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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

4.1 Visualize Fractions

  • Property of One
    • Any number, except zero, divided by itself is one.
      aa=1aa=1, where a0a0.
  • Mixed Numbers
    • A mixed number consists of a whole number aa and a fraction bcbc where c0c0.
    • It is written as follows: abcc0abcc0
  • Proper and Improper Fractions
    • The fraction abab is a proper fraction if a<ba<b and an improper fraction if abab.
  • Convert an improper fraction to a mixed number.
    1. Step 1. Divide the denominator into the numerator.
    2. Step 2. Identify the quotient, remainder, and divisor.
    3. Step 3. Write the mixed number as quotientremainderdivisorremainderdivisor.
  • Convert a mixed number to an improper fraction.
    1. Step 1. Multiply the whole number by the denominator.
    2. Step 2. Add the numerator to the product found in Step 1.
    3. Step 3. Write the final sum over the original denominator.
  • Equivalent Fractions Property
    • If a, b,a, b, and cc are numbers where b0b0, c0c0, then ab=acbcab=acbc.

4.2 Multiply and Divide Fractions

  • Equivalent Fractions Property
    • If a, b, ca, b, c are numbers where b0b0, c0c0, then ab=acbcab=acbc and acbc=abacbc=ab.
  • Simplify a fraction.
    1. Step 1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
    2. Step 2. Simplify, using the equivalent fractions property, by removing common factors.
    3. Step 3. Multiply any remaining factors.
  • Fraction Multiplication
    • If a, b, c,a, b, c, and dd are numbers where b0b0and d0d0, then abcd=acbdabcd=acbd.
  • Reciprocal
    • A number and its reciprocal have a product of 11. abba=1abba=1
    • Opposite Absolute Value Reciprocal
      has opposite sign is never negative has same sign, fraction inverts
      Table 4.4
  • Fraction Division
    • If a, b, c,a, b, c, and dd are numbers where b0b0, c0c0 and d0d0 , then
      ab÷cd=abdcab÷cd=abdc
    • To divide fractions, multiply the first fraction by the reciprocal of the second.

4.3 Multiply and Divide Mixed Numbers and Complex Fractions

  • Multiply or divide mixed numbers.
    1. Step 1. Convert the mixed numbers to improper fractions.
    2. Step 2. Follow the rules for fraction multiplication or division.
    3. Step 3. Simplify if possible.
  • Simplify a complex fraction.
    1. Step 1. Rewrite the complex fraction as a division problem.
    2. Step 2. Follow the rules for dividing fractions.
    3. Step 3. Simplify if possible.
  • Placement of negative sign in a fraction.
    • For any positive numbers aa and bb, -ab=a-b=-ab-ab=a-b=-ab.
  • Simplify an expression with a fraction bar.
    1. Step 1. Simplify the numerator.
    2. Step 2. Simplify the denominator.
    3. Step 3. Simplify the fraction.

4.4 Add and Subtract Fractions with Common Denominators

  • Fraction Addition
    • If a, b,a, b, and cc are numbers where c0c0, then ac+bc=a+ccac+bc=a+cc.
    • To add fractions, add the numerators and place the sum over the common denominator.
  • Fraction Subtraction
    • If a, b,a, b, and cc are numbers where c0c0, then ac-bc=a-ccac-bc=a-cc.
    • To subtract fractions, subtract the numerators and place the difference over the common denominator.

4.5 Add and Subtract Fractions with Different Denominators

  • Find the least common denominator (LCD) of two fractions.
    1. Step 1. Factor each denominator into its primes.
    2. Step 2. List the primes, matching primes in columns when possible.
    3. Step 3. Bring down the columns.
    4. Step 4. Multiply the factors. The product is the LCM of the denominators.
    5. Step 5. The LCM of the denominators is the LCD of the fractions.
  • Equivalent Fractions Property
    • If a, ba, b, and cc are whole numbers where b0b0, c0c0 then
      ab = acbcab=acbc and acbc=abacbc=ab
  • Convert two fractions to equivalent fractions with their LCD as the common denominator.
    1. Step 1. Find the LCD.
    2. Step 2. For each fraction, determine the number needed to multiply the denominator to get the LCD.
    3. Step 3. Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2.
    4. Step 4. Simplify the numerator and denominator.
  • Add or subtract fractions with different denominators.
    1. Step 1. Find the LCD.
    2. Step 2. Convert each fraction to an equivalent form with the LCD as the denominator.
    3. Step 3. Add or subtract the fractions.
    4. Step 4. Write the result in simplified form.
  • Summary of Fraction Operations
    • Fraction multiplication: Multiply the numerators and multiply the denominators.
      abcd=acbdabcd=acbd
    • Fraction division: Multiply the first fraction by the reciprocal of the second.
      ab+cd=abdcab+cd=abdc
    • Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.
      ac+bc=a+bcac+bc=a+bc
    • Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.
      ac-bc=a-bcac-bc=a-bc
  • Simplify complex fractions.
    1. Step 1. Simplify the numerator.
    2. Step 2. Simplify the denominator.
    3. Step 3. Divide the numerator by the denominator.
    4. Step 4. Simplify if possible.

4.6 Add and Subtract Mixed Numbers

  • Add mixed numbers with a common denominator.
    1. Step 1. Add the whole numbers.
    2. Step 2. Add the fractions.
    3. Step 3. Simplify, if possible.
  • Subtract mixed numbers with common denominators.
    1. Step 1. Rewrite the problem in vertical form.
    2. Step 2. Compare the two fractions.
      If the top fraction is larger than the bottom fraction, go to Step 3.
      If not, in the top mixed number, take one whole and add it to the fraction part, making a mixed number with an improper fraction.
    3. Step 3. Subtract the fractions.
    4. Step 4. Subtract the whole numbers.
    5. Step 5. Simplify, if possible.
  • Subtract mixed numbers with common denominators as improper fractions.
    1. Step 1. Rewrite the mixed numbers as improper fractions.
    2. Step 2. Subtract the numerators.
    3. Step 3. Write the answer as a mixed number, simplifying the fraction part, if possible.

4.7 Solve Equations with Fractions

  • 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.
  • Addition, Subtraction, and Division Properties of Equality
    • For any numbers a, b, and c,
      if a=ba=b, then a+c=b+ca+c=b+c. Addition Property of Equality
    • if a=ba=b, then a-c=b-ca-c=b-c. Subtraction Property of Equality
    • if a=ba=b, then ac=bcac=bc, c0c0. Division Property of Equality
  • The Multiplication Property of Equality
    • For any numbers abab and c, a=bc, a=b, then ac=bcac=bc.
    • If you multiply both sides of an equation by the same quantity, you still have equality.
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