Developmental mathematics

1286 Pages·172.629 MB·English

by  Bittinger, Marvin L;Beecher, Judith A

Pearson New International Edition

Developmental Mathematics
Marvin L. Bittinger Judith A. Beecher
Eighth Edition
International_PCL_TP.indd 1 7/29/13 11:23 AM
ISBN 10: 1-292-04016-5
ISBN 13: 978-1-292-04016-5
Pearson Education Limited
Edinburgh Gate
Harlow
Essex CM20 2JE
England and Associated Companies throughout the world
Visit us on the World Wide Web at: www.pearsoned.co.uk
© Pearson Education Limited 2014
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without either the
prior written permission of the publisher or a licence permitting restricted copying in the United Kingdom
issued by the Copyright Licensing Agency Ltd, Saffron House, 6–10 Kirby Street, London EC1N 8TS.
All trademarks used herein are the property of their respective owners. The use of any trademark
in this text does not vest in the author or publisher any trademark ownership rights in such
trademarks, nor does the use of such trademarks imply any affi liation with or endorsement of this
book by such owners.
ISBN 10: 1-292-04016-5
ISBN 10: 1-269-37450-8
ISBN 13: 978-1-292-04016-5
ISBN 13: 978-1-269-37450-7
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Printed in the United States of America
Copyright_Pg_7_24.indd 1 7/29/13 11:28 AM
11234556789186600986541739919733797
P E A R S O N C U S T O M L I B R AR Y
Table of Contents
Glossary
Marvin L. Bittinger/Judith A Beecher 1
1. Whole numbers
Marvin L. Bittinger/Judith A Beecher 7
2. Fraction Notation
Marvin L. Bittinger/Judith A Beecher 113
3. Decimal Notation
Marvin L. Bittinger/Judith A Beecher 189
4. Percent Notation
Marvin L. Bittinger/Judith A Beecher 269
5. Data, Graphs, and Statistics
Marvin L. Bittinger/Judith A Beecher 361
6. Geometry
Marvin L. Bittinger/Judith A Beecher 409
7. Introduction to Real Numbers and Algebraic Expressions
Marvin L. Bittinger/Judith A Beecher 507
8. Solving Equations and Inequalities
Marvin L. Bittinger/Judith A Beecher 593
9. Graphs of Linear Equations
Marvin L. Bittinger/Judith A Beecher 683
10. Polynomials: Operations
Marvin L. Bittinger/Judith A Beecher 767
11. Polynomials: Factoring
Marvin L. Bittinger/Judith A Beecher 859
12. Rational Expressions and Equations
Marvin L. Bittinger/Judith A Beecher 947
I
1111111111111111111001111112222222222239567899011233456779339953951513973135
13. Systems of Equations
Marvin L. Bittinger/Judith A Beecher 1039
14. Radical Expressions and Equations
Marvin L. Bittinger/Judith A Beecher 1093
Appendix: Linear Measures: American Units and Metric Units
Marvin L. Bittinger/Judith A Beecher 1153
Appendix: Weight and Mass; Medical Applications
Marvin L. Bittinger/Judith A Beecher 1169
Appendix: Capacity; Medical Applications
Marvin L. Bittinger/Judith A Beecher 1179
Appendix: Time and Temperature
Marvin L. Bittinger/Judith A Beecher 1185
Appendix: Sets
Marvin L. Bittinger/Judith A Beecher 1193
Appendix: Factoring Sums or Differences of Cubes
Marvin L. Bittinger/Judith A Beecher 1199
Appendix: Finding Equations of Lines: Point-Slope Equation
Marvin L. Bittinger/Judith A Beecher 1205
Appendix: Equations Involving Absolute Value
Marvin L. Bittinger/Judith A Beecher 1211
Appendix: The Distance Formula and Midpoints
Marvin L. Bittinger/Judith A Beecher 1215
Appendix: Higher Roots, and Rational Numbers as Exponents
Marvin L. Bittinger/Judith A Beecher 1221
Appendix: Inequalities and Interval Notation
Marvin L. Bittinger/Judith A Beecher 1233
Appendix: Nonlinear Inequalities
Marvin L. Bittinger/Judith A Beecher 1239
Appendix: Systems of Linear Inequalities
Marvin L. Bittinger/Judith A Beecher 1247
Appendix: Probability
Marvin L. Bittinger/Judith A Beecher 1253
Appendix: The Complex Numbers
Marvin L. Bittinger/Judith A Beecher 1261
Geometric Formulas; Fraction, Decimal, and Percent Equivalents
Marvin L. Bittinger/Judith A Beecher 1273
Index 1275
II
Glossary
A Average A center point of a set of numbers found by
adding the numbers and dividing by the number of
Abscissa The first coordinate in an ordered pair of
items of data; also called the meanor the arithmetic
numbers
mean
Absolute value The distance that a number is from 0 on
Axes Two perpendicular number lines used to identify
the number line
points in a plane
ac-method A method for factoring trinomials of the
type ax2 + bx + c, a Z 1,involving the product, ac, of B
the leading coefficient aand the last term c; also
called the grouping method Bar graph A graphic display of data using bars
Acute angle An angle whose measure is greater than 0° proportional in length to the numbers represented
and less than 90° Base In exponential notation, the number being raised
Acute triangle A triangle in which all three angles are to a power
acute Binomial A polynomial containing two terms
Addends In addition, the numbers being added
C
Additive identity The number 0
Additive inverse A number’s opposite; two numbers are Celsius A temperature scale in which water freezes at 0°
additive inverses of each other if their sum is zero. and water boils at 100°
