The Humongous Book of Algebra Problems
iii
Contents
Introduction
Chapter 1: Arithmetic Fundamentals 1
Number Classification ................................................................................................ 2
Expressions Containing Signed Numbers ....................................................................... 5
Grouping Symbols ...................................................................................................... 8
Algebraic Properties .................................................................................................. 11
Chapter 2: Rational Numbers 17
Rational Number Notation ........................................................................................ 18
Simplifying Fractions ................................................................................................ 23
Combining Fractions ................................................................................................ 26
C
hapter 3: Basic Algebraic Expressions 37
Translating Expressions ............................................................................................ 38
Exponential Expressions ............................................................................................ 40
Distributive Property ................................................................................................. 45
Order of Operations .................................................................................................. 48
Evaluating Expressions ............................................................................................. 51
C
hapter 4: Linear Equations in One Variable 55
Adding and Subtracting to Solve an Equation ............................................................. 56
Multiplying and Dividing to Solve an Equation ........................................................... 59
Solving Equations Using Multiple Steps ...................................................................... 61
Absolute Value Equations .......................................................................................... 70
Equations Containing Multiple Variables .................................................................... 73
Your one-stop shop for a review of numbers
Numbers fall into different groups
Add, subtract, multiply, and divide positive and negative numbers
When numbers band together, deal with them rst
Basic assumptions about algebra
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Proper and improper fractions, decimals, and mixed numbers
Add, subtract, multiply, and divide fractions
Reducing fractions to lowest terms, like
1
/
2
instead of
5
/
10
Time for x to make its stunning debut
The alchemy of turning words into math
Rules for simplifying expressions that contain powers
Multiply one thing by a bunch of things in parentheses
My dear Aunt Sally is eternally excused
Replace variables with numbers
How to solve basic equations
Add to/subtract from both sides
Multiply/divide both sides
Nothing new here, just more steps
Most of them have two solutions
Equations with TWO variables (like x and y) or more
Table of Contents
The Humongous Book of Algebra Problems
iv
Chapter 5: Graphing Linear Equations in Two Variables 77
Number Lines and the Coordinate Plane ..................................................................... 78
Graphing with a Table of Values ................................................................................ 83
Graphing Using Intercepts ......................................................................................... 90
Calculating Slope of a Line ....................................................................................... 93
Graphing Absolute Value Equations ..........................................................................100
Chapter 6: Linear Equations in Two Variables 105
Point-Slope Form of a Linear Equation ......................................................................106
Slope-Intercept Form of a Linear Equation ..................................................................110
Graphing Lines in Slope-Intercept Form ......................................................................113
Standard Form of a Linear Equation .........................................................................118
Creating Linear Equations .......................................................................................121
C
hapter 7: Linear Inequalities 127
Inequalities in One Variable .....................................................................................128
Graphing Inequalities in One Variable .......................................................................132
Compound Inequalities ............................................................................................135
Absolute Value Inequalities .......................................................................................137
Set Notation ...........................................................................................................140
Graphing Inequalities in Two Variables .....................................................................142
C
hapter 8: Systems of Linear Equations and Inequalities 147
Graphing Linear Systems .........................................................................................148
The Substitution Method ..........................................................................................153
Variable Elimination ...............................................................................................162
Systems of Inequalities .............................................................................................168
Linear Programming ...............................................................................................173
C
hapter 9: Matrix Operations and Calculations 181
Anatomy of a Matrix ...............................................................................................182
Adding and Subtracting Matrices .............................................................................183
Multiplying Matrices ...............................................................................................188
Calculating Determinants ........................................................................................192
Cramer’s Rule .........................................................................................................200
Which should you use to graph?
Plug in some x’s, plot some points, call it a day
The easiest way to plot two points on a line quickly
Figure out how slanty a line is
Don’t miss the point in these graphs (Get it?)
