Intermediate Algebra

Cover......Page 1
Title Page......Page 9
Copyright......Page 10
ABOUT THE AUTHORS......Page 12
CONTENTS......Page 13
PREFACE......Page 17
INDEX OF APPLICATIONS......Page 27
CHAPTER 1 A Review of Basic Algebra......Page 33
1 Write Verbal and Mathematical Models......Page 34
2 Use Equations to Construct Tables of Data......Page 38
1 Define the Set of Natural Numbers, Whole Numbers, and Integers......Page 41
2 Define the Set of Rational Numbers......Page 43
3 Define the Set of Irrational Numbers......Page 44
4 Classify Real Numbers......Page 45
5 Graph Real Numbers......Page 46
6 Order the Real Numbers......Page 47
7 Find the Opposite and the Absolute Value of a Real Number......Page 48
SECTION 1.2 STUDY SET......Page 49
1 Add and Subtract Real Numbers......Page 52
2 Multiply and Divide Real Numbers......Page 54
3 Find Powers and Square Roots of Real Numbers......Page 56
4 Use the Order of Operations Rule......Page 58
5 Evaluate Algebraic Expressions......Page 60
SECTION 1.3 STUDY SET......Page 62
1.4 Simplifying Algebraic Expressions Using Properties of Real Numbers......Page 64
2 Identify and Use Properties of Real Numbers......Page 65
4 Use the Distributive Property......Page 68
5 Combine Like Terms......Page 70
SECTION 1.4 STUDY SET......Page 72
1 Determine Whether a Number Is a Solution......Page 75
2 Use Properties of Equality to Solve Equations......Page 76
3 Simplify Expressions to Solve Equations......Page 79
4 Clear Equations of Fractions and Decimals......Page 81
5 Identify Identities and Contradictions......Page 83
SECTION 1.5 STUDY SET......Page 84
1 Use Formulas from Geometry......Page 87
2 Solve for a Specified Variable......Page 90
3 Solve Application Problems Using Formulas......Page 92
SECTION 1.6 STUDY SET......Page 93
1 Apply the Steps of a Problem-Solving Strategy......Page 97
2 Solve Number-Value Problems......Page 100
3 Solve Geometry Problems......Page 101
4 Use Formulas to Solve Problems......Page 102
SECTION 1.7 STUDY SET......Page 103
1 Solve Percent Problems......Page 107
2 Solve Investment Problems......Page 110
3 Solve Uniform Motion Problems......Page 111
4 Solve Mixture Problems......Page 112
SECTION 1.8 STUDY SET......Page 115
CHAPTER SUMMARY AND REVIEW......Page 120
CHAPTER TEST......Page 130
GROUP PROJECT......Page 132
CHAPTER 2 Graphs, Equations of Lines, and Functions......Page 133
1 Plot Ordered Pairs and Determine the Coordinates of a Point......Page 134
3 Read Graphs......Page 136
4 Find the Midpoint of a Line Segment......Page 138
SECTION 2.1 STUDY SET......Page 140
1 Determine Whether an Ordered Pair Is a Solution of an Equation......Page 145
3 Graph Linear Equations by Plotting Points......Page 146
4 Graph Linear Equations by Finding Intercepts......Page 148
5 Graph Horizontal and Vertical Lines......Page 150
6 Use Linear Models to Solve Applied Problems......Page 151
SECTION 2.2 STUDY SET......Page 154
1 Calculate an Average Rate of Change......Page 158
2 Find the Slope of a Line from Its Graph......Page 159
3 Find the Slope of a Line Given Two Points......Page 160
4 Find the Slope of Horizontal and Vertical Lines......Page 161
5 Solve Applications of Slope......Page 162
6 Determine Whether Lines Are Parallel or Perpendicular Using Slope......Page 163
SECTION 2.3 STUDY SET......Page 165
1 Use the Slope–Intercept Form to Write the Equation of a Line......Page 170
2 Write a linear equation model......Page 172
3 Use the Point–Slope Form to Write the Equation of a Line......Page 173
4 Write a linear depreciation equation......Page 174
5 Use Slope as an Aid When Graphing......Page 175
6 Recognize Parallel and Perpendicular Lines......Page 176
SECTION 2.4 STUDY SET......Page 178
1 Define Relation, Domain, and Range......Page 184
2 Identify Functions......Page 185
