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Intermediate Algebra

Key Concepts

Intermediate AlgebraKey Concepts
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. 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
    12. Chapter 12
  15. Index

1.1 Use the Language of Algebra

  • Divisibility Tests
    A number is divisible by:
      2 if the last digit is 0, 2, 4, 6, or 8.
      3 if the sum of the digits is divisible by 3.
      5 if the last digit is 5 or 0.
      6 if it is divisible by both 2 and 3.
      10 if it ends with 0.
  • How to find the prime factorization of a composite number.
    1. Step 1. Find two factors whose product is the given number, and use these numbers to create two branches.
    2. Step 2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
    3. Step 3. If a factor is not prime, write it as the product of two factors and continue the process.
    4. Step 4. Write the composite number as the product of all the circled primes.
  • How To Find the least common multiple using the prime factors method.
    1. Step 1. Write each number as a product of primes.
    2. Step 2. List the primes of each number. Match primes vertically when possible.
    3. Step 3. Bring down the columns.
    4. Step 4. Multiply the factors.
  • Equality Symbol
    a=ba=b is read “a is equal to b.”
    The symbol “=” is called the equal sign.
  • Inequality
    For a less than b, a is to the left of b on the number line. For a greater than b, a is to the right of b on the number line.
  • Inequality Symbols
    Inequality Symbols Words
    abab a is not equal to b.
    a<ba<b a is less than b.
    abab a is less than or equal to b.
    a>ba>b a is greater than b.
    abab a is greater than or equal to b.
    Table 1.4
  • Grouping Symbols
    Parentheses()Brackets[]Braces{}Parentheses()Brackets[]Braces{}
  • Exponential Notation
    anan means multiply a by itself, n times.
    The expression anan is read a to the nthnth power.
  • Simplify an Expression
    To simplify an expression, do all operations in the expression.
  • How to use the order of operations.
    1. Step 1. Parentheses and Other Grouping Symbols
      • Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
    2. Step 2. Exponents
      • Simplify all expressions with exponents.
    3. Step 3. Multiplication and Division
      • Perform all multiplication and division in order from left to right. These operations have equal priority.
    4. Step 4. Addition and Subtraction
      • Perform all addition and subtraction in order from left to right. These operations have equal priority.
  • How to combine like terms.
    1. Step 1. Identify like terms.
    2. Step 2. Rearrange the expression so like terms are together.
    3. Step 3. Add or subtract the coefficients and keep the same variable for each group of like terms.
    Operation Phrase Expression
    Addition a plus b
    the sum of aa and b
    a increased by b
    b more than a
    the total of a and b
    b added to a
    a+ba+b
    Subtraction a minus bb
    the difference of a and b
    a decreased by b
    b less than a
    b subtracted from a
    abab
    Multiplication a times b
    the product of aa and bb
    twice a
    a·b,ab,a(b),(a)(b)a·b,ab,a(b),(a)(b)


    2a2a
    Division a divided by b

    the quotient of a and b
    the ratio of a and b
    b divided into a
    a÷b,a/b,ab,baa÷b,a/b,ab,ba
    Table 1.5

1.2 Integers

  • Opposite Notation
    ameans the opposite of the numbera The notationais read as “the opposite ofa.”ameans the opposite of the numbera The notationais read as “the opposite ofa.”
  • Absolute Value
    The absolute value of a number is its distance from 0 on the number line.
    The absolute value of a number n is written as |n||n| and |n|0|n|0 for all numbers.
    Absolute values are always greater than or equal to zero.
  • Grouping Symbols
    Parentheses()Braces{} Brackets[]Absolute value||Parentheses()Braces{} Brackets[]Absolute value||
  • Subtraction Property
    ab=a+(b)ab=a+(b)
    Subtracting a number is the same as adding its opposite.
  • Multiplication and Division of Signed Numbers
    For multiplication and division of two signed numbers:
    Same signs Result
    •  Two positives Positive
    •  Two negatives Positive

    If the signs are the same, the result is positive.
    Different signs Result
    •  Positive and negative Negative
    •  Negative and positive Negative

    If the signs are different, the result is negative.
  • Multiplication by −1−1
    −1a=a−1a=a
    Multiplying a number by −1−1 gives its opposite.
  • How to Use Integers in Applications.
    1. Step 1. Read the problem. Make sure all the words and ideas are understood
    2. Step 2. Identify what we are asked to find.
    3. Step 3. Write a phrase that gives the information to find it.
    4. Step 4. Translate the phrase to an expression.
    5. Step 5. Simplify the expression.
    6. Step 6. Answer the question with a complete sentence.

