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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

3.1 Use a Problem-Solving Strategy

  • Problem-Solving Strategy
    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 all the important information. Then, translate the English sentence into an algebra equation.
    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.
  • Consecutive Integers
    Consecutive integers are integers that immediately follow each other.
    n1stintegern+12ndinteger consecutive integern+23rdconsecutive integer . . . etc.n1stintegern+12ndinteger consecutive integern+23rdconsecutive integer . . . etc.

    Consecutive even integers are even integers that immediately follow one another.
    n1stintegern+22ndinteger consecutive integern+43rdconsecutive integer . . . etc.n1stintegern+22ndinteger consecutive integern+43rdconsecutive integer . . . etc.

    Consecutive odd integers are odd integers that immediately follow one another.
    n1stintegern+22ndinteger consecutive integern+43rdconsecutive integer . . . etc.n1stintegern+22ndinteger consecutive integern+43rdconsecutive integer . . . etc.

3.2 Solve Percent Applications

  • Percent Increase To find the percent increase:
    1. Step 1. Find the amount of increase. increase=new amountoriginalamountincrease=new amountoriginalamount
    2. Step 2. Find the percent increase. Increase is what percent of the original amount?
  • Percent Decrease To find the percent decrease:
    1. Step 1. Find the amount of decrease. decrease=original amountnewamountdecrease=original amountnewamount
    2. Step 2. Find the percent decrease. Decrease is what percent of the original amount?
  • Simple Interest If an amount of money, P, called the principal, is invested for a period of t years at an annual interest rate r, the amount of interest, I, earned is
    I=PrtwhereI=interestP=principalr=ratet=timeI=PrtwhereI=interestP=principalr=ratet=time
  • Discount
    • amount of discount is discount rate ·· original price
    • sale price is original price – discount
  • Mark-up
    • amount of mark-up is mark-up rate ·· original cost
    • list price is original cost + mark up

3.3 Solve Mixture Applications

  • Total Value of Coins For the same type of coin, the total value of a number of coins is found by using the model.
    number·value=totalvaluenumber·value=totalvalue where number is the number of coins and value is the value of each coin; total value is the total value of all the coins
  • Problem-Solving Strategy—Coin Word Problems
    1. Step 1. Read the problem. Make all the words and ideas are understood. Determine the types of coins involved.
      • Create a table to organize the information.
      • Label the columns type, number, value, total value.
      • List the types of coins.
      • Write in the value of each type of coin.
      • Write in the total value of all the coins.
    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.
      Use variable expressions to represent the number of each type of coin and write them in the table.
      Multiply the number times the value to get the total value of each type of coin.
    4. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the sentence into an equation.
      Write the equation by adding the total values of all the types of coins.
    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.

3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem

  • Problem-Solving Strategy for Geometry Applications
    1. Step 1. Read the problem and make 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.
  • Triangle Properties For ABCABC
    Angle measures:
    • mA+mB+mC=180mA+mB+mC=180
    Perimeter:
    • P=a+b+cP=a+b+c
    Area:
    • A=12bh,b=base,h=heightA=12bh,b=base,h=height
    A right triangle has one 90°90° angle.
  • The Pythagorean Theorem In any right triangle, a2+b2=c2a2+b2=c2 where c is the length of the hypotenuse and a and b are the lengths of the legs.
  • Properties of Rectangles
    • Rectangles have four sides and four right (90°) angles.
    • The lengths of opposite sides are equal.
    • The perimeter of a rectangle is the sum of twice the length and twice the width: P=2L+2W.P=2L+2W. The area of a rectangle is the length times the width: A=LW.A=LW.

3.5 Solve Uniform Motion Applications

  • Distance, Rate, and Time
    • D = rt where D = distance, r = rate, t = time
  • Problem-Solving Strategy—Distance, Rate, and Time Applications
    1. Step 1. Read the problem. Make sure all the words and ideas are understood.
      Draw a diagram to illustrate what it happening.
      Create a table to organize the information: Label the columns rate, time, distance. List the two scenarios. Write in the information you know.
    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.
      Complete the chart.
      Use variable expressions to represent that quantity in each row.
      Multiply the rate times the time to get the distance.
    4. Step 4. Translate into an equation.
      Restate the problem in one sentence with all the important information.
      Then, translate the sentence into an equation.
    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.

3.6 Solve Applications with Linear Inequalities

  • Solving inequalities
    1. Step 1. Read the problem.
    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. Write a sentence that gives the information to find it. Translate into an inequality.
    5. Step 5. Solve the inequality.
    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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