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Prealgebra 2e

Key Concepts

Prealgebra 2eKey Concepts
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Table of contents
  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. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    7. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    7. 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

Key Concepts

11.1 Use the Rectangular Coordinate System

  • Sign Patterns of the Quadrants
    Quadrant I Quadrant II Quadrant III Quadrant IV
    (x,y) (x,y) (x,y) (x,y)
    (+,+) (−,+) (−,−) (+,−)
  • Coordinates of Zero
    • Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates ( a, 0).
    • Points with a x-coordinate equal to 0 are on the y-axis, and have coordinates ( 0, b).
    • The point (0, 0) is called the origin. It is the point where the x-axis and y-axis intersect.

11.2 Graphing Linear Equations

  • Graph a linear equation by plotting points.
    1. Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
    2. Step 2. Plot the points on a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
    3. Step 3. Draw the line through the points. Extend the line to fill the grid and put arrows on both ends of the line.
  • Graph of a Linear Equation:The graph of a linear equation ax+by=cax+by=c is a straight line.
    • Every point on the line is a solution of the equation.
    • Every solution of this equation is a point on this line.

11.3 Graphing with Intercepts

  • Intercepts
    • The x-intercept is the point, (a,0)(a,0), where the graph crosses the x-axis. The x-intercept occurs when y is zero.
    • The y-intercept is the point, (0,b)(0,b), where the graph crosses the y-axis. The y-intercept occurs when x is zero.
    • The x-intercept occurs when y is zero.
    • The y-intercept occurs when x is zero.
  • Find the x and y intercepts from the equation of a line
    • To find the x-intercept of the line, let y=0y=0 and solve for x.
    • To find the y-intercept of the line, let x=0x=0 and solve for y.
      x y
      0
      0
  • Graph a line using the intercepts
    1. Step 1.
      Find the x- and y- intercepts of the line.
      • Let y=0y=0 and solve for x.
      • Let x=0x=0 and solve for y.
    2. Step 2. Find a third solution to the equation.
    3. Step 3. Plot the three points and then check that they line up.
    4. Step 4. Draw the line.
  • Choose the most convenient method to graph a line
    1. Step 1. Determine if the equation has only one variable. Then it is a vertical or horizontal line.
      x=ax=a is a vertical line passing through the x-axis at a.
      y=by=b is a horizontal line passing through the y-axis at b.
    2. Step 2. Determine if y is isolated on one side of the equation. The graph by plotting points.
       Choose any three values for x and then solve for the corresponding y- values.
    3. Step 3. Determine if the equation is of the form Ax+By=CAx+By=C, find the intercepts.
      Find the x- and y- intercepts and then a third point.

11.4 Understand Slope of a Line

  • Find the slope from a graph
    1. Step 1. Locate two points on the line whose coordinates are integers.
    2. Step 2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
    3. Step 3. Count the rise and the run on the legs of the triangle.
    4. Step 4. Take the ratio of rise to run to find the slope, m=riserunm=riserun
  • Slope of a Horizontal Line
    • The slope of a horizontal line, y=by=b, is 0.
  • Slope of a Vertical Line
    • The slope of a vertical line, x=ax=a, is undefined.
  • Slope Formula
    • The slope of the line between two points (x1,y1)(x1,y1) and (x2,y2)(x2,y2) is m=y2-y1x2-x1m=y2-y1x2-x1
  • Graph a line given a point and a slope.
    1. Step 1. Plot the given point.
    2. Step 2. Use the slope formula to identify the rise and the run.
    3. Step 3. Starting at the given point, count out the rise and run to mark the second point.
    4. Step 4. Connect the points with a line.
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