Additive inverse of a polynomial Two polynomials are Circle The set of all points in a plane that are a given
additive inverses, or opposites, of each other if their distance (radius) from a given point (center)
sum is zero. Circle graph A graphic display of data using a divided
Algebraic expression A number or variable or a circle to show the percent of a quantity in each of
collection of numbers and variables on which several categories; also called pie chart
operations are performed Circumference The distance around a circle
Angle A set of points consisting of two rays (half-lines) Coefficient The numeric multiplier of a variable
with a common endpoint (vertex) Commission A percent of total sales paid to a
Area The number of square units that fill a plane region salesperson
Arithmetic mean A center point of a set of numbers Commutative law of addition The statement that when
found by adding the numbers and dividing by the two numbers are added, changing the order in which
number of items of data; also called the averageor the the numbers are added does not affect the sum
mean Commutative law of multiplication The statement
Arithmetic numbers The set of whole numbers and posi- that when two numbers are multiplied, changing the
tive fractions; also called nonnegative rational numbers order in which the numbers are multiplied does not
Ascending order When a polynomial is written with the affect the product
exponents of the variable increasing as read from left Complementary angles Two angles for which the sum
to right, it is said to be in ascending order. of their measures is 90°
Associative law of addition The statement that when Completing the square Adding a particular constant to
three numbers are added, regrouping the addends an expression so that the resulting sum is a perfect
gives the same sum square
Associative law of multiplication The statement that Complex fraction expression A rational expression that
when three numbers are multiplied, regrouping the has one or more rational expressions within its
factors gives the same product numerator and/or denominator
From the Glossary of Developmental Mathematics:College Mathematics and Introductory Algebra, 8/e. Marvin L. Bittinger.
Judith A. Beecher. Copyright © 2012 by Pearson Education. Published by Pearson Addison-Wesley. All rights reserved.
1
Glossary
Complex number Any number that can be named Diameter A line segment that passes through the
a + bi,where aand bare any real numbers center of a circle and has its endpoints on the circle
Complex-number system A number system that Difference The result of subtracting one number from
contains the real-number system and is designed so another
that negative numbers have defined square roots Difference of cubes An expression that can be written
Complex rational expression A rational expression that in the form A3 - B3
has one or more rational expressions within its Difference of squares An expression that can be written
numerator and/or denominator in the form A2 - B2
Composite number A natural number, other than 1, Digit A number 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 that names a
that is not prime place-value location
Compound interest Interest computed on the sum of Direct variation A situation that translates to an
an original principal and the interest previously equation of the form y = kx,with ka positive constant
accrued by that principal Discount The amount subtracted from the original
Congruent angles Two angles that have the same price of an item to find the sale price
measure Discriminant The radicand, b2 - 4ac,from the
Congruent segments Two line segments that have the quadratic formula
same length Distance formula The equation
Congruent triangles Triangles in which corresponding d = 2 x - x 2 + y - y 2that represents
1 2 12 1 2 12
angles and sides are congruent the distance between two points, 1x1, y12and 1x2, y22,
Conjugate of a complex number The conjugate of on the coordinate plane
a + biis a - bi, and the conjugate of a - biis a + bi. Distributive law of multiplication over addition The
Conjugates Pairs of radical terms, like 1a + 1band statement that multiplying a factor by the sum of two
1a - 1bor c + 1dand c - 1d,for which the numbers gives the same result as multiplying the
product does not have a radical term factor by each of the two numbers and then adding
Consecutive even integers Even integers that are two Distributive law of multiplication over subtraction The
units apart statement that multiplying a factor by the difference
Consecutive integers Integers that are one unit apart of two numbers gives the same result as multiplying
Consecutive odd integers Odd integers that are two the factor by each of the two numbers and then
units apart subtracting
Constant A number or letter that stands for just one Dividend In division, the number being divided
number Divisible The number bis said to be divisible by
Constant of proportionality The constant in an another number aif bis a multiple of a.
