The Quick Math Review
2 4 = 5 a 10 2
x
2
2
3(11) - 5 (4)
6 4
Diana Gafford Texas State Technical College Harlingen
Dr. Mike Hosseinpour Texas State Technical College Harlingen
Š 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 Publishing Texas State Technical College Waco 3801 Campus Drive Waco, TX 76705 http://publishing.tstc.edu/ Publisher: Mark Long Graphics specialist: Grace Arsiaga Editor: Todd Glasscock Printing production: Bill Evridge Cover design: Stacie Buterbaugh Graphics intern: Joe Miller
Manufactured in the United States of America First edition
Table of Contents Chapter 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
I. Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 II. Using Proportions to Solve Word Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Chapter 4: Proportions
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
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 8: Polynomials
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 and q ≠ 0 . Examples:
p in which p and q are integers q
−5 1 , 7 5
o Whole numbers are rational numbers that can be written with a denominator of one.
0 7 -13 , -13 or , 0 or 1 1 1 o Repeating or terminating decimals are rational numbers that can be written in the Examples: 7 or
form
p . q
Examples: 0 .3333... =
1 3 or 0.75 = 3 4
• Irrational Numbers o Any number found on a number line that is not rational is irrational. Examples:
2, �
Introduction
• Prime Numbers o Any number that can be divided evenly only by itself and one is a prime number. Examples: 2, 3, 5, 13, 37 • Composite Numbers o Numbers 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
-4
-3 3 = 3
-2
-1
0
1
2
3
4
-1
0
1
2
3 units from zero
Example 2
-6
-5
-4
-3
− 5 = 5 5 units from zero
-2
Introduction
Example 3
-6 -6
-5 − 1.5 = 1.5
-4
-3
-2
-1
units from zero
Examples of Absolute Value 1 1 = 2 2
− 4 = −4
-0.37 = 0.37
− − 6 = −6
2−9 = −7 = 7
Exercises Find the absolute values and simplify. 1.
14
2.
− 7.3
3.
−3
4.
7 − −4
5.
11 - 6
6.
−2 3
7.
6 -11
8.
-15 + -7
0
1
2
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 2
Example 5 2
3(4 + 7) - 5 (7 - 3)
2
2
3(11) - 5 (4)
Example 6
9 - 4 (7 - 2 · 5)
62 + 5 (8 - 3) 2 · 7 -5
9 - 4 (7 -10)
62 + 5 (5) 14 - 5
3(121) - 5 (16)
9 - 4 (-3)
36 + 5(5) 9
363 – 80
9 + 12
36 + 25 9
283
21
61 9
Introduction
Exercises Simplify. 1.
32 - 5 · 4
2.
8 (-2) ¸ (-8)2
3.
(3 + 7)
4.
32 + 7 2
5.
243 ¸ 81 ¸ 3
6.
243 ¸ (81 ¸ 3)
7.
49 - 5 · 23
8.
7 + 4 (2 · 5 -1)
9.
3 + (14 - 5) + 62 3+ 7 ·3
10.
4 · 12 - (7 + 8) ¸ 5 - (7 - 4)
11.
61- 2 {3 éë 12 - (13 - 4) ¸ 3ùû
12.
6 éë 2 (3 + 7) - 5ùû
2
}
Introduction
IV. Properties of Real Numbers • Commutative Property o Addition Order in which two numbers are added does not affect the sum:
a+b =b+a
o Multiplication Order in which two numbers are multiplied does not affect the product:
a•b = b•a
• 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 + 0 = a or 0 + a = a
o Multiplication
a · 1 = a or 1 · a = a
Introduction
• Multiplication Property of Zero o Multiplication of any whole number by 0 = 0 (product).
a · 0 = 0 or 0 · a = 0
• Division with Zero and One o For any whole number a, a ≠ 0
a = 1 a
a ¸ a = 1 or
0 ¸ a = 0 or
0 =0 a
o For any whole number a
a ¸ 1 = a or
a =a 1
o For any whole number a, a ¸ 0 or • 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
a is undefined. 0
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) 6•5
=
30 =
2 · 15
30
Distributive Property of Multiplication over Addition
a (b + c) = ab + ac 5 · (6 + 3) =
5•6 + 5•3
5•9
=
30 + 15
45
=
45
Multiplication Property of Zero a • 0 = 0 or 0 • a = 0
3• 0 = 0
0•7 = 0
Introduction
Division with Zero and One
a ¸ a = 1 or
8 ÷ 8 = 1,
a =1 a
−5 =1 −5
0 ¸ 3 = 0 or
a ¸ 1 = a or
a =a 1
6 ¸ 1 = 6 or
-7 = -7 1
a ¸ 0 Undefined
7 ¸ 0 Undefined
-13 Undefined 0
0 ¸ a = 0 or
0 =0 -7
0 =0 a
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