The Humongous Book of Algebra Problems

578 Pages·2008·20.01 MB·English

by  W. Michael Kelley

A member of Penguin Group (USA) Inc.
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Contents
Introduction
Chapter 1: Arithmetic Fundamentals Your one-stop shop for a review of numbers 1
Numbers fall into different groups
Number Classification ................................................................................................2
Expressions Containing Signed Numbers .A.d..d.,. s.u.b..t.r.a.c..t,. .m.u..lt.i.p.ly.,. a..n.d.. d..iv.i.d.e.. p.o.s.i.t.iv..e. a..n.d.. n.e..g.a.t.i.v.e. .n.u.m..b.e..r.s...5
When numbers band together, deal with them first
Grouping Symbols ......................................................................................................8
Algebraic Properties ...................B..a..s..ic.. .a..s.s..u.m...p.t..io..n.s. .a..b..o..u.t.. .a..l.g.e..b..r.a.................................. 11
Chapter 2: Rational Numbers Understanding fractions sure beats being afraid of t h e m 17
Rational Number Notation ..P.r.o..p.e..r. .a..n..d.. .i.m..p.r.o..p.e..r. .f..r.a..c..t.i.o..n.s.,. .d..e..c..im...a..l.s., .a..n..d.. .m..i.x..e..d.. .n.u..m..b..e..r.s.. 18
Reducing fractions to lowest terms, like 1/ instead of 5/
Simplifying Fractions ..................................................................2...................1.0.......... 23
Combining Fractions ............A.d..d..., .s.u..b..t.r..a..c..t., .m...u.l.t..ip..ly.., .a..n..d.. .d..i.v.i.d..e.. .f..r.a..c..t.i.o..n.s...................... 26
Chapter 3: Basic Algebraic Expressions Time for x to make its stunning debut 37
The alchemy of turning words into math
Translating Expressions ............................................................................................ 38
Rules for simplifying expressions that contain powers
Exponential Expressions ............................................................................................ 40
Distributive Property ...........M..u..lt..i.p.l.y. .o..n.e.. .t.h..i.n..g. .b..y.. a... b..u..n..c.h.. .o..f. .t..h.i.n..g.s. .i.n.. .p.a..r.e..n.t..h..e..s.e..s........... 45
My dear Aunt Sally is eternally excused
Order of Operations .................................................................................................. 48
Evaluating Expressions ........................R..e..p.la...c.e.. .v..a..r.i.a..b.l.e..s. .w..i.t.h.. .n..u..m..b..e..r.s......................... 51
Chapter 4: Linear Equations in One Variable How to solve basic equations 55
Adding and Subtracting to Solve an Equation .........A.d...d.. .t.o./..s.u..b.t..r.a..c..t.. f..r.o..m.. .b..o.t..h.. .s.i.d..e.s....... 56
Multiplying and Dividing to Solve an Equation ............M..u..l.t.i.p.l.y./..d..iv..i.d..e.. .b.o..t.h.. .s.i.d..e..s............ 59
Solving Equations Using Multiple Steps ..............N.o..t.h..i.n.g.. .n.e..w.. .h..e..r.e..,. j.u..s.t.. .m..o.r..e.. s..t.e..p.s........... 61
Absolute Value Equations .......................M..o..s.t. .o..f. .t..h..e.m... .h.a...v.e.. .t.w..o.. .s.o.l.u..t.i.o..n.s....................... 70
Equations Containing Multiple Variables ..E..q.u..a..t.i.o.n.s. .w..i.t.h.. T..W..O.. .v.a..r.i.a..b.l.e..s. .(.li.k.e.. .x. .a..n.d.. .y.). .o.r. .m..o..r.e.. 73
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Chapter 5: Graphing Linear EquationsI dine nTtwifoy tVhaer ipaobinltess t hat make an equation true 77
Which should you use to graph?
Number Lines and the Coordinate Plane ..................................................................... 78
Plug in some x’s, plot some points, call it a day
Graphing with a Table of Values ................................................................................ 83
Graphing Using Intercepts ...........T..h..e.. .e..a..s.i.e..s.t. .w..a..y.. .t.o.. .p.l.o.t.. .t.w..o.. .p.o.i.n..t.s. .o..n. .a.. .l.i.n.e.. .q..u..ic..k..ly........ 90
Figure out how slanty a line is
Calculating Slope of a Line ....................................................................................... 93
Don’t miss the point in these graphs (Get it?)