equation of direct or inverse variation Divisor In division, the number dividing another
Coordinates The numbers in an ordered pair number
Coplanar lines Lines in the same plane Domain The set of all first coordinates of the ordered
Cross products Given an equation with a single pairs in a function
fraction on each side, the products formed by
multiplying the left numerator and the right E
denominator, and the left denominator and the
Elimination method An algebraic method that uses the
right numerator
addition principle to solve a system of equations
Cube root The number cis called a cube root of a,
Empty set The set without members
written 23 a,if c3 = a. Equation A number sentence that says that the
expressions on either side of the equals sign, =,
represent the same number
Decimal notation A representation of a number Equation of direct variation An equation, described by
containing a decimal point y = kxwith ka positive constant, used to represent
Degree of a polynomial The degree of the term of direct variation
highest degree in a polynomial Equation of inverse variation An equation, described
Degree of a term The sum of the exponents of the k
by y = with ka positive constant, used to represent
variables x
Denominator The number below the fraction bar in a inverse variation
fraction Equiangular triangle A triangle in which all angles are
Descending order When a polynomial is written with congruent
the exponents of the variable decreasing as read from Equilateral triangle A triangle in which all sides are
left to right, it is said to be in descending order. the same length
Diagonal of a quadrilateral A line segment that joins Equivalent equations Equations with the same
two opposite vertices solutions
2
Glossary
Equivalent expressions Expressions that have the same I
value for all allowable (or meaningful) replacements
Identity property of 1 The statement that the product
Equivalent fractions Two different fractions that
of a number and 1 is always the original number
represent the same number
Identity property of 0 The statement that the sum of a
Equivalent inequalities Inequalities that have the same
number and 0 is always the original number
solution set
Imaginary number A number that can be named bi,
Evaluate To substitute a value for each occurrence of a
where bis some real number and b Z 0
variable in an expression Index In the expression 2k a, the number kis called the
Even root A root with an even index
index.
Event A set of outcomes
Inequality A mathematical sentence using 6, 7, …, Ú,
Exponent In expressions of the form an,the number n
or Z
is an exponent.
