MTH05LectureNotes

MTH 05 Lecture Notes

Andrew McInerney Fall 2016

c 2016 Andrew McInerney All rights reserved.

This work may be distributed and/or modified under the conditions of the Copyleft License.

Andrew McInerney Department of Mathematics and Computer Science Bronx Community College of the City University of New York

2155 University Ave. Bronx, NY 10453

andrew.mcinerney@bcc.cuny.edu

First edition: January 2016

Revisions:

August 2016

Contents

I Preparing for algebra

1

1 Review of fractions

3

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Decimal representation . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Mixed numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Graphing fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Equivalent fractions . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5.1 Writing fractions with common denominators . . . . . . . 8

1.5.2 Writing fractions in simplest form . . . . . . . . . . . . . 9

1.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.6 Multiplying and dividing fractions . . . . . . . . . . . . . . . . . 10

1.6.1 Multiplying fractions . . . . . . . . . . . . . . . . . . . . . 10

1.6.2 Dividing fractions . . . . . . . . . . . . . . . . . . . . . . 11

1.6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.7 Adding and subtracting fractions . . . . . . . . . . . . . . . . . . 14

1.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.8 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Signed numbers

19

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Graphing signed numbers . . . . . . . . . . . . . . . . . . . . . . 20

2.3 Adding and subtracting signed numbers . . . . . . . . . . . . . . 21

2.3.1 Adding signed numbers . . . . . . . . . . . . . . . . . . . 22

2.3.2 Subtracting signed numbers . . . . . . . . . . . . . . . . . 25

2.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Multiplying and dividing signed numbers . . . . . . . . . . . . . 29

2.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Exponents and roots with signed numbers . . . . . . . . . . . . . 31

2.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.6 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . 32

iii

3 Introduction to algebra

35

3.1 The order of operations . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Algebraic expressions . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Evaluating algebraic expressions . . . . . . . . . . . . . . . . . . 40

3.3.1 Function notation . . . . . . . . . . . . . . . . . . . . . . 42

3.3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4 Translating algebraic expressions . . . . . . . . . . . . . . . . . . 45

3.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . 47

II Linear equations and inequalities

49

4 Linear statements in one variable

51

4.1 Algebraic statements and solutions . . . . . . . . . . . . . . . . . 51

4.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Solving linear equations in one variable . . . . . . . . . . . . . . 56

4.2.1 The rules of the game . . . . . . . . . . . . . . . . . . . . 57

4.2.2 Applying the rules: Solving linear equations . . . . . . . . 57

4.2.3 Some unusual cases: Linear equations in one variable that

do not have exactly one solution . . . . . . . . . . . . . . 61

4.2.4 Another use of the multiplication principle: Equations

involving fractions . . . . . . . . . . . . . . . . . . . . . . 62

4.2.5 Some word problems . . . . . . . . . . . . . . . . . . . . . 64

4.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3 A detour: "Solving" literal equations . . . . . . . . . . . . . . . . 68

4.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.4 Solving linear inequalities in one variable . . . . . . . . . . . . . 71

4.4.1 Solving linear inequalities in one variable: Test value method 73

4.4.2 Solving linear inequalities in one variable: Standard form

method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5 Linear statements in two variables

85

5.1 Solving linear equations in two variables . . . . . . . . . . . . . . 85

5.1.1 A method for producing solutions . . . . . . . . . . . . . 86

5.1.2 Graphing linear equations in two variables . . . . . . . . . 88

5.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2 A detour: Slope and the geometry of lines . . . . . . . . . . . . . 95

5.2.1 Using the slope as an aid in graphing . . . . . . . . . . . 101

5.2.2 Finding an equation of a given line . . . . . . . . . . . . . 105

5.2.3 Special cases: Horizontal and vertical lines . . . . . . . . . 112

5.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.3 Solving linear inequalities in two variables . . . . . . . . . . . . . 116

5.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.4 Solving systems of linear equations . . . . . . . . . . . . . . . . . 125

5.4.1 Systems that do not have exactly one solution . . . . . . 131 5.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . 136

III Polynomials

137

6 Polynomials

139

6.1 Introduction to polynomials . . . . . . . . . . . . . . . . . . . . . 139

6.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

6.2 Adding and subtracting polynomials . . . . . . . . . . . . . . . . 144

6.2.1 Adding polynomials . . . . . . . . . . . . . . . . . . . . . 145

6.2.2 Subtracting polynomials . . . . . . . . . . . . . . . . . . . 146

6.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.3 Properties of exponents . . . . . . . . . . . . . . . . . . . . . . . 149

6.3.1 Integer exponents . . . . . . . . . . . . . . . . . . . . . . . 152

6.3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

6.4 A detour: Scientific notation . . . . . . . . . . . . . . . . . . . . 155

6.4.1 Multiplication and division of numbers in scientific notation157

6.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.5 Multiplying polynomials . . . . . . . . . . . . . . . . . . . . . . . 160

6.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.6 Dividing a polynomial by a monomial . . . . . . . . . . . . . . . 165

6.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.7 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7 Factoring

169

7.1 Introduction to factoring . . . . . . . . . . . . . . . . . . . . . . . 169

7.2 "Factoring out" the greatest common factor . . . . . . . . . . . . 172

7.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

7.3 Differences of squares . . . . . . . . . . . . . . . . . . . . . . . . 177

7.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

7.4 Quadratic trinomials I. Monic trinomials . . . . . . . . . . . . . . 181

7.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

7.5 Quadratic trinomials II. The ac-method . . . . . . . . . . . . . . 185

7.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

7.6 Factoring by grouping . . . . . . . . . . . . . . . . . . . . . . . . 192

7.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.7 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . 195

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