3 Use Function Notation......Page 187
4 Find the Domain of a Function......Page 189
5 Graph Linear Functions......Page 190
6 Write Equations of Linear Functions......Page 191
SECTION 2.5 STUDY SET......Page 194
1 Find Function Values Graphically......Page 198
2 Find the Domain and Range of a Function Graphically......Page 199
3 Graph Nonlinear Functions......Page 200
4 Translate Graphs of Functions......Page 203
5 Reflect Graphs of Functions......Page 205
6 Use the Vertical Line Test......Page 206
SECTION 2.6 STUDY SET......Page 207
CHAPTER SUMMARY AND REVIEW......Page 211
CHAPTER TEST......Page 220
CUMULATIVE REVIEW......Page 222
CHAPTER 3 Systems of Equations......Page 225
3.1 Solving Systems of Equations by Graphing......Page 226
1 Determine Whether an Ordered Pair Is a Solution of a System......Page 227
2 Solve Systems of Linear Equations by Graphing......Page 228
3 Use Graphing to Identify Inconsistent Systems and Dependent Equations......Page 229
4 Solve Equations Graphically......Page 233
SECTION 3.1 STUDY SET......Page 234
3.2 Solving Systems of Equations Algebraically......Page 237
1 Solve Systems of Linear Equations by Substitution......Page 238
2 Solve Systems of Linear Equations by the Elimination (Addition) Method......Page 240
3 Use Substitution and Elimination (Addition) to Identify Inconsistent Systems and Dependent Equations......Page 244
4 Determine the Most Efficient Method to Use to Solve a Linear System......Page 245
SECTION 3.2 STUDY SET......Page 246
1 Determine Whether an Ordered Triple Is a Solution of a System......Page 249
2 Solve Systems of Three Linear Equations in Three Variables......Page 250
3 Solve Systems of Equations with Missing Variable Terms......Page 252
4 Identify Inconsistent Systems and Dependent Equations......Page 255
SECTION 3.3 STUDY SET......Page 257
2 Write the Augmented Matrix for a System......Page 260
3 Perform Elementary Row Operations on Matrices......Page 261
4 Use Matrices to Solve a System of Two Equations......Page 263
5 Use Matrices to Solve a System of Three Equations......Page 264
6 Use Matrices to Identify Inconsistent Systems and Dependent Equations......Page 266
SECTION 3.4 STUDY SET......Page 268
3.5 Solving Systems of Equations Using Determinants......Page 270
1 Evaluate 2 x 2 and 3 x 3 Determinants......Page 271
2 Use Cramer’s Rule to Solve Systems of Two Equations......Page 274
3 Use Cramer’s Rule to Solve Systems of Three Equations......Page 276
SECTION 3.5 STUDY SET......Page 278
1 Assign Variables to Two Unknowns......Page 281
2 Use Systems to Solve Geometry Problems......Page 282
3 Use Systems to Solve Number-Value Problems......Page 283
4 Use Systems to Find the Break Point......Page 285
5 Use Systems to Solve Interest, Uniform Motion, and Mixture Problems......Page 286
SECTION 3.6 STUDY SET......Page 291
1 Assign Variables to Three Unknowns......Page 297
2 Use Systems to Solve Curve-Fitting Problems......Page 301
SECTION 3.7 STUDY SET......Page 302
CHAPTER SUMMARY AND REVIEW......Page 305
CHAPTER TEST......Page 314
CUMULATIVE REVIEW......Page 316
CHAPTER 4 Inequalities......Page 319
4.1 Solving Linear Inequalities in One Variable......Page 320
2 Graph Intervals and Use Interval and Set-Builder Notation......Page 321
3 Solve Linear Inequalities Using Properties of Inequality......Page 323
4 Use Linear Inequalities to Solve Problems......Page 328
SECTION 4.1 STUDY SET......Page 330
4.2 Solving Compound Inequalities......Page 334
2 Solve Compound Inequalities Containing the Word And......Page 335
3 Solve Double Linear Inequalities......Page 338
4 Solve Compound Inequalities Containing the Word Or......Page 340
SECTION 4.2 STUDY SET......Page 342
1 Solve Equations of the Form......Page 346
2 Solve Equations with Two Absolute Values......Page 349
