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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

2.1 Use the Language of Algebra

Operation Notation Say: The result is…
Addition a+ba+b aplusbaplusb the sum of aa and bb
Multiplication a·b,(a)(b),(a)b,a(b)a·b,(a)(b),(a)b,a(b) atimesbatimesb The product of aa and bb
Subtraction abab aminusbaminusb the difference of aa and bb
Division a÷b,a/b,ab,baa÷b,a/b,ab,ba aa divided by bb The quotient of aa and bb
  • Equality Symbol
    • a=ba=b is read as aa is equal to bb
    • The symbol == is called the equal sign.
  • Inequality
    • a<ba<b is read aa is less than bb
    • aa is to the left of bb on the number line
      ..
    • a>ba>b is read aa is greater than bb
    • aa is to the right of bb on the number line
      ..
Algebraic Notation Say
a=ba=b aa is equal to bb
abab aa is not equal to bb
a<ba<b aa is less than bb
a>ba>b aa is greater than bb
abab aa is less than or equal to bb
abab aa is greater than or equal to bb
Table 2.14
  • Exponential Notation
    • For any expression anan is a factor multiplied by itself nn times, if nn is a positive integer.
    • anan means multiply nn factors of aa
      ..
    • The expression of anan is read aa to the nthnth power.

Order of Operations When simplifying mathematical expressions perform the operations in the following order:

  • Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
  • Exponents: Simplify all expressions with exponents.
  • Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
  • Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.

2.2 Evaluate, Simplify, and Translate Expressions

  • Combine like terms.
    1. Step 1. Identify like terms.
    2. Step 2. Rearrange the expression so like terms are together.
    3. Step 3. Add the coefficients of the like terms

2.3 Solving 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 Property of Equality
    • For any numbers aa, bb, and cc,
      if a=ba=b
      then ac=bcac=bc
  • Solve an equation using the Subtraction Property of Equality.
    1. Step 1. Use the Subtraction Property of Equality to isolate the variable.
    2. Step 2. Simplify the expressions on both sides of the equation.
    3. Step 3. Check the solution.
  • Addition Property of Equality
    • For any numbers aa, bb, and cc,
      if a=ba=b
      then a+c=b+ca+c=b+c
  • Solve an equation using the Addition Property of Equality.
    1. Step 1. Use the Addition Property of Equality to isolate the variable.
    2. Step 2. Simplify the expressions on both sides of the equation.
    3. Step 3. Check the solution.

2.4 Find Multiples and Factors

Divisibility Tests
A number is divisible by
2 if the last digit is 0, 2, 4, 6, or 8
3 if the sum of the digits is divisible by 3
5 if the last digit is 5 or 0
6 if divisible by both 2 and 3
10 if the last digit is 0
  • Factors If ab=mab=m, then aa and bb are factors of mm, and mm is the product of aa and bb.
  • Find all the factors of a counting number.
    1. Step 1. Divide the number by each of the counting numbers, in order, until the quotient is smaller than the divisor.
      1. If the quotient is a counting number, the divisor and quotient are a pair of factors.
      2. If the quotient is not a counting number, the divisor is not a factor.
    2. Step 2. List all the factor pairs.
    3. Step 3. Write all the factors in order from smallest to largest.
  • Determine if a number is prime.
    1. Step 1. Test each of the primes, in order, to see if it is a factor of the number.
    2. Step 2. Start with 2 and stop when the quotient is smaller than the divisor or when a prime factor is found.
    3. Step 3. If the number has a prime factor, then it is a composite number. If it has no prime factors, then the number is prime.

2.5 Prime Factorization and the Least Common Multiple

  • Find the prime factorization of a composite number using the tree method.
    1. Step 1. Find any factor pair of the given number, and use these numbers to create two branches.
    2. Step 2. If a factor is prime, that branch is complete. Circle the prime.
    3. Step 3. If a factor is not prime, write it as the product of a factor pair and continue the process.
    4. Step 4. Write the composite number as the product of all the circled primes.
  • Find the prime factorization of a composite number using the ladder method.
    1. Step 1. Divide the number by the smallest prime.
    2. Step 2. Continue dividing by that prime until it no longer divides evenly.
    3. Step 3. Divide by the next prime until it no longer divides evenly.
    4. Step 4. Continue until the quotient is a prime.
    5. Step 5. Write the composite number as the product of all the primes on the sides and top of the ladder.
  • Find the LCM by listing multiples.
    1. Step 1. List the first several multiples of each number.
    2. Step 2. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.
    3. Step 3. Look for the smallest number that is common to both lists.
    4. Step 4. This number is the LCM.
  • Find the LCM using the prime factors method.
    1. Step 1. Find the prime factorization of each number.
    2. Step 2. Write each number as a product of primes, matching primes vertically when possible.
    3. Step 3. Bring down the primes in each column.
    4. Step 4. Multiply the factors to get the LCM.
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