The Quick MaTh Review

Diana GaffordTexas State Technical College Harlingen

Dr. Mike HosseinpourTexas State Technical College Harlingen

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3 11 5 42 2( ) - ( )

© 2007 Diana Gafford & Dr. Mike Hosseinpour

ISBN 978-1-934302-06-4

All rights reserved, including the right to reproduce this book or any portion thereof in any form. Requests for such permissions should be addressed to:

TSTC PublishingTexas State Technical College Waco3801 Campus DriveWaco, TX 76705

http://publishing.tstc.edu/

Publisher: Mark LongGraphics specialist: Grace ArsiagaEditor: Todd GlasscockPrinting production: Bill EvridgeCover design: Stacie ButerbaughGraphics intern: Joe Miller

Manufactured in the United States of America

First edition

Table of ConTenTschapter 1: Introduction I. Number Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. Absolute Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 III. Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 IV. Properties of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Chapter 2: Fractions I. Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 II. Converting Improper Fractions to Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 III. Converting Mixed Numbers to Improper Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 IV. Equivalent Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 V. Multiplying Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 VI. Multiplying Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 VII. Reciprocals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 VIII. Dividing Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 IX. Adding and Subtracting Fractions with Like Denominators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 X. Finding the Least Common Denominator (LCD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 XI. Adding or Subtracting Fractions with Unlike Denominators . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 XII. Adding Mixed Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 XIII. Subtracting Mixed Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Chapter 3: Decimals I. Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 II. Adding Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 III. Subtracting Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 IV. Multiplying Decimals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 V. Dividing Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 VI. Converting Fractions to Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 VII. Common Decimal Equivalents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 VIII. Converting Mixed Numbers to Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 IX. Converting Decimals to Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Chapter 4: Proportions I. Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 II. Using Proportions to Solve Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Chapter 5: Percent I. Percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 II. Changing a Percent to a Decimal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 III. Changing a Decimal or a Fraction to a Percent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 IV. Identifying Base, Rate, and Amount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Chapter 6: Real Numbers I. Real Numbers, Applications, and the Negative of a Number . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 II. Addition and Subtraction of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 III. Multiplication and Division of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Chapter 7: Exponents and Scientific Notation I. Exponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 II. Scientific Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 III. Expanded Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 IV. Multiplying and Dividing in Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Chapter 8: Polynomials I. Evaluating Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 II. Numerical Coefficients and Like Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 III. Adding and Subtracting Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 IV. Multiplying Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 V. Multiplying Two Binomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 VI. Multiplying Two Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 VII. Dividing Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 VIII. Dividing a Polynomial by a Monomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 IX. Dividing a Polynomial by Another Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Chapter 9: Equations and Inequalities I. Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 II. Addition Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 III. Multiplication Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 IV. Multiplication and Addition Properties Together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 V. Solving Strategy for Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 VI. Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 VII. Application Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 VIII. Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 IX. Inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 X. Graphing Linear Equations: Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 XI. Graph Using Slope-Intercept Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 XII. Graph Using Intercept Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Chapter 10: Factoring I. Greatest Common Factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 II. Factor by Grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 III. Factoring Trinomials of the Form x2 + bx + c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 IV. Factoring Trinomials of the Form x2 + bx + c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 V. Factoring Trinomials of the Form ax2 + bx + c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 VI. Special Factoring: Perfect Square Trinomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 VII. Special Factoring: Difference of Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 VIII. General Approach to Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 IX. Solving Quadratic Equations by Factoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Chapter 11: Rational Expressions and Equations I. Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 II. Reducing Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 III. Simplifying Rational Expressions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 IV. Multiplying Fractional Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 V. Dividing Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 VI. Adding or Subtracting Rational Expressions: Like Denominators . . . . . . . . . . . . . . . . . . . . . . 124 VII. Adding or Subracting with Unlike Denominators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 VIII. Complex Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 IX. Rational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Chapter 12 Radical Expressions I. Square Root and Common Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 II. Positive and Negative Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 III. Converting a Radical to a Fractional Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 IV. Radical Properties and Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 V. Multiplying and Simplifying Radicals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 VI. Adding and Subtracting Radicals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 VII. Radical Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 VIII. Radical Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 IX. Simplified Form for Radicals and Rationalizing Denominators . . . . . . . . . . . . . . . . . . . . . . . . 155 X. Simplifying Radical Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 XI. Rationalizing Denominators When the Index is ≥ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 XII. Rationalizing Denominators with Two Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 XIII. Solving Radical Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Chapter 13: Systems of Equations I. The Graphing Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 II. The Substitution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 III. The Elimination Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Chapter 14: Quadratic Equations and Graphs I. Standard Form of a Quadratic Equation: ax2 + bx + c = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 II. Discriminant: b2 – 4ac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 III. Graphing Quadratic Equations—Parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 IV. Quadratic Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Chapter 15: Functions I. Relations and Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 II. Vertical Line Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 III. Function Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Answer Keys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197-216

About TSTC Publishing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Introduction �

Chapter 1: Introduction

I. Number Sets

• Natural Numbers or Counting Numbers

o 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

• Whole Numbers

o 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

• Integers

o …, -3, -2, -1, 0, 1, 2, 3, …

• Rational Numbers

o Any number is rational that can be written qp in which p and q are integers

and 0≠q .

Examples: 51,

75−

o Whole numbers are rational numbers that can be written with a denominator of one.