3 Solve Inequalities of the Form [X] 4 Solve Inequalities of the Form [X] =k......Page 352
SECTION 4.3 STUDY SET......Page 355
1 Graph Linear Inequalities......Page 358
2 Graph Inequalities with a Boundary through the Origin......Page 361
4 Solve Applied Problems Involving Linear Inequalities in Two Variables......Page 362
SECTION 4.4 STUDY SET......Page 364
1 Solve Systems of Linear Inequalities......Page 367
2 Graph Compound Inequalities......Page 370
3 Solve Problems Involving Systems of Linear Inequalities......Page 371
SECTION 4.5 STUDY SET......Page 372
CHAPTER SUMMARY AND REVIEW......Page 375
CHAPTER TEST......Page 382
CUMULATIVE REVIEW......Page 384
CHAPTER 5 Exponents, Polynomials, and Polynomial Functions......Page 387
1 Identify Bases and Exponents......Page 388
2 Use the Product and Power Rules for Exponents......Page 389
3 Use the Zero and Negative-Integer Exponent Rules......Page 392
4 Use the Quotient Rule for Exponents......Page 395
5 Simplify Quotients Raised to Negative Powers......Page 396
SECTION 5.1 STUDY SET......Page 398
5.2 Scientific Notation......Page 400
1 Write Numbers in Scientific Notation......Page 401
2 Convert from Scientific Notation to Standard Notation......Page 402
3 Perform Calculations with Scientific Notation......Page 403
SECTION 5.2 STUDY SET......Page 406
5.3 Polynomials and Polynomial Functions......Page 408
1 Define and Classify Polynomials......Page 409
2 Evaluate Polynomial Functions......Page 411
3 Find Function Values and the Domain and Range of Polynomial Functions Graphically......Page 412
4 Simplify Polynomials by Combining Like Terms......Page 413
5 Add Polynomials......Page 414
6 Subtract Polynomials......Page 415
SECTION 5.3 STUDY SET......Page 417
2 Multiply a Polynomial by a Monomial......Page 422
3 Multiply Two Binomials......Page 423
4 Multiply Any Two Polynomials......Page 425
5 Multiply Three Polynomials......Page 426
6 Find Special Products......Page 427
SECTION 5.4 STUDY SET......Page 430
5.5 The Greatest Common Factor and Factoring by Grouping......Page 433
2 Factor Out the Greatest Common Factor......Page 434
3 Factor by Grouping......Page 437
4 Use Factoring to Solve Formulas for a Specified Variable......Page 440
SECTION 5.5 STUDY SET......Page 441
5.6 Factoring Trinomials......Page 443
2 Factor Trinomials of the Form x[Sup(2)]+bx+c......Page 444
3 Factor Trinomials of the Form x[Sup(2)]+bx+c......Page 447
5 Use the Grouping Method to Factor Trinomials......Page 451
SECTION 5.6 STUDY SET......Page 454
1 Factor the Difference of Two Squares......Page 457
2 Factor the Sum and Difference of Two Cubes......Page 460
SECTION 5.7 STUDY SET......Page 463
5.8 Summary of Factoring Techniques......Page 465
1 Factor Random Polynomials......Page 466
SECTION 5.8 STUDY SET......Page 469
1 Solve Quadratic Equations Using the Zero-Factor Property......Page 470
2 Solve Higher-Degree Polynomial Equations by Factoring......Page 475
3 Use Quadratic Equations to Solve Problems......Page 476
SECTION 5.9 STUDY SET......Page 479
CHAPTER SUMMARY AND REVIEW......Page 484
CHAPTER TEST......Page 493
GROUP PROJECT......Page 495
CUMULATIVE REVIEW......Page 496
CHAPTER 6 Rational Expressions and Equations......Page 499
1 Define Rational Expressions and Rational Functions......Page 500
2 Evaluate Rational Functions......Page 501
4 Recognize the Graphs of Rational Functions......Page 502
5 Simplify Rational Expressions......Page 503
6 Simplify Rational Expressions That Have Factors That Are Opposites......Page 508
SECTION 6.1 STUDY SET......Page 509
6.2 Multiplying and Dividing Rational Expressions......Page 512
1 Multiply Rational Expressions......Page 513
2 Find Powers of Rational Expressions......Page 516
3 Divide Rational Expressions......Page 517