Input A member of the domain of a function
Exponential notation A representation of a number
Integers The whole numbers and their opposites;
using a base raised to a power
Á, -4, -3, -2, -1, 0, 1, 2, 3, 4,Á
Intercept The point at which a graph intersects the
F
x- or y-axis
Factor Verb: to write an equivalent expression that is a Interest A percentage of an amount invested or
product; Noun: a multiplier borrowed
Factoring Writing an expression as a product Interest rate The percent at which interest is calculated
Factorization A number expressed as a product of two on a principal
or more numbers Intersecting lines Lines that cross each other at a
Factorization of a polynomial An expression that common point
names the polynomial as a product Intersection of setsAandB The set of all elements that
Fahrenheit A temperature scale in which water freezes are common to both Aand B, denoted AxB

at 32°and water boils at 212° Inverse variation A situation that translates to an
FOIL To multiply two binomials by multiplying the k
First terms, the Outside terms, the Inside terms, and equation of the form y = ,with ka positive constant
x
then the Last terms
Irrational number A real number that cannot be
Formula An equation that uses numbers or letters to
named as a ratio of two integers
represent a relationship between two or more quantities
Isosceles triangle A triangle in which two or more sides
Fraction equation An equation containing one or more
are the same length
rational expressions; also called a rational equation
Fraction expression A quotient, or ratio, of two L
polynomials; also called a rational expression
Leading coefficient The coefficient of the term of
Fraction notation A number written using a numerator
highest degree in a polynomial
and a denominator
Leading term The term of highest degree in a
Function A correspondence between a first set, called
polynomial
the domain, and a second set, called the range, such
Least common denominator (LCD) The least
that each member of the domain corresponds to
common multiple of the denominators of two or
exactly onemember of the range
more fractions
G Least common multiple (LCM) The smallest number
that is a multiple of two or more numbers
Geometric figure A set of points
Legs In a right triangle, the two sides that form the right
Grade The measure of a road’s steepness
angle
Grade point average (GPA) The average of the grade
Like radicals Radicals that have the same index and the
point values for each credit hour taken
same radicand
Graph A picture or diagram of the data in a table; a line,
Like terms Terms that have exactly the same variable
curve, or collection of points that represents all the
factors
solutions of an equation
Line Two rays that have common points and continue
Greatest common factor (GCF) The common factor of
in opposite directions
a polynomial with the largest possible coefficient and
Line of symmetry A line that can be drawn through a
the largest possible exponent(s)
graph such that the part of the graph on one side of
the line is an exact reflection of the part on the
H
opposite side
Hypotenuse In a right triangle, the side opposite the Line graph A graph in which quantities are represented
right angle as points connected by straight-line segments
3
Glossary
Linear equation Any equation that can be written in Opposite The opposite, or additive inverse, of a
the form Ax + By = C,where xand yare variables number xis written -x.Opposites are the same
Linear function A function that can be described by an distance from 0 on the number line but on different
equation of the form y = mx + b,where xand yare sides of 0.
variables Opposite of a polynomial Two polynomials are
Linear inequality An inequality whose related equation opposites, or additive inverses, of each other if their
is a linear equation sum is zero.
Ordered pair A pair of numbers of the form a, b for
1 2
which the order in which the numbers are listed is
important
Marked price The original price of an item
Ordinate The second coordinate in an ordered pair of
Mean A center point of a set of numbers found by
numbers
dividing the sum of the numbers by the number of
items of data; also called the averageor the arithmetic Origin The point 10, 02on a graph where the two axes
intersect
mean
Original price The price of an item before a discount is
Median In a set of data listed in order from smallest to
deducted
largest, the middle number if there is an odd number
Outcome The result of an experiment
of data items, or the average of the two middle
Output A member of the range of a function
numbers if there is an even number of data items
Metric system A measurement system used in most
countries of the world, but very little in the United States P
x + x y + y
1 2 1 2
Midpoint formula The formula , . Palindrone prime A prime number that remains a
a 2 2 b
prime number when its digits are reversed
If the endpoints of a segment are 1x1, y12and 1x2, y22, Parabola The graph of a quadratic equation
then this represents the coordinates of the midpoint.
Parallel lines Lines in the same plane that never
Minuend The number from which another number is
intersect; two lines are parallel if they have the same
being subtracted
slope.
Mixed numeral A number represented by a whole
Parallelogram A four-sided polygon with two pairs of
number and a fraction less than 1
parallel sides
Mode The number or numbers that occur most often in
Percent notation A representation of a number as n
a set of data
parts per 100; n%
Monomial An expression of the type axn,where ais a
Perfect square A rational number pfor which there
real-number constant and nis a nonnegative integer
exists a number afor which a2 = p
Multiple of a number A product of the number and
Perfect-square trinomial A trinomial that is the square
some natural number
of a binomial
Multiplication property of 0 The statement that the
Perimeter The distance around an object or the sum of
product of 0 and any real number is 0
the lengths of its sides
Multiplicative identity The number 1
Periods Groups of three digits, separated by commas
Multiplicative inverses Reciprocals; two numbers
Perpendicular lines Two lines that intersect to form a
whose product is 1
right angle; two lines are perpendicular if the product
of their slopes is -1.