Graphing Absolute Value Equations ..........................................................................100
Chapter 6: Linear Equations in Two Variables Generating equations of lines 105
Point-Slope Form of a Linear Equation .....................P..o.i.n.t.. .+.. .s.l.o.p.e.. .=.. .e..q.u..a..t..io..n.................106
Lines that look like y = mx + b
Slope-Intercept Form of a Linear Equation ..................................................................110
Graphing equations that are solved for y
Graphing Lines in Slope-Intercept Form ......................................................................113
Write equations of lines in a uniform way
Standard Form of a Linear Equation .........................................................................118
Creating Linear Equations ..................P..r.a..c..t.i.c..e.. .a..l.l .t..h..e. .s..k.i.l.ls.. .f.r.o..m.. .t..h.i.s. .c..h..a..p.t..e..r.............121
Chapter 7: Linear Inequalities They’re like equations without the equal sign 127
Dust off your equation-solving skills from Chapter 4
Inequalities in One Variable .....................................................................................128
Graphing Inequalities in One Variable .................S..h..o.o.t.. .a..r.r.o..w..s. .i.n.t..o. .n..u.m...b..e.r.. .li.n..e.s.............132
Compound Inequalities ........................T..w..o. .i.n.e..q..u..a..l.it..i.e.s.. .f.o..r. .t.h..e.. .p.r.i.c..e.. .o.f.. .o.n.e...................135
Break these into two inequalities
Absolute Value Inequalities .......................................................................................137
Set Notation ..................................A. .f..a..n.c..y.. .w..a..y.. .t.o.. w...r.i.t.e.. .s.o.l.u..t.i.o..n.s................................140
Graphing Inequalities in Two Variables .L..i.n.e..s. .t.h..a..t. .g.i.v.e.. .o.f..f. .s.h..a..d..e. .i.n. .t..h.e.. .c.o..o.r.d..i.n.a..t.e.. .p.l.a..n.e....142
Chapter 8: Systems of Linear Equations and InWeoqruk awlitithi emso r e than one equation at a time 147
Graph two lines at once
Graphing Linear Systems .........................................................................................148
Solve one equation for a variable and plug it into the other
The Substitution Method ..........................................................................................153
Variable Elimination ..........M..a..k..e.. o..n.e.. .v..a..r.i.a..b..le.. .d..i.s.a..p..p.e..a..r. .a..n.d... .s.o.l.v.e.. .f..o.r.. t..h..e.. .o.t.h..e..r. .o..n.e......162
The answer is where the shading overlaps
Systems of Inequalities .............................................................................................168
Linear Programming .............U.s..e. .t..h..e. .s..h.a...r.p. .p.o..in..t.s.. .a..t. .t..h.e.. .e..d..g..e. .o..f. .a.. .s..h.a...d..e..d.. .r.e..g.i.o.n.........173
Chapter 9: Matrix Operations and Calculations Numbers in rows and columns 181
The order of a matrix and identifying elements
Anatomy of a Matrix ...............................................................................................182
Combine numbers in matching positions
Adding and Subtracting Matrices .............................................................................183
Not as easy as adding or subtracting them
Multiplying Matrices ...............................................................................................188
Values defined for square matrices only
Calculating Determinants ........................................................................................192
Cramer’s Rule ......................D..o.u..b..l.e.-.d..e..c..k..e.r.. .m..a..t..r.ic..e..s. .t..h.a...t. .s.o..lv..e.. .s.y..s.t.e..m..s.......................200
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Chapter 10: Applications of Matrix Algebra Advanced matrix stuff 207
Augmented and Identity Matrices .............E..x..t.r.a... .c.o..lu..m...n.s. .a..n..d.. .l.o.t..s. .o.f.. .0..s. .a..n.d... .1.s...............208
Swap rows, add rows, or multiply by a number
Matrix Row Operations ...........................................................................................211
Row and Reduced-Row Echelon Form ..M..o..r.e.. .m..a..t..r.ic..e..s. .f..u..ll. .o.f.. .0..s. .w..i.t.h.. .a.. .d..i.a..g.o..n.a..l. .o..f. .1..s......216
Inverse Matrices ..................M...a..t.r.i.c..e..s. .t.h..a..t.. .c.a..n..c..e..l .o..t.h..e..r. .m..a..t..r.i.c..e.s.. .o.u..t..........................228
