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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

1.1 Introduction to Whole Numbers

A chart titled 'Place Value' with fifteen columns and 4 rows, with the columns broken down into five groups of three. The header row shows Trillions, Billions, Millions, Thousands, and Ones. The next row has the values 'Hundred trillions', 'Ten trillions', 'trillions', 'hundred billions', 'ten billions', 'billions', 'hundred millions', 'ten millions', 'millions', 'hundred thousands', 'ten thousands', 'thousands', 'hundreds', 'tens', and 'ones'. The first 8 values in the next row are blank. Starting with the ninth column, the values are '5', '2', '7', '8', '1', '9', and '4'.
Figure 1.16
  • Name a whole number in words.
    1. Step 1. Starting at the digit on the left, name the number in each period, followed by the period name. Do not include the period name for the ones.
    2. Step 2. Use commas in the number to separate the periods.
  • Use place value to write a whole number.
    1. Step 1. Identify the words that indicate periods. (Remember the ones period is never named.)
    2. Step 2. Draw three blanks to indicate the number of places needed in each period.
    3. Step 3. Name the number in each period and place the digits in the correct place value position.
  • Round a whole number to a specific place value.
    1. Step 1. Locate the given place value. All digits to the left of that place value do not change.
    2. Step 2. Underline the digit to the right of the given place value.
    3. Step 3. Determine if this digit is greater than or equal to 5. If yes—add 1 to the digit in the given place value. If no—do not change the digit in the given place value.
    4. Step 4. Replace all digits to the right of the given place value with zeros.

1.2 Add Whole Numbers

  • Addition Notation To describe addition, we can use symbols and words.
    Operation Notation Expression Read as Result
    Addition ++ 3+43+4 three plus four the sum of 33 and 44
  • Identity Property of Addition
    • The sum of any number aa and 00 is the number. a+0=aa+0=a 0+a=a0+a=a
  • Commutative Property of Addition
    • Changing the order of the addends aa and bb does not change their sum. a+b=b+aa+b=b+a.
  • Add whole numbers.
    1. Step 1. Write the numbers so each place value lines up vertically.
    2. Step 2. Add the digits in each place value. Work from right to left starting with the ones place. If a sum in a place value is more than 9, carry to the next place value.
    3. Step 3. Continue adding each place value from right to left, adding each place value and carrying if needed.

1.3 Subtract Whole Numbers

Operation Notation Expression Read as Result
Subtraction 7373 seven minus three the difference of 77 and 33
  • Subtract whole numbers.
    1. Step 1. Write the numbers so each place value lines up vertically.
    2. Step 2. Subtract the digits in each place value. Work from right to left starting with the ones place. If the digit on top is less than the digit below, borrow as needed.
    3. Step 3. Continue subtracting each place value from right to left, borrowing if needed.
    4. Step 4. Check by adding.

1.4 Multiply Whole Numbers

Operation Notation Expression Read as Result
MultiplicationMultiplication ××
··
()()
3×83×8
3·83·8
3(8)3(8)
three times eightthree times eight the product of 3 and 8the product of 3 and 8
  • Multiplication Property of Zero
    • The product of any number and 0 is 0.
      a0=0a0=0
      0a=00a=0
  • Identity Property of Multiplication
    • The product of any number and 1 is the number.
      1a=a1a=a
      a1=aa1=a
  • Commutative Property of Multiplication
    • Changing the order of the factors does not change their product.
      ab=baab=ba
  • Multiply two whole numbers to find the product.
    1. Step 1. Write the numbers so each place value lines up vertically.
    2. Step 2. Multiply the digits in each place value.
    3. Step 3. Work from right to left, starting with the ones place in the bottom number.
    4. Step 4. Multiply the bottom number by the ones digit in the top number, then by the tens digit, and so on.
    5. Step 5. If a product in a place value is more than 9, carry to the next place value.
    6. Step 6. Write the partial products, lining up the digits in the place values with the numbers above. Repeat for the tens place in the bottom number, the hundreds place, and so on.
    7. Step 7. Insert a zero as a placeholder with each additional partial product.
    8. Step 8. Add the partial products.

1.5 Divide Whole Numbers

Operation Notation Expression Read as Result
DivisionDivision ÷÷
abab
baba
a/ba/b
12÷412÷4
124124
412412
12/412/4
Twelve divided by fourTwelve divided by four the quotient of 12 and 4the quotient of 12 and 4
  • Division Properties of One
    • Any number (except 0) divided by itself is one. a÷a=1a÷a=1
    • Any number divided by one is the same number. a÷1=aa÷1=a
  • Division Properties of Zero
    • Zero divided by any number is 0. 0÷a=00÷a=0
    • Dividing a number by zero is undefined. a÷0a÷0 undefined
  • Divide whole numbers.
    1. Step 1. Divide the first digit of the dividend by the divisor.
      If the divisor is larger than the first digit of the dividend, divide the first two digits of the dividend by the divisor, and so on.
    2. Step 2. Write the quotient above the dividend.
    3. Step 3. Multiply the quotient by the divisor and write the product under the dividend.
    4. Step 4. Subtract that product from the dividend.
    5. Step 5. Bring down the next digit of the dividend.
    6. Step 6. Repeat from Step 1 until there are no more digits in the dividend to bring down.
    7. Step 7. Check by multiplying the quotient times the divisor.
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