Pi P The number that results when the circumference
Natural numbers The counting numbers: 1, 2, 3, 4, 5,Á 1of a2 circle is divided by its diameter; p L 3.14,or 22
7
Negative integers Integers to the left of zero on the Pictograph A graphic means of displaying information
number line using symbols to represent the amounts
Nonnegative rational numbers The set of whole numbers Pie chart A graphic display of data using a divided
and positive fractions; also called arithmetic numbers circle to show the percent of a quantity in each of
Numerator The number above the fraction bar in a several categories; also called circle graph
fraction Point–slope equation The equation
y - y1 = m1x - x12,where x1, y1,and mare real
numbers and mis the slope and 1x1, y12is a point that
Obtuse angle An angle whose measure is greater than lies on the graph of the equation
90°and less than 180° Polygon A closed geometric figure with three or more
Obtuse triangle A triangle in which one angle is an line segments as sides
obtuse angle Polynomial A monomial or a combination of sums
Odd root A root with an odd index and/or differences of monomials
4
Glossary
Polynomial equation An equation in which two Radius A line segment with one endpoint on the center
polynomials are set equal to each other of a circle and the other endpoint on the circle
Positive integers Integers to the right of zero on the Range The set of all second coordinates of the ordered
number line pairs in a function
Prime factorization A factorization of a composite Rate A ratio used to compare two different kinds of
number as a product of prime numbers measure
Prime number A natural number that has exactly two Ratio The quotient of two quantities; the ratio of ato b
differentfactors: itself and 1 a
is ,also written a:b
Prime polynomial A polynomial that cannot be b
factored using only integer coefficients Rational equation An equation containing one or more
Principal An amount of money that is invested or rational expressions; also called a fraction equation
borrowed Rational expression A quotient, or ratio, of two
Principal square root The nonnegative square root of a polynomials; also called a fraction expression
number Rational numbers Any number that can be written as
Principle of zero products The statement that an a
equation ab = 0is true if and only if a = 0is true or the ratio of two integers b,where b Z 0
b = 0is true, or both are true Rationalizing the denominator A procedure for
Product The result when one number is multiplied by finding an equivalent expression without a radical in
another the denominator
Proportion An equation stating that two ratios are Ray A part of a line consisting of one endpoint and all
equal the points on the line on one side of the endpoint
Proportional numbers Two pairs of numbers having Real numbers All rational and irrational numbers; the
the same ratio set of all numbers corresponding to points on the
Protractor A device used to measure and draw angles number line
Purchase price The price of an item before sales tax is Reciprocal A multiplicative inverse; two numbers are
added reciprocals if their product is 1
Pythagorean equation The equation a2 + b2 = c2, Rectangle A four-sided polygon with four 90°angles
where aand bare lengths of the legs of a right triangle Relation A correspondence between a first set, called
and cis the length of the hypotenuse the domain, and a second set, called the range, such
Pythagorean theorem In any right triangle, if aand b that each member of the domain corresponds to at
are the lengths of the legs and cis the length of the least onemember of the range
hypotenuse, then a2 + b2 = c2. Repeating decimal A decimal in which a number
pattern repeats indefinitely
Q
Right angle An angle whose measure is 90°
Quadrants The four regions into which the axes divide Right triangle A triangle in which one angle is a right
a plane angle
Quadratic equation An equation of the form Rise The change in the second coordinate between two
ax2 + bx + c = 0,where a Z 0 points on a line
Quadratic formula The solutions of ax2 + bx + c = 0, Roster notation A way of naming sets by listing all the
a Z 0,are given by the equation elements in the set
-b ; 2b2 - 4ac Rounding Approximating the value of a number; used
x = . when estimating
2a
Run The change in the first coordinate between two
Quadratic function A second-degree polynomial
points on a line
function in one variable
Quadratic inequality An inequality whose related
S
equation is a quadratic equation
Quotient The result when one number is divided by Sale price The price of an item after a discount has
another been deducted
Sales tax A tax added to the purchase price of an item
R Sample space The set of all possible outcomes
Scalene triangle A triangle in which all sides are of
Radical equation An equation that has variables in one
different lengths
or more radicands
Scientific notation A number written in the form
Radical expression An algebraic expression in which a
M * 10n,where nis an integer, 1 … M 6 10,and
radical symbol appears
Radical symbol The symbol 1 Mis expressed in decimal notation
Segment A geometric figure consisting of two points,
Radicand The expression under the radical
called endpoints, and all points between them
5