Math 121 Notes

 

Chapter 0

 

 

 

 

Section 0.1  (Inequalities)

 

 

Definitions

 

Rational Numbers:  A number that can be expressed as the quotient of two integers

 

Examples:  1, 5, -4, -8 ,, 3.45, 2.346

 

Irrational Numbers:  A number that can not be expressed as the quotient of two integers,

 

Examples: 

 

 

Examples for the textbook page 0-7

 

2)      - 3678  ( This is a rational number)

 

 

      4)     (This number is irrational)

 

 

      6)        (This number is rational)

 

     10)      (This number is irrational)

 

 

Inequalities

 

Symbols

 

 

 

 

 

Solving Inequalities

 

Inequality Properties

 

Transitive Property    

 

Addition Property for Inequalities  

 

Multiplication properties for Inequalities  

 

(If c is positive, then )

(If c is negative, then )

 

Subtraction property for Inequalities  

 

 

Examples of solving inequalities

 

1)      Solve 

 

 

 

 

2)      Solve 

 

 

 

 

 

3)      Solve

 

 

 

 

Compound Inequalities

 

4)      Solve

 

          

 

 

 Examples from the book

 

#20  page 0-7

 

Solve

 

         

 

#22      Solve

 

                

 

 

 

 

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Section 0.2

 

Absolute Values and Inequalities

 

 

 

 

 

 

Introduction to Interval Notation

 

- Positive Infinity

 

- Negative Infinity

 

( ) –open point

 

[ ] – closed point

 

 

Inequality

Graph

Interval

 

 

 

                  |

         -3     0        4

 

[-3,4]

 

                             

 

                  |                  

         -2     0        5                                                            

 

[-2,5]

 

 

 

            |

           0         3

 

 

 

 

 

                    °        |

                   2       0

 

 

 

Rules for absolute value expressions with inequalities

 

 

 

 

 

 

 

Examples

 

1)      Solve

 

              

                                |

                      -3      0        3  

         Interval:

 

 

2)  Solve

               

 

 

3)      Solve

                

 

4)      Solve 

                 

 

 

 

 

 

5)      Solve

                

 

Examples from page 0-12

 

10)   Write (-7,1) as an inequality expression with absolute values.

 

                             

                                                 d=3  d=3

                                                |       |       |

                                                ş      |       ş

                                               -7    -4     1

 

                                                                                   

 

 

12)  Write  as an inequality expression with absolute values.

 

                                                 

                                                   |----|----|

                                                         |

                                     20  22  24

 

                                  

 

#6 page 0-12

 

(-3,3) as an inequality expression with absolute values

 

 

 

Examples

 

1)      Solve

 

                  

 

                               

 

 

 

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Section 0.3

 

Exponents and Radicals

 

 

 

 

 

Definition of an Exponent

 

Define

 

Examples

 

 

Properties for Exponents

 

 

Examples

 

1)  Simplify   

 

2)  Simplify 

 

3) Simplify 

 

 

 

 

 

 

 

 

Operations with Exponents

 

 

 

Examples

 

1)      Simplify      (Rule 9)

 

 

2)      Simplify    (Rule 7)

 

 

 

3)      Simplify     (Rules 7 and 9)

 

 

 

4)      Simplify     (Rule 6)

 

 

 

 

5)      Simplify        (Rule 8)

 

 

 

 

 

More Examples

 

6)      Simplify 

 

 

 

 

7)      Simplify 

 

 

 

 

8)      Simplify 

 

 

 

 

9)      Simplify 

 

 

 

 

 

 

Examples from the book page 0-18

 

4) Evaluate where x = 4

 

 

 

 

 

 

 

8)      Evaluate   where x = 3

 

 

 

 

 

14) Evaluate   where x = 16

 

 

 

 

 

 

22) Simplify

 

 

 

 

26) Simplify

 

 

 

 

 

28) Simplify 

 

 

 

 

34) Simplify

 

 

 

 

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Section 0.4

 

Factoring Polynomials

 

 

 

 

 

Types of Factoring

 

1)      Factor by taking out a Greatest Common Factor (GFC)

2)      Factor a trinomial as two binomials.

3)      Factor a binomial as a difference of two squares

4)      Factor a binomial as a difference of cubes or a sum of cubes

5)      Factor by grouping

 

 

 

Factoring out a Greatest Common Factor

 

Taking out a greatest common factor is essentially the same as working the distributive property backwards.

 

Review of distributive property

 

 

Examples of factoring out a greatest common factor

 

1)      Factor   

                

 

 

 

2)      Factor  

 

 

 

 

 

3)      Factor

 

 

 

 

4)  Factor 

 

     

 

 

 

Factoring a trinomial as two binomials

 

Factoring a trinomial is the same as working the FOIL process

 

So, here is a short review of FOIL

 

 

Examples of factoring a trinomial as two binomials

 

1)      Factor

            

In this example you want to find two numbers that multiply to get 25 and add to get

 

10.    By using x as the first entry in each binomial you get:

 

      

 

Here are some similar examples

 

2)      Factor

 

Answer:      Hint:  Find two numbers that multiply to get 24 and add to get -10

 

 

 

3)      Factor  

 

       Answer:       Hint:  Find two integers that multiply two get -35 and add

       to get -2.      -7,5

 

 

 

 

4)      Factor  

 

      Answer: 

 

 

 

 

5)      Factor

 

Answer: 

 

 

 

 

Factor a binomial as a difference of squares

 

Examples

 

1)      Factor

 

       Answer:    Hint:  Basically use the FOIL process backwards again and find two integers that multiply to get -4 and add to get zero.  This process will cancel out the x-terms.

 

Check:

 

 

 

Other similar examples

 

2)      Factor   

 

      Answer:

 

 

 

3)      Factor

 

       Answer: 

 

 

 

4)      Factor

 

       Answer:

 

      

 

 

General Form of a Difference of Squares:

 

 

 

 

 

 

 

Difference of Cubes and sum of Cubes

 

Main formulas

 

Difference of Two Cubes

 

Sum of Two Cubes

 

Examples

 

1)      Factor 

 

 Answer:

           

 

 

2)      Factor 

 

       Answer:     

                    

 

 

      3)

             

 

        

      

 

 

     

 

4)

             

 

 

 

 

 

 

 

Factoring by Grouping

 

Examples

 

1)      Factor

 

Solution:

 

 Factor out a from the first two terms and -4 from the last two terms

Factor out (x – 2) form both terms

 

Similar types of examples

 

 

2)      Factor  

 

Solution:

 

 

 

 

 

 

 

 

3)      Factor

 

Solution:

 

    

 

Don’t forget that is a difference of squares

 

 

 

Solving Polynomial and Quadratic Equations

 

1)      Solve

 

 

 

2)      Solve

 

 

 

3)      Solve 

 

 

4)      Solve 

 

 

 

 

 

 

5)      Solve 2

 

 

 

6)      Solve

 

      

 

7)      Solve 

 

 

 

 

 

 

 

 

 

 

 

 

 

8)      Solve  

 

 

9) Solve