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Elementary Algebra

Key Concepts

Elementary AlgebraKey Concepts
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Table of contents
  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. Chapter Review
      1. Key Terms
      2. Key Concepts
    13. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    10. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    9. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    12. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    11. 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. Chapter Review
      1. Key Terms
      2. Key Concepts
    8. 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

Key Concepts

2.1 Solve Equations Using the Subtraction and Addition Properties of Equality

  • To Determine Whether a Number is a Solution to an Equation
    1. Step 1. Substitute the number in 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 statement is true.
      • If it is true, the number is a solution.
      • If it is not true, the number is not a solution.
  • Addition Property of Equality
    • For any numbers a, b, and c, if a=ba=b, then a+c=b+ca+c=b+c.
  • Subtraction Property of Equality
    • For any numbers a, b, and c, if a=ba=b, then a−c=b−ca−c=b−c.
  • To Translate a Sentence to an Equation
    1. Step 1. Locate the “equals” word(s). Translate to an equal sign (=).
    2. Step 2. Translate the words to the left of the “equals” word(s) into an algebraic expression.
    3. Step 3. Translate the words to the right of the “equals” word(s) into an algebraic expression.
  • To Solve an Application
    1. Step 1. Read the problem. Make sure all the words and ideas are understood.
    2. Step 2. Identify what we are looking for.
    3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
    4. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with the important information.
    5. Step 5. Solve the equation using good algebra techniques.
    6. Step 6. Check the answer in the problem and make sure it makes sense.
    7. Step 7. Answer the question with a complete sentence.

2.2 Solve Equations using the Division and Multiplication Properties of Equality

  • The Division Property of Equality—For any numbers a, b, and c, and c≠0c≠0, if a=ba=b, then ac=bcac=bc.
     When you divide both sides of an equation by any non-zero number, you still have equality.
  • The Multiplication Property of Equality—For any numbers a, b, and c, if a=ba=b, then ac=bcac=bc.
     If you multiply both sides of an equation by the same number, you still have equality.

2.3 Solve Equations with Variables and Constants on Both Sides

  • Beginning Strategy for Solving an Equation with Variables and Constants on Both Sides of the Equation
    1. Step 1. Choose which side will be the “variable” side—the other side will be the “constant” side.
    2. Step 2. Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality.
    3. Step 3. Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality.
    4. Step 4. Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality.
    5. Step 5. Check the solution by substituting it into the original equation.

2.4 Use a General Strategy to Solve Linear Equations

  • General Strategy for Solving Linear Equations
    1. Step 1. Simplify each side of the equation as much as possible.
       Use the Distributive Property to remove any parentheses.
       Combine like terms.
    2. Step 2. Collect all the variable terms on one side of the equation.
       Use the Addition or Subtraction Property of Equality.
    3. Step 3. Collect all the constant terms on the other side of the equation.
       Use the Addition or Subtraction Property of Equality.
    4. Step 4. Make the coefficient of the variable term to equal to 1.
       Use the Multiplication or Division Property of Equality.
       State the solution to the equation.
    5. Step 5. Check the solution.
       Substitute the solution into the original equation.

2.5 Solve Equations with Fractions or Decimals

  • Strategy to Solve an Equation with Fraction Coefficients
    1. Step 1. Find the least common denominator of all the fractions in the equation.
    2. Step 2. Multiply both sides of the equation by that LCD. This clears the fractions.
    3. Step 3. Solve using the General Strategy for Solving Linear Equations.

2.6 Solve a Formula for a Specific Variable

  • To Solve an Application (with a formula)
    1. Step 1. Read the problem. Make sure all the words and ideas are understood.
    2. Step 2. Identify what we are looking for.
    3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
    4. Step 4. Translate into an equation. Write the appropriate formula for the situation. Substitute in the given information.
    5. Step 5. Solve the equation using good algebra techniques.
    6. Step 6. Check the answer in the problem and make sure it makes sense.
    7. Step 7. Answer the question with a complete sentence.
  • Distance, Rate and Time
     For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula: d=rtd=rt where d = distance, r = rate, t = time.
  • To solve a formula for a specific variable means to get that variable by itself with a coefficient of 1 on one side of the equation and all other variables and constants on the other side.

2.7 Solve Linear Inequalities

  • Subtraction Property of Inequality
     For any numbers a, b, and c,
     if a<ba<b then a−c<b−ca−c<b−c and
     if a>ba>b then a−c>b−c.a−c>b−c.
  • Addition Property of Inequality
     For any numbers a, b, and c,
     if a<ba<b then a+c<b+ca+c<b+c and
     if a>ba>b then a+c>b+c.a+c>b+c.
  • Division and Multiplication Properties of Inequality
     For any numbers a, b, and c,
     if a<ba<b and c>0c>0, then ac<bcac<bc and ac>bcac>bc.
     if a>ba>b and c>0c>0, then ac>bcac>bc and ac>bcac>bc.
     if a<ba<b and c<0c<0, then ac>bcac>bc and ac>bcac>bc.
     if a>ba>b and c<0c<0, then ac<bcac<bc and ac<bcac<bc.
  • When we divide or multiply an inequality by a:
    • positive number, the inequality stays the same.
    • negative number, the inequality reverses.
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