Examples: 7 71

or , - -13 131

or , 0 01

or

o Repeating or terminating decimals are rational numbers that can be written in the

form qp .

Examples: 0. ... .3333 13

0 75 34

= =or

• Irrational Numbers

o Any number found on a number line that is not rational is irrational.

Examples: ,2 �

� Introduction

• Prime Numbers

oAny number that can be divided evenly only by itself and one is a prime number.

Examples: 2, 3, 5, 13, 37

• Composite Numbers

oNumbers that are not prime are composite numbers. Examples: 4, 9, 12

• The number 1 is considered to be neither prime nor composite.

II. Absolute Value

• The distance from zero on a number line is the absolute value.

ExamplE 1

33 = 3 units from zero

ExamplE 2

55 =− 5 units from zero

0 1 3 4-1 -2 -3 -4 2

-2 -1 1 2-3 -4 -5 -6 0

Introduction �

ExamplE 3

5.15.1 =− units from zero

Examples of absolute Value

21

21= 44 −=−

- . .0 37 0 37= 66 −=−−

7792 =−=−

Exercises

Find the absolute values and simplify.

1. 14 2. 3.7−

3. 3− 4. 47 −−

5. 11 6- 6.32−

7. 6 11- 8. - + -15 7

-2 -1 1 2-3 -4 -5 -6 0 -6

� Introduction

III. Order of Operations

• First, do all operations inside a set of grouping symbols: ( ) , [ ] , or { }.

o If more than one set of grouping symbols is present, work from the innermost set to the outermost set.

• Second, evaluate exponents and roots.

• Third, perform multiplication and division as either occurs from left to right.

• Fourth, perform addition and subtraction as either occurs from left to right.

• Note: Simplify numerator and denominator separately if fractions are present.

ExamplE 4 ExamplE 5 ExamplE 6

3 4 7 5 7 32 2+( ) - -( ) 9 4 7 2 5- - ·( ) 6 5 8 3

2 7 5

2 + -( )· -

3 11 5 42 2( ) - ( ) 9 4 7 10- -( ) 6 5 514 5

2 + ( )-

3 121 5 16( )- ( ) 9 4 3- -( ) 36 5 59

+ ( )

363 – 80 9 12+ 36 259+

283 21 619

Introduction �

Exercises

Simplify.

1. 32 5 4- · 2. 8 2 8 2- -( )¸( )

3. 3 7 2+( ) 4. 22 73 +

5. 243 81 3¸ ¸ 6. 243 81 3¸ ¸( )

7. 49 5 23- · 8. 7 4 2 5 1+ · -( )

9. 3 14 5 63 7 3

2+ -( )++ ·

10. 4 12 7 8 5 7 4· - +( )¸ - -( )

11. 61 2 3 12 13 4 3- - -( )¸éë ùû{ } 12. 6 2 3 7 5+( )-éë ùû

� Introduction

IV. Properties of Real Numbers

• Commutative Property

o Addition

Order in which two numbers are added does not affect the sum: abba +=+

o Multiplication

Order in which two numbers are multiplied does not affect the product:

abba •=•

• Associative Property

o Addition

Order in which a group of numbers are added does not affect the sum:

a b c a b c+( )+ = + +( )

o Multiplication

Order in which a group of numbers are multiplied does not affect the product:

ab c a bc( ) = ( )

• Distributive Property of Multiplication over Addition

o Either add first and then multiply or multiply first and then add.

a b c ab ac+( )= +

• Identity Property

o Addition a a a a+ = + =0 0or

o Multiplication a a a a· = · =1 1or

Introduction �

• Multiplication Property of Zero

o Multiplication of any whole number by 0 = 0 (product).

a a· = · =0 0 0 0or

• Division with Zero and One

o For any whole number 0, ≠aa

a a aa

¸ = =1 1or 0 0 0 0¸ = =aa

or

o For any whole number a

a a a a¸ = =11

or

o For any whole number a a a, ¸00

or is undefined.

• Note: Division by zero is not allowed.

Examples

Commutative Property of Addition

a b b a+ = +

8 7 7 8+ = + 10 3 3 10+ = +

15 15= 13 13=

Associative Property of Addition

a b c a b c+( )+ = + +( )

7 3 5+( )+ = 7 3 5+ +( )

10 5+ = 7 8+

15 = 15

� Introduction

Commutative Property of Multiplication

a b b a· = ·

4 5• = 5 4•

20 = 20

Associative Property of Multiplication

ab c a bc( ) = ( )

2 3 5·( )· = 2 3 5· ·( )

56 • = 2 15·

30 = 30

Distributive Property of Multiplication over Addition

a b c ab ac+( )= +

5 6 3· +( ) = 3565 •+•

95 • = 30 15+

45 = 45

Multiplication Property of Zero0 0 or 0 0a a• = • =

003 =• 070 =•

Introduction �

Division with Zero and One

a a aa

¸ = =1 1or

0 0 0 0¸ = =aa

or

155,188 =

−−

=÷

0 3 0 07

0¸ =-

=or

a a a a¸ = =11

or

6 1 6 71

7¸ =-

=-or

a¸0 Undefined

7 0¸ Undefined

-130

Undefined

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