4 Perform Mixed Operations......Page 519
SECTION 6.2 STUDY SET......Page 520
1 Add and Subtract Rational Expressions with Like Denominators......Page 523
2 Find the Least Common Denominator......Page 524
3 Add and Subtract Rational Expressions with Unlike Denominators......Page 525
4 Add and Subtract Rational Expressions That Have Denominators That Are Opposites......Page 528
5 Perform Mixed Operations......Page 529
SECTION 6.3 STUDY SET......Page 530
1 Simplify Complex Fractions Using Division......Page 534
2 Simplify Complex Fractions Using the LCD......Page 535
SECTION 6.4 STUDY SET......Page 541
1 Divide a Monomial by a Monomial......Page 544
2 Divide a Polynomial by a Monomial......Page 545
3 Divide a Polynomial by a Polynomial......Page 546
4 Divide Polynomials with Missing Terms......Page 549
SECTION 6.5 STUDY SET......Page 550
1 Perform Synthetic Division......Page 553
2 Use the Remainder Theorem to Evaluate Polynomials......Page 556
3 Use the Factor Theorem to Factor Polynomials......Page 557
SECTION 6.6 STUDY SET......Page 558
6.7 Solving Rational Equations......Page 560
1 Solve Rational Equations......Page 561
2 Solve Rational Equations with Extraneous Solutions......Page 565
3 Solve Formulas for a Specified Variable......Page 566
SECTION 6.7 STUDY SET......Page 567
1 Solve Shared-Work Problems......Page 571
2 Solve Uniform-Motion Problems......Page 574
SECTION 6.8 STUDY SET......Page 577
1 Identify Ratios, Rates, and Proportions......Page 580
2 Solve Proportions......Page 581
3 Use Proportions to Solve Problems......Page 582
4 Solve Problems Involving Similar Triangles......Page 583
5 Solve Problems Involving Direct Variation......Page 585
6 Solve Problems Involving Inverse Variation......Page 586
7 Solve Problems Involving Joint Variation......Page 588
8 Solve Problems Involving Combined Variation......Page 589
SECTION 6.9 STUDY SET......Page 590
CHAPTER SUMMARY AND REVIEW......Page 594
CHAPTER TEST......Page 605
CUMULATIVE REVIEW......Page 607
CHAPTER 7 Radical Expressions and Equations......Page 611
1 Find Square Roots......Page 612
2 Find Square Roots of Expressions Containing Variables......Page 614
3 Graph the Square Root Function......Page 615
4 Evaluate Radical Functions......Page 617
5 Find Cube Roots......Page 618
6 Graph the Cube Root Function......Page 620
7 Find nth Roots......Page 621
SECTION 7.1 STUDY SET......Page 623
1 Simplify Expressions of the Form a[(Sub)1/n]......Page 627
2 Simplify Expressions of the Form [(Sub)m/n]......Page 629
4 Simplify Expressions with Negative Rational Exponents......Page 632
5 Use Rules for Exponents to Simplify Expressions......Page 633
6 Simplify Radical Expressions......Page 634
SECTION 7.2 STUDY SET......Page 635
7.3 Simplifying and Combining Radical Expressions......Page 638
1 Use the Product Rule to Simplify Radical Expressions......Page 639
2 Use Prime Factorization to Simplify Radical Expressions......Page 641
3 Use the Quotient Rule to Simplify Radical Expressions......Page 642
4 Add and Subtract Radical Expressions......Page 644
SECTION 7.3 STUDY SET......Page 646
1 Multiply Radical Expressions......Page 649
2 Find Powers of Radical Expressions......Page 651
3 Rationalize Denominators......Page 652
4 Rationalize Denominators That Have Two Terms......Page 656
SECTION 7.4 STUDY SET......Page 657
1 Solve Equations Containing One Radical......Page 661
2 Solve Equations Containing Two Radicals......Page 666
3 Solve Formulas Containing Radicals......Page 668
SECTION 7.5 STUDY SET......Page 669
1 Use the Pythagorean Theorem to Solve Problems......Page 672
2 Solve Problems Involving 45°– 45°– 90° Triangles......Page 674
3 Solve Problems Involving 30°– 60°– 90° Triangles......Page 675
4 Use the Distance Formula to Solve Problems......Page 677
SECTION 7.6 STUDY SET......Page 678