Chapter 11: Polynomials Clumps of numbers and variables raised to powers 237
Labeling them based on the exponent and total terms
Classifying Polynomials ...........................................................................................238
Only works for like terms
Adding and Subtracting Polynomials ........................................................................239
FOIL and beyond
Multiplying Polynomials ..........................................................................................244
A lot like long dividing integers
Long Division of Polynomials ...................................................................................246
Divide using only the coefficients
Synthetic Division of Polynomials ..............................................................................251
Chapter 12: Factoring Polynomials The opposite of multiplying polynomials 257
Greatest Common Factors ..........L.a..r.g..e..s.t. .f..a..c..t.o..r. .t.h..a..t.. .d..i.v.i.d..e..s. .i.n.t..o. .e..v.e..r.y..t..h.i.n..g. .e..v..e.n..l.y........258
Factoring by Grouping ..........................Y.o..u.. c..a..n.. .f.a..c..t..o.r. .o..u..t. .b..i.n.o.m...i.a..ls.., .t..o.o......................265
Common Factor Patterns ..D..i.f.f..e.r..e.n..c..e.. .o.f.. p..e..r.f.e..c..t. .s..q.u..a..r.e..s./.c..u..b..e..s., .s..u.m... .o.f.. .p.e..r.f.e..c..t.. .c.u..b..e..s....267
Factoring Quadratic Trinomials ...............T..u..r.n. .o..n.e.. .t..r.i.n.o..m..i.a..l. i.n..t.o.. t..w..o.. b..i.n.o..m..i.a..l.s...............270
Chapter 13: Radical Expressions and SEqquuaartei oronost s , cube roots, and fractional exponents 275
Moving things out from under the radical
Simplifying Radical Expressions ................................................................................276
Fractional powers are radicals in disguise
Rational Exponents .................................................................................................281
Add, subtract, multiply, and divide roots
Radical Operations .................................................................................................283
Solving Radical Equations .....................U..s.e.. .e..x.p.o..n.e..n..t.s. .t..o. .c..a..n..c.e..l. .o.u..t. .r.a...d..ic..a..l.s...............288
—
Complex Numbers .........................N..u..m..b..e..r.s. .t..h.a..t.. .c..o.n..t.a..i.n. .i.,. .w..h.i.c..h.. .e.q..u..a..l.s. .√..–..1.................290
Chapter 14: Quadratic Equations and Inequalities Solve equations containing x2 295
Use techniques from Chapter 12 to solve equations
Solving Quadratics by Factoring ...............................................................................296
Make a trinomial into a perfect square
Completing the Square .............................................................................................300
Quadratic Formula ......U..s.e.. .a..n. .e..q..u..a..t.i.o.n..’s. .c..o..e.f..fi..c..i.e.n..t.s.. .t.o. .c..a..l.c..u.l.a..t..e.. t..h..e.. .s.o.l.u..t.i.o.n..............305
Applying the Discriminant .............W...h..a..t. .b..2. .–. .4..a..c.. .t.e..l.ls. .y..o.u.. .a..b..o..u.t.. .a..n. .e..q..u..a..t.i.o.n...............312
One-Variable Quadratic Inequalities ...................I.n..e..q.u..a..l.i.t.i.e..s. .t.h..a..t.. .c.o..n.t..a..i.n. .x..2................316
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Chapter 15: Functions Named expressions that give one output per input 323
Relations and Functions ........................W..h..a..t.. m...a..k..e.s.. .a.. .f.u..n.c..t..i.o.n.. .a.. .f.u..n.c..t..io..n.?.................324
+, –, ·, and ÷ functions
Operations on Functions ..........................................................................................326
Composition of Functions .........................P.l.u..g. .o..n.e.. .f.u..n..c..t.i.o.n.. .i.n.t.o.. .a..n.o..t.h..e..r.......................330
Inverse Functions .........................F.u..n..c..t.i.o.n..s. .t.h..a..t.. .c.a...n.c..e..l .e..a..c..h.. .o.t..h.e..r.. .o.u..t......................335
Piecewise-Defined Functions .......F..u.n..c..t.i.o.n.. .r.u..l.e.s.. .t.h..a..t. .c..h..a..n..g.e.. .b..a..s.e..d.. .o.n.. .t.h..e.. .x..-.in..p.u..t...........343
Chapter 16: Graphing Functions Drawing graphs that aren’t lines 347
Graphing with a Table of Values ......................P..l.u.g.. .i.n. .a.. .b..u..n.c..h.. .o.f.. .t.h..i.n.g..s. .f..o.r. .x................348
