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Elementary Algebra 2e

Key Concepts

Elementary Algebra 2eKey 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 Solving 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 Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    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 in Two Variables
    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

8.1 Simplify Rational Expressions

  • Determine the Values for Which a Rational Expression is Undefined
    1. Step 1. Set the denominator equal to zero.
    2. Step 2. Solve the equation, if possible.
  • Simplified Rational Expression
    • A rational expression is considered simplified if there are no common factors in its numerator and denominator.
  • Simplify a Rational Expression
    1. Step 1. Factor the numerator and denominator completely.
    2. Step 2. Simplify by dividing out common factors.
  • Opposites in a Rational Expression
    • The opposite of a−ba−b is b−ab−a.
      a−bb−a=−1a≠0,b≠0,a≠ba−bb−a=−1a≠0,b≠0,a≠b

8.2 Multiply and Divide Rational Expressions

  • Multiplication of Rational Expressions
    • If p,q,r,sp,q,r,s are polynomials where q≠0,s≠0q≠0,s≠0, then pq·rs=prqspq·rs=prqs.
    • To multiply rational expressions, multiply the numerators and multiply the denominators
  • Multiply a Rational Expression
    1. Step 1. Factor each numerator and denominator completely.
    2. Step 2. Multiply the numerators and denominators.
    3. Step 3. Simplify by dividing out common factors.
  • Division of Rational Expressions
    • If p,q,r,sp,q,r,s are polynomials where q≠0,r≠0,s≠0q≠0,r≠0,s≠0, then pq÷rs=pq·srpq÷rs=pq·sr.
    • To divide rational expressions multiply the first fraction by the reciprocal of the second.
  • Divide Rational Expressions
    1. Step 1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
    2. Step 2. Factor the numerators and denominators completely.
    3. Step 3. Multiply the numerators and denominators together.

    4. Step 4. Simplify by dividing out common factors.

8.3 Add and Subtract Rational Expressions with a Common Denominator

  • Rational Expression Addition
    • If p,q,andrp,q,andr are polynomials where r≠0r≠0, then
      pr+qr=p+qrpr+qr=p+qr
    • To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.
  • Rational Expression Subtraction
    • If p,q,andrp,q,andr are polynomials where r≠0r≠0, then
      pr−qr=p−qrpr−qr=p−qr
    • To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

8.4 Add and Subtract Rational Expressions with Unlike Denominators

  • Find the Least Common Denominator of Rational Expressions
    1. Step 1. Factor each expression completely.
    2. Step 2. List the factors of each expression. Match factors vertically when possible.
    3. Step 3. Bring down the columns.
    4. Step 4. Multiply the factors.
  • Add or Subtract Rational Expressions
    1. Step 1.
      Determine if the expressions have a common denominator.
       Yes – go to step 2.
       No – Rewrite each rational expression with the LCD.
      • Find the LCD.
      • Rewrite each rational expression as an equivalent rational expression with the LCD.
    2. Step 2. Add or subtract the rational expressions.
    3. Step 3. Simplify, if possible.

8.5 Simplify Complex Rational Expressions

  • To Simplify a Rational Expression by Writing it as Division
    1. Step 1. Simplify the numerator and denominator.
    2. Step 2. Rewrite the complex rational expression as a division problem.
    3. Step 3. Divide the expressions.
  • To Simplify a Complex Rational Expression by Using the LCD
    1. Step 1. Find the LCD of all fractions in the complex rational expression.
    2. Step 2. Multiply the numerator and denominator by the LCD.
    3. Step 3. Simplify the expression.

8.6 Solve Rational Equations

  • Strategy to Solve Equations with Rational Expressions
    1. Step 1. Note any value of the variable that would make any denominator zero.
    2. Step 2. Find the least common denominator of all denominators in the equation.
    3. Step 3. Clear the fractions by multiplying both sides of the equation by the LCD.
    4. Step 4. Solve the resulting equation.
    5. Step 5. Check.
    • If any values found in Step 1 are algebraic solutions, discard them.
    • Check any remaining solutions in the original equation.

8.7 Solve Proportion and Similar Figure Applications

  • Property of Similar Triangles
    • If ΔABCΔABC is similar to ΔXYZΔXYZ, then their corresponding angle measures are equal and their corresponding sides are in the same ratio.
  • Problem Solving Strategy for Geometry Applications
    1. Step 1. Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.
    2. Step 2. Identify what we are looking for.
    3. Step 3. Name what we are looking for by choosing a variable to represent it.
    4. Step 4. Translate into an equation by writing the appropriate formula or model 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.
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