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Elementary Algebra

Key Concepts

Elementary AlgebraKey Concepts
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope–Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solve Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 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
  13. Index

4.1 Use the Rectangular Coordinate System

  • Sign Patterns of the Quadrants
    Quadrant IQuadrant IIQuadrant IIIQuadrant IV(x,y)(x,y)(x,y)(x,y)(+,+)(,+)(,)(+,)Quadrant IQuadrant IIQuadrant IIIQuadrant IV(x,y)(x,y)(x,y)(x,y)(+,+)(,+)(,)(+,)
  • Points on the Axes
    • On the x-axis, y=0y=0. Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates (a,0)(a,0).
    • On the y-axis, x=0x=0. Points with an x-coordinate equal to 0 are on the y-axis, and have coordinates (0,b).(0,b).
  • Solution of a Linear Equation
    • An ordered pair (x,y)(x,y) is a solution of the linear equation Ax+By=CAx+By=C, if the equation is a true statement when the x- and y- values of the ordered pair are substituted into the equation.

4.2 Graph Linear Equations in Two Variables

  • 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 in 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 three points. Extend the line to fill the grid and put arrows on both ends of the line.

4.3 Graph with Intercepts

  • Find the x- and y- Intercepts from the Equation of a Line
    • Use the equation of the line to find the x- intercept of the line, let y=0y=0 and solve for x.
    • Use the equation of the line to find the y- intercept of the line, let x=0x=0 and solve for y.
  • Graph a Linear Equation 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.



  • Strategy for Choosing the Most Convenient Method to Graph a Line:
    • Consider the form of the equation.
    • If it only has one variable, it is a vertical or horizontal line.
      x=ax=a is a vertical line passing through the x- axis at aa
      y=by=b is a horizontal line passing through the y- axis at bb.
    • If y is isolated on one side of the equation, graph by plotting points.
    • Choose any three values for x and then solve for the corresponding y- values.
    • 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.

4.4 Understand Slope of a Line

  • Find the Slope of a Line from its Graph using m=riserunm=riserun
    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.



  • Graph a Line Given a Point and the Slope
    1. Step 1. Plot the given point.
    2. Step 2. Use the slope formula m= rise run m= rise run 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.



  • 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

4.5 Use the Slope–Intercept Form of an Equation of a Line

  • The slope–intercept form of an equation of a line with slope mm and y-intercept, (0,b)(0,b) is, y=mx+by=mx+b.
  • Graph a Line Using its Slope and y-Intercept
    1. Step 1. Find the slope-intercept form of the equation of the line.
    2. Step 2. Identify the slope and y-intercept.
    3. Step 3. Plot the y-intercept.
    4. Step 4. Use the slope formula m=riserunm=riserun to identify the rise and the run.
    5. Step 5. Starting at the y-intercept, count out the rise and run to mark the second point.
    6. Step 6. Connect the points with a line.



  • Strategy for Choosing the Most Convenient Method to Graph a Line: Consider the form of the equation.
    • If it only has one variable, it is a vertical or horizontal line.
      x=ax=a is a vertical line passing through the x-axis at aa.
      y=by=b is a horizontal line passing through the y-axis at bb.
    • If yy is isolated on one side of the equation, in the form y=mx+by=mx+b, graph by using the slope and y-intercept.
      Identify the slope and y-intercept and then graph.
    • If the equation is of the form Ax+By=CAx+By=C, find the intercepts.
      Find the x- and y-intercepts, a third point, and then graph.
  • Parallel lines are lines in the same plane that do not intersect.
    • Parallel lines have the same slope and different y-intercepts.
    • If m1 and m2 are the slopes of two parallel lines then m1=m2.m1=m2.
    • Parallel vertical lines have different x-intercepts.
  • Perpendicular lines are lines in the same plane that form a right angle.
    • If m1andm2m1andm2 are the slopes of two perpendicular lines, then m1·m2=−1m1·m2=−1 and m1=−1m2m1=−1m2.
    • Vertical lines and horizontal lines are always perpendicular to each other.

4.6 Find the Equation of a Line

  • To Find an Equation of a Line Given the Slope and a Point
    1. Step 1. Identify the slope.
    2. Step 2. Identify the point.
    3. Step 3. Substitute the values into the point-slope form, yy1=m(xx1)yy1=m(xx1).
    4. Step 4. Write the equation in slope-intercept form.



  • To Find an Equation of a Line Given Two Points
    1. Step 1. Find the slope using the given points.
    2. Step 2. Choose one point.
    3. Step 3. Substitute the values into the point-slope form, yy1=m(xx1)yy1=m(xx1).
    4. Step 4. Write the equation in slope-intercept form.



  • To Write and Equation of a Line
    • If given slope and y-intercept, use slope–intercept form y=mx+by=mx+b.
    • If given slope and a point, use point–slope form yy1=m(xx1)yy1=m(xx1).
    • If given two points, use point–slope form yy1=m(xx1)yy1=m(xx1).



  • To Find an Equation of a Line Parallel to a Given Line
    1. Step 1. Find the slope of the given line.
    2. Step 2. Find the slope of the parallel line.
    3. Step 3. Identify the point.
    4. Step 4. Substitute the values into the point-slope form, yy1=m(xx1)yy1=m(xx1).
    5. Step 5. Write the equation in slope-intercept form.



  • To Find an Equation of a Line Perpendicular to a Given Line
    1. Step 1. Find the slope of the given line.
    2. Step 2. Find the slope of the perpendicular line.
    3. Step 3. Identify the point.
    4. Step 4. Substitute the values into the point-slope form, yy1=m(xx1)yy1=m(xx1).
    5. Step 5. Write the equation in slope-intercept form.

4.7 Graphs of Linear Inequalities

  • To Graph a Linear Inequality
    1. Step 1. Identify and graph the boundary line.
      If the inequality is oror, the boundary line is solid.
      If the inequality is < or >, the boundary line is dashed.
    2. Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality?
    3. Step 3. Shade in one side of the boundary line.
      If the test point is a solution, shade in the side that includes the point.
      If the test point is not a solution, shade in the opposite side.
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