Domain and Range of a Function ..........W...h..a..t. .c..a..n. .y..o.u.. .p.l.u..g. .i.n..?. .W..h..a..t. .c..o..m..e..s. .o.u..t..?............354
Symmetry ...................P..ie..c..e..s. .o.f.. .a.. .g.r..a..p.h.. .a..r.e.. .r.e..fl..e..c..t.i.o.n..s. .o.f.. .e..a..c..h.. o..t.h..e..r.........................360
Fundamental Function Graphs ..............T..h..e.. .g.r.a..p..h..s. .y.o..u.. n..e..e..d.. .t.o.. u..n..d..e..r.s.t..a..n.d.. .m...o.s.t...........365
Graphing Functions Using Transformations ....M..o.v..e.,. .s.t..r.e..t.c..h.., .s.q..u..i.s.h.., .a..n..d.. .fl..i.p. .g..r.a..p.h..s.........369
Absolute Value Functions ..................T..h..e..s.e.. .g.r.a...p.h..s. .m..i.g.h..t.. .h.a...v.e.. .s.h..a..r.p. .p..o.i.n.t..s....................374
Chapter 17: Calculating Roots of Functions Roots = solutions = x-intercepts 379
Identifying Rational Roots ...........F.a..c..t..o.r.i.n..g. .p.o..l.y.n..o.m...ia...ls. .g..i.v.e..n.. .a.. .h.e..a..d.. .s..t.a..r.t....................380
The ends of a function describe the ends of its graph
Leading Coefficient Test ...........................................................................................384
Descartes’ Rule of Signs ...............S..ig..n.. c..h..a..n..g.e..s. .h..e..l.p. .e..n.u..m..e..r.a..t..e.. .r.e..a..l. r..o.o.t..s.....................388
Rational Root Test ..............F.i.n..d.. .p.o.s..s.i.b.l.e.. .r.o..o.t..s. .g.i.v.e..n.. .n.o..t.h..i.n.g.. .b..u.t.. .a.. .f.u..n..c..t.i.o.n....................390
Synthesizing Root Identification Strategies .F.a..c..t..o.r.i.n..g. .b..i.g. .p.o..ly..n..o.m..i.a..l.s. .f..r.o.m... .t.h..e.. .g.r..o.u..n.d... u..p....394
Chapter 18: Logarithmic Functions Contains enough logs to build yourself a cabin 399
Given log b = c, find a, b, or c
Evaluating Logarithmic Expressions ...............................a...........................................400
Graphs of Logarithmic Functions ......A..l.l. .lo..g. .f..u.n..c..t.i.o..n.s. .h..a..v..e. .t..h..e.. s..a..m..e.. .b..a..s.i.c.. .s.h..a..p.e...........402
What the bases equal when no bases are written
Common and Natural Logarithms ............................................................................406
Change of Base Formula ............C..a..lc..u..l.a..t.e.. .l.o.g.. .v.a..l.u..e.s.. .t.h..a..t. .h..a..v..e.. .w..e.i.r.d.. .b..a..s..e.s.................409
Logarithmic Properties .....E..x..p.a..n..d..i.n.g.., .c..o.n..t.r..a..c..t.i.n.g.., .a..n..d.. .s.i.m...p.l.if..y.i.n..g. .l.o.g.. .e..x.p..r.e..s.s.i.o.n..s.............412
Chapter 19: Exponential Functions Functions with a variable in the exponent 417
Graphs that start close to y = 0 and climb fast
Graphing Exponential Functions ..............................................................................418
Composing Exponential and Logarithmic Functions .T..h..e..y. .c..a..n..c.e..l. .e..a..c..h. .o..t.h..e..r. .o.u..t............423
Exponential and Logarithmic Equations .C..a..n..c.e..l. .l.o.g..s. .w..i.t.h.. .e..x.p..o.n.e..n..t.i.a..l.s. .a..n..d.. .v.i.c..e.. .v.e..r.s.a......426
Exponential Growth and Decay ....U..s.e.. .f.(.t.). .=.. .N..e..k.t .t..o. .m..e..a..s..u.r..e. .t..h..in..g.s.. .li.k..e.. .p.o.p.u..l.a..t..io..n............433
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Chapter 20: Rational Expressions Fractions with lots of variables in them 439
Reducing fractions by factoring
Simplifying Rational Expressions ..............................................................................440
Adding and Subtracting Rational Expressions ...U..s.e.. .c.o..m..m...o.n.. d...e.n..o.m...i.n.a..t..o.r.s......................444
Multiplying and Dividing Rational Expressions .C.o..m..m...o.n.. .d..e..n.o..m..i.n.a..t..o.r..s. .n.o..t. .n..e.c..e..s.s.a..r..y.......452
Simplifying Complex Fractions .........R..e..d..u.c..e.. .f..r.a..c..t.i.o.n..s. .t..h.a..t.. .c..o.n..t.a..i.n.s.. .f.r.a..c..t..io..n..s..............457
Graphing Rational Functions ............R..a..t..io..n.a..l. .f..u.n..c..t.i.o.n..s. .h..a..v.e.. .a..s..y.m...p.t.o..t.e..s.....................459
Chapter 21: Rational Equations and Inequalities Solve equations using the skills from C h ap t e4r6 250
Proportions and Cross Multiplication W..h..e.n.. .t.w..o. .f.r.a..c..t.i.o.n.s.. a..r.e.. .e.q..u.a..l., .“.X..”. .m..a..r.k.s.. t..h.e.. .s.o.l.u.t..io.n.......466
Ditch the fractions or cross multiply to solve
Solving Rational Equations .....................................................................................470