P
oint + slope = equation
Lines that look like y = mx + b
Graphing equations that are solved for y
Write equations of lines in a uniform way
Practice all the skills from this chapter
Generating equations of lines
They’re like equations without the equal sign
Dust off your equation-solving skills from Chapter 4
Shoot arrows into number lines
Two inequalities for the price of one
Break these into two inequalities
A fancy way to write solutions
Lines that give off shade in the coordinate plane
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Graph two lines at once
Solve one equation for a variable and plug it into the other
Make one variable disappear and solve for the other one
The answer is where the shading overlaps
Use the sharp points at the edge of a shaded region
Numbers in rows and columns
The order of a matrix and identifying elements
Combine numbers in matching positions
Not as easy as adding or subtracting them
Values dened for square matrices only
Double-decker matrices that solve systems
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Table of Contents
The Humongous Book of Algebra Problems
v
Chapter 10: Applications of Matrix Algebra 207
Augmented and Identity Matrices ..............................................................................208
Matrix Row Operations ...........................................................................................211
Row and Reduced-Row Echelon Form ........................................................................216
Inverse Matrices ......................................................................................................228
Chapter 11: Polynomials 237
Classifying Polynomials ...........................................................................................238
Adding and Subtracting Polynomials ........................................................................239
Multiplying Polynomials ..........................................................................................244
Long Division of Polynomials ...................................................................................246
Synthetic Division of Polynomials ..............................................................................251
C
hapter 12: Factoring Polynomials 257
Greatest Common Factors .........................................................................................258
Factoring by Grouping .............................................................................................265
Common Factor Patterns ..........................................................................................267
Factoring Quadratic Trinomials ................................................................................270
C
hapter 13: Radical Expressions and Equations 275
Simplifying Radical Expressions ................................................................................276
Rational Exponents .................................................................................................281
Radical Operations .................................................................................................283
Solving Radical Equations .......................................................................................288
Complex Numbers ....................................................................................................290
C
hapter 14: Quadratic Equations and Inequalities 295
Solving Quadratics by Factoring ...............................................................................296
Completing the Square .............................................................................................300
Quadratic Formula .................................................................................................305
Applying the Discriminant .......................................................................................312
One-Variable Quadratic Inequalities ..........................................................................316
Extra columns and lots of 0s and 1s
Advanced matrix stuff
Swap rows, add rows, or multiply by a number
More matrices full of 0s with a diagonal of 1s
Matrices that cancel other matrices out
Clumps of numbers and variables raised to powers
Labeling them based on the exponent and total terms
Only works for like terms
FOIL and beyond
A lot like long dividing integers
Divide using only the coefcients
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Moving things out from under the radical
Fractional powers are radicals in disguise
Add, subtract, multiply, and divide roots
Use exponents to cancel out radicals
Numbers that contain i, which equals
√
—
–1
Solve equations containing x
2
Use techniques from Chapter 12 to solve equations
Make a trinomial into a perfect square
Use an equation’s coefcients to calculate the solution
What b
2
– 4ac tells you about an equation
Inequalities that contain x
2
Largest factor that divides into everything evenly
The opposite of multiplying polynomials
You can factor out binomials, too
Difference of perfect squares/cubes, sum of perfect cubes
Turn one trinomial into two binomials
Table of Contents
The Humongous Book of Algebra Problems
vi
Chapter 15: Functions 323
Relations and Functions ..........................................................................................324
Operations on Functions ..........................................................................................326
Composition of Functions .........................................................................................330
Inverse Functions ....................................................................................................335
Piecewise-Defined Functions ......................................................................................343
Chapter 16: Graphing Functions 347
Graphing with a Table of Values ...............................................................................348
Domain and Range of a Function .............................................................................354
Symmetry ...............................................................................................................360
Fundamental Function Graphs ..................................................................................365
Graphing Functions Using Transformations ...............................................................369
Absolute Value Functions ..........................................................................................374
C
hapter 17: Calculating Roots of Functions 379
Identifying Rational Roots .......................................................................................380
Leading Coefficient Test ...........................................................................................384
Descartes’ Rule of Signs ...........................................................................................388
Rational Root Test ..................................................................................................390
Synthesizing Root Identification Strategies ..................................................................394
C
hapter 18: Logarithmic Functions 399
Evaluating Logarithmic Expressions ..........................................................................400
Graphs of Logarithmic Functions ..............................................................................402
Common and Natural Logarithms ............................................................................406
Change of Base Formula ..........................................................................................409
Logarithmic Properties .............................................................................................412
C
hapter 19: Exponential Functions 417
Graphing Exponential Functions ..............................................................................418
Composing Exponential and Logarithmic Functions ....................................................423
Exponential and Logarithmic Equations ....................................................................426
Exponential Growth and Decay .................................................................................433
Named expressions that give one output per input
What makes a function a function?