1.3 Fractions

  • Equivalent Fractions Property
    If a, b, and c are numbers where b0,c0,b0,c0, then
    ab=a·cb·canda·cb·c=ab.ab=a·cb·canda·cb·c=ab.
  • How to simplify a fraction.
    1. Step 1. Rewrite the numerator and denominator to show the common factors.
      If needed, factor the numerator and denominator into prime numbers first.
    2. Step 2. Simplify using the Equivalent Fractions Property by dividing out common factors.
    3. Step 3. Multiply any remaining factors.
  • Fraction Multiplication
    If a, b, c, and d are numbers where b0,b0, and d0,d0, then
    ab·cd=acbd.ab·cd=acbd.
    To multiply fractions, multiply the numerators and multiply the denominators.
  • Fraction Division
    If a, b, c, and d are numbers where b0,c0,b0,c0, and d0,d0, then
    ab÷cd=ab·dc.ab÷cd=ab·dc.
    To divide fractions, we multiply the first fraction by the reciprocal of the second.
  • Fraction Addition and Subtraction
    If a, b, and c are numbers where c0,c0, then
    ac+bc=a+bcandacbc=abc.ac+bc=a+bcandacbc=abc.
    To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
  • How to add or subtract fractions.
    1. Step 1. Do they have a common denominator?
      • Yes—go to step 2.
      • No—rewrite each fraction with the LCD (least common denominator).
        • Find the LCD.
        • Change each fraction into an equivalent fraction with the LCD as its denominator.
    2. Step 2. Add or subtract the fractions.
    3. Step 3. Simplify, if possible.
  • How to simplify an expression with a fraction bar.
    1. Step 1. Simplify the expression in the numerator. Simplify the expression in the denominator.
    2. Step 2. Simplify the fraction.
  • Placement of Negative Sign in a Fraction
    For any positive numbers a and b,
    ab=ab=ab.ab=ab=ab.
  • How to simplify complex fractions.
    1. Step 1. Simplify the numerator.
    2. Step 2. Simplify the denominator.
    3. Step 3. Divide the numerator by the denominator. Simplify if possible.