Direct and Indirect Variation .......T..u..r.n.. .a.. .w..o..r.d.. .p.r.o..b.l.e..m.. .i.n..t.o. .a.. .r..a..t.i.o.n..a..l. .e.q..u..a..t..io..n..............475
Critical numbers, test points, and shading
Solving Rational Inequalities ...................................................................................479
Chapter 22: Conic Sections Parabolas, Circles, Ellipses, and Hyperbolas 487
Parabolas ................V..e..r.t.e..x.., .a...x.i.s. .o.f.. .s.y..m..m...e..t.r.y.., .f..o.c..u..s.,. .a..n.d.. .d...ir.e..c..t..r.i.x.............................488
Center, radius, and diameter
Circles ...................................................................................................................494
Major and minor axes, center, foci, and eccentricity
Ellipses ..................................................................................................................499
Transverse and conjugate axes, foci, vertices, and asymptotes
Hyperbolas .............................................................................................................506
Chapter 23: Word Problems I f two trains leave the station full of consecutive integers, how much inte r5e1st5 is earned?
Determining Unknown Values ................In..t.e..g..e.r.. .a..n.d... a...g.e.. .p.r.o..b..le..m..s................................516
Simple, compound, and continuously compounding
Calculating Interest .................................................................................................521
Area, volume, perimeter, and so on
Geometric Formulas .................................................................................................525
Distance equals rate times time
Speed and Distance .................................................................................................529
Mixture and Combination ............M...e.a..s..u.r..in..g. .i.n..g.r.e..d..i.e..n.t..s. .i.n. .a.. .m...i.x.t..u..r.e...........................534
Work ......................H.o..w.. .m..u..c..h.. .t.i.m..e.. .d..o..e.s.. .it.. .s.a..v.e.. .t..o. .w..o..r.k.. t..o.g..e..t.h..e..r.?...............................538
Appendix A: Algebraic Properties 545
Appendix B: Important Graphs and Graph Transformations 547
Appendix C: Key Algebra Formulas 551
Index 555
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Introduction
Are you in an algebra class? Yes? Then you NEED this book. Here’s why:
Fact #1: The best way to learn algebra is by working out algebra problems.
There’s no denying it. If you could figure this class out just by reading the
textbook or taking good notes in class, everybody would pass with flying colors.
Unfortunately, the harsh truth is that you have to buckle down and work
problems out until your fingers are numb.
Fact #2: Most textbooks only tell you WHAT the answers to their practice
problems are, but not HOW to do them! Sure, your textbook may have 175
problems for every topic, but most of them only give you the answers. That
means if you don’t get the answer right you’re totally out of luck! Knowing you’re
wrong is no help at all if you don’t know why you’re wrong. Math textbooks sit on a
huge throne like the Great and Terrible Oz and say, “Nope, try again,” and we
do. Over and over. And we keep getting the problem wrong. What a delightful
way to learn! (Let’s not even get into why they only tell you the answers to the
odd problems. Does that mean the book’s actual author didn’t even feel like
working out the even ones?)
Fact #3: Even when math books try to show you the steps for a problem, they
do a lousy job. Math people love to skip steps. You’ll be following along fine with
an explanation and then all of a sudden BAM, you’re lost. You’ll think to yourself,
“How did they do that?” or “Where the heck did that 42 come from? It wasn’t
there in the last step!” Why do almost all of these books assume that in order to
work out a problem on page 200, you’d better know pages 1 through 199 like the
back of your hand? You don’t want to spend the rest of your life on homework!
You just want to know why you keep getting a negative number when you’re
calculating the minimum cost of building a pool whose length is four times the
sum of its depth plus the rate at which the water is leaking out of a train that
left Chicago at 4:00 a.m. traveling due west at the same speed carbon decays.
The Humongous Book of Algebra Problems
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