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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

5.1 Decimals

  • Name a decimal number.
    1. Step 1. Name the number to the left of the decimal point.
    2. Step 2. Write “and” for the decimal point.
    3. Step 3. Name the “number” part to the right of the decimal point as if it were a whole number.
    4. Step 4. Name the decimal place of the last digit.
  • Write a decimal number from its name.
    1. Step 1. Look for the word “and”—it locates the decimal point.
      Place a decimal point under the word “and.” Translate the words before “and” into the whole number and place it to the left of the decimal point.
      If there is no “and,” write a “0” with a decimal point to its right.
    2. Step 2. Mark the number of decimal places needed to the right of the decimal point by noting the place value indicated by the last word.
    3. Step 3. Translate the words after “and” into the number to the right of the decimal point. Write the number in the spaces—putting the final digit in the last place.
    4. Step 4. Fill in zeros for place holders as needed.
  • Convert a decimal number to a fraction or mixed number.
    1. Step 1. Look at the number to the left of the decimal.
      If it is zero, the decimal converts to a proper fraction.
      If it is not zero, the decimal converts to a mixed number.
      Write the whole number.
    2. Step 2. Determine the place value of the final digit.
    3. Step 3. Write the fraction. numerator—the ‘numbers’ to the right of the decimal point denominator—the place value corresponding to the final digit
    4. Step 4. Simplify the fraction, if possible.
  • Order decimals.
    1. Step 1. Check to see if both numbers have the same number of decimal places. If not, write zeros at the end of the one with fewer digits to make them match.
    2. Step 2. Compare the numbers to the right of the decimal point as if they were whole numbers.
    3. Step 3. Order the numbers using the appropriate inequality sign.
  • Round a decimal.
    1. Step 1. Locate the given place value and mark it with an arrow.
    2. Step 2. Underline the digit to the right of the given place value.
    3. Step 3. Is this digit greater than or equal to 5?
      Yes - add 1 to the digit in the given place value.
      No - do not change the digit in the given place value
    4. Step 4. Rewrite the number, removing all digits to the right of the given place value.

5.2 Decimal Operations

  • Add or subtract decimals.
    1. Step 1. Write the numbers vertically so the decimal points line up.
    2. Step 2. Use zeros as place holders, as needed.
    3. Step 3. Add or subtract the numbers as if they were whole numbers. Then place the decimal in the answer under the decimal points in the given numbers.
  • Multiply decimal numbers.
    1. Step 1. Determine the sign of the product.
    2. Step 2. Write the numbers in vertical format, lining up the numbers on the right.
    3. Step 3. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
    4. Step 4. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors. If needed, use zeros as placeholders.
    5. Step 5. Write the product with the appropriate sign.
  • Multiply a decimal by a power of 10.
    1. Step 1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
    2. Step 2. Write zeros at the end of the number as placeholders if needed.
  • Divide a decimal by a whole number.
    1. Step 1. Write as long division, placing the decimal point in the quotient above the decimal point in the dividend.
    2. Step 2. Divide as usual.
  • Divide decimal numbers.
    1. Step 1. Determine the sign of the quotient.
    2. Step 2. Make the divisor a whole number by moving the decimal point all the way to the right. Move the decimal point in the dividend the same number of places to the right, writing zeros as needed.
    3. Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
    4. Step 4. Write the quotient with the appropriate sign.
  • Strategy for Applications
    1. Step 1. Identify what you are asked to find.
    2. Step 2. Write a phrase that gives the information to find it.
    3. Step 3. Translate the phrase to an expression.
    4. Step 4. Simplify the expression.
    5. Step 5. Answer the question with a complete sentence.

5.3 Decimals and Fractions

  • Convert a Fraction to a Decimal To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.
  • Properties of Circles ... rr is the length of the radius
    dd is the length of the diameter
    The circumference is 2πr2πr. C=2πrC=2πr
    The area is πr2πr2. A=πr2A=πr2

5.4 Solve Equations with Decimals

  • Determine whether a number is a solution to an equation.
    • Substitute the number for the variable in the equation.
    • Simplify the expressions on both sides of the equation.
    • Determine whether the resulting equation is true.
      If so, the number is a solution.
      If not, the number is not a solution.
  • Properties of Equality
Subtraction Property of Equality Addition Property of Equality
For any numbers aa, bb, and cc,
Ifa=b thenac=bcIfa=b thenac=bc
For any numbers aa, bb, and cc,
Ifa=b thena+c=b+cIfa=b thena+c=b+c
Division of Property of Equality Multiplication Property of Equality
For any numbers aa, bb, and c0c0,
Ifa=b thenac=bcIfa=b thenac=bc
For any numbers aa, bb, and cc,
Ifa=b thenac=bcIfa=b thenac=bc
Table 5.8

5.5 Averages and Probability

  • Calculate the mean of a set of numbers.
    1. Step 1. Write the formula for the mean mean=sum of values in data setnmean=sum of values in data setn
    2. Step 2. Find the sum of all the values in the set. Write the sum in the numerator.
    3. Step 3. Count the number, n, of values in the set. Write this number in the denominator.
    4. Step 4. Simplify the fraction.
    5. Step 5. Check to see that the mean is reasonable. It should be greater than the least number and less than the greatest number in the set.
  • Find the median of a set of numbers.
    1. Step 1. List the numbers from least to greatest.
    2. Step 2. Count how many numbers are in the set. Call this nn.
    3. Step 3. Is nn odd or even?
      If nn is an odd number, the median is the middle value.
      If nn is an even number, the median is the mean of the two middle values
  • Identify the mode of a set of numbers.
    1. Step 1. List the data values in numerical order.
    2. Step 2. Count the number of times each value appears.
    3. Step 3. The mode is the value with the highest frequency.

5.7 Simplify and Use Square Roots

  • Square Root Notation mm is read ‘the square root of mm
    If m=n2m=n2, then m=nm=n, for n0n0. .
  • Use a strategy for applications with square roots.
    • Identify what you are asked to find.
    • Write a phrase that gives the information to find it.
    • Translate the phrase to an expression.
    • Simplify the expression.
    • Write a complete sentence that answers the question.
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