+, –, ·, and ÷ functions
Plug one function into another
Functions that cancel each other out
Function rules that change based on the x-input
Drawing graphs that aren’t lines
Plug in a bunch of things for x
What can you plug in? What comes out?
Pieces of a graph are reections of each other
The graphs you need to understand most
Move, stretch, squish, and ip graphs
These graphs might have sharp points
R
oots = solutions = x-intercepts
Factoring polynomials given a head start
The ends of a function describe the ends of its graph
Sign changes help enumerate real roots
Find possible roots given nothing but a function
Factoring big polynomials from the ground up
Contains enough logs to build yourself a cabin
Given log
a
b = c, nd a, b, or c
All log functions have the same basic shape
What the bases equal when no bases are written
Calculate log values that have weird bases
Expanding, contracting, and simplifying log expressions
Functions with a variable in the exponent
G
raphs that start close to y = 0 and climb fast
They cancel each other out
Cancel logs with exponentials and vice versa
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kt
to measure things like population
Table of Contents
The Humongous Book of Algebra Problems
vii
Chapter 20: Rational Expressions 439
Simplifying Rational Expressions ..............................................................................440
Adding and Subtracting Rational Expressions ............................................................444
Multiplying and Dividing Rational Expressions ..........................................................452
Simplifying Complex Fractions ..................................................................................457
Graphing Rational Functions ...................................................................................459
Chapter 21: Rational Equations and Inequalities 465
Proportions and Cross Multiplication ........................................................................466
Solving Rational Equations .....................................................................................470
Direct and Indirect Variation ....................................................................................475
Solving Rational Inequalities ...................................................................................479
C
hapter 22: Conic Sections 487
Parabolas ..............................................................................................................488
Circles ...................................................................................................................494
Ellipses ..................................................................................................................499
Hyperbolas .............................................................................................................506
C
hapter 23: Word Problems 515
Determining Unknown Values ..................................................................................516
Calculating Interest .................................................................................................521
Geometric Formulas .................................................................................................525
Speed and Distance .................................................................................................529
Mixture and Combination .......................................................................................534
Work .....................................................................................................................538
Appendix A: Algebraic Properties 545
Appendix B: Important Graphs and Graph Transformations 547
Appendix C: Key Algebra Formulas 551
Index 555
Fractions with lots of variables in them
Reducing fractions by factoring
Use common denominators
Common denominators not necessary
Reduce fractions that contains fractions
Rational functions have asymptotes
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When two fractions are equal, “X” marks the solution
Ditch the fractions or cross multiply to solve
Turn a word problem into a rational equation
Critical numbers, test points, and shading
Parabolas, Circles, Ellipses, and Hyperbolas
Vertex, axis of symmetry, focus, and directrix
Center, radius, and diameter
Major and minor axes, center, foci, and eccentricity
Transverse and conjugate axes, foci, vertices, and asymptotes
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Integer and age problems
Simple, compound, and continuously compounding
Area, volume, perimeter, and so on
Distance equals rate times time
Measuring ingredients in a mixture
How much time does it save to work together?
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