1.4 Decimals

  • How to round decimals.
    1. Step 1. Locate the given place value and mark it with an arrow.
    2. Step 2. Underline the digit to the right of the place value.
    3. Step 3. Is the underlined digit greater than or equal to 5?5?
      • Yes: add 1 to the digit in the given place value.
      • No: do not change the digit in the given place value
    4. Step 4. Rewrite the number, deleting all digits to the right of the rounding digit.
  • How to add or subtract decimals.
    1. Step 1. Determine the sign of the sum or difference.
    2. Step 2. Write the numbers so the decimal points line up vertically.
    3. Step 3. Use zeros as placeholders, as needed.
    4. Step 4. Add or subtract the numbers as if they were whole numbers. Then place the decimal point in the answer under the decimal points in the given numbers.
    5. Step 5. Write the sum or difference with the appropriate sign
  • How to multiply decimals.
    1. Step 1. Determine the sign of the product.
    2. Step 2. Write in vertical format, lining up the numbers on the right. Multiply the numbers as if they were whole numbers, temporarily ignoring the decimal points.
    3. Step 3. Place the decimal point. The number of decimal places in the product is the sum of the number of decimal places in the factors.
    4. Step 4. Write the product with the appropriate sign.
  • How to multiply a decimal by a power of ten.
    1. Step 1. Move the decimal point to the right the same number of places as the number of zeros in the power of 10.
    2. Step 2. Add zeros at the end of the number as needed.
  • How to divide decimals.
    1. Step 1. Determine the sign of the quotient.
    2. Step 2. Make the divisor a whole number by “moving” the decimal point all the way to the right. “Move” the decimal point in the dividend the same number of places—adding zeros as needed.
    3. Step 3. Divide. Place the decimal point in the quotient above the decimal point in the dividend.
    4. Step 4. Write the quotient with the appropriate sign.
  • How to convert a decimal to a proper fraction and a fraction to a decimal.
    1. Step 1. To convert a decimal to a proper fraction, determine the place value of the final digit.
    2. Step 2. Write the fraction.
      • numerator—the “numbers” to the right of the decimal point
      • denominator—the place value corresponding to the final digit
    3. Step 3. To convert a fraction to a decimal, divide the numerator of the fraction by the denominator of the fraction.
  • How to convert a percent to a decimal and a decimal to a percent.
    1. Step 1. To convert a percent to a decimal, move the decimal point two places to the left after removing the percent sign.
    2. Step 2. To convert a decimal to a percent, move the decimal point two places to the right and then add the percent sign.
  • Square Root Notation
    mm is read “the square root of m.”
    If m=n2,m=n2, then m=n,m=n, for n0.n0.
    The square root of m, m,m, is the positive number whose square is m.
  • Rational or Irrational
    If the decimal form of a number
    • repeats or stops, the number is a rational number.
    • does not repeat and does not stop, the number is an irrational number.
  • Real Numbers
    A chart shows that counting numbers 1, 2, 3 are a part of whole numbers 0, 1, 2, 3. Whole numbers are a part of integers minus 2, minus 1, 0, 1, 2. Integers are a part of rational numbers. Rational numbers along with irrational numbers form the set of real numbers.
    Figure 1.9

1.5 Properties of Real Numbers

Commutative Property
When adding or multiplying, changing the order gives the same result

of additionIfa,bare real numbers, thena+b=b+a of multiplicationIfa,bare real numbers, thena·b=b·aof additionIfa,bare real numbers, thena+b=b+a of multiplicationIfa,bare real numbers, thena·b=b·a
Associative Property
When adding or multiplying, changing the grouping gives the same result.

of additionIfa,b,andcare real numbers, then(a+b)+c=a+(b+c) of multiplicationIfa,b,andcare real numbers, then(a·b)·c=a·(b·c) of additionIfa,b,andcare real numbers, then(a+b)+c=a+(b+c) of multiplicationIfa,b,andcare real numbers, then(a·b)·c=a·(b·c)
Distributive Property

Ifa,b,andcare real numbers, thena(b+c)=ab+ac (b+c)a=ba+ca a(bc)=abac (bc)a=baca Ifa,b,andcare real numbers, thena(b+c)=ab+ac (b+c)a=ba+ca a(bc)=abac (bc)a=baca
Identity Property

of additionFor any real numbera:a+0=a 0is theadditive identity0+a=a of multiplicationFor any real numbera:a·1=a 1is themultiplicative identity1·a=a of additionFor any real numbera:a+0=a 0is theadditive identity0+a=a of multiplicationFor any real numbera:a·1=a 1is themultiplicative identity1·a=a
Inverse Property

of additionFor any real numbera,a+(a)=0 ais theadditive inverseofa A number and itsoppositeadd to zero. of multiplicationFor any real numbera,a0a·1a=1 1ais themultiplicative inverseofa A number and itsreciprocalmultiply to one. of additionFor any real numbera,a+(a)=0 ais theadditive inverseofa A number and itsoppositeadd to zero. of multiplicationFor any real numbera,a0a·1a=1 1ais themultiplicative inverseofa A number and itsreciprocalmultiply to one.
Properties of Zero
For any real numbera,a·0=0 0·a=0 For any real numbera,a0,0a=0 For any real numbera,a0is undefined For any real numbera,a·0=0 0·a=0 For any real numbera,a0,0a=0 For any real numbera,a0is undefined
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