both sides of the equation by that LCD. This clears the fractions.
    3. Step 3. Solve using the General Strategy for Solving Linear Equations.
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