7.7 Complex Numbers......Page 682
1 Express Square Roots of Negative Numbers in Terms of i......Page 683
2 Write Complex Numbers in the Form a+bi......Page 684
4 Multiply Complex Numbers......Page 685
5 Divide Complex Numbers......Page 687
6 Perform Operations Involving Powers of i......Page 690
SECTION 7.7 STUDY SET......Page 691
CHAPTER SUMMARY AND REVIEW......Page 694
CHAPTER TEST......Page 703
CUMULATIVE REVIEW......Page 705
CHAPTER 8 Quadratic Equations, Functions, and Inequalities......Page 709
8.1 The Square Root Property and Completing the Square......Page 710
1 Use the Square Root Property to Solve Quadratic Equations......Page 711
2 Solve Quadratic Equations by Completing the Square......Page 714
SECTION 8.1 STUDY SET......Page 720
8.2 The Quadratic Formula......Page 722
1 Derive the Quadratic Formula......Page 723
2 Solve Quadratic Equations Using the Quadratic Formula......Page 724
3 Write Equivalent Equations to Make Quadratic Formula Calculations Easier......Page 726
4 Use the Quadratic Formula to Solve Application Problems......Page 727
SECTION 8.2 STUDY SET......Page 732
8.3 The Discriminant and Equations That Can Be Written in Quadratic Form......Page 735
1 Use the Discriminant to Determine Number and Type of Solutions......Page 736
2 Solve Equations That Are Quadratic in Form......Page 737
3 Solve Application Problems Using Quadratic Equations......Page 741
SECTION 8.3 STUDY SET......Page 742
1 Graph Functions of the Form f(x ) =ax[(sub)2] and f(x)= ax[(Sub)2] + k......Page 745
2 Graph Functions of the Form f(x)= a(x -h)[(Sub)2] and f(x) = a(x-h)[(Sub)2] + k......Page 748
3 Graph Functions of the Form f(x) = ax[(Sub)2] + bx +c by Completing the Square......Page 749
4 Find the Vertex Using -b/2a......Page 751
5 Determine Minimum and Maximum Values......Page 754
6 Solve Quadratic Equations Graphically......Page 755
SECTION 8.4 STUDY SET......Page 756
8.5 Quadratic and Other Nonlinear Inequalities......Page 759
1 Solve Quadratic Inequalities......Page 760
2 Solve Rational Inequalities......Page 762
3 Graph Nonlinear Inequalities in Two Variables......Page 766
SECTION 8.5 STUDY SET......Page 768
CHAPTER SUMMARY AND REVIEW......Page 770
CHAPTER TEST......Page 776
GROUP PROJECT......Page 778
CUMULATIVE REVIEW......Page 779
CHAPTER 9 Exponential and Logarithmic Functions......Page 783
1 Add, Subtract, Multiply, and Divide Functions......Page 784
2 Find the Composition of Functions......Page 787
3 Use Graphs to Evaluate Functions......Page 789
4 Use Composite Functions to Solve Problems......Page 790
SECTION 9.1 STUDY SET......Page 791
1 Determine Whether a Function Is a One-to-One Function......Page 795
2 Use the Horizontal Line Test to Determine Whether a Function Is One-to-One......Page 796
3 Find the Equation of the Inverse of a Function......Page 798
4 Find the Composition of a Function and Its Inverse......Page 799
5 Graph a Function and Its Inverse......Page 800
SECTION 9.2 STUDY SET......Page 802
9.3 Exponential Functions......Page 805
1 Define Exponential Functions......Page 806
2 Graph Exponential Functions......Page 807
3 Use Exponential Functions in Applications Involving Growth or Decay......Page 811
SECTION 9.3 STUDY SET......Page 815
9.4 Logarithmic Functions......Page 819
1 Define Logarithm......Page 820
3 Write Exponential Equations as Logarithmic Equations......Page 821
4 Evaluate Logarithmic Expressions......Page 823
5 Graph Logarithmic Functions......Page 825
6 Use Logarithmic Formulas and Functions in Applications......Page 827
SECTION 9.4 STUDY SET......Page 829
9.5 Base-e Exponential and Logarithmic Functions......Page 832
2 Graph the Natural Exponential Function......Page 833
3 Use Base-e Exponential Formulas and Functions in Applications......Page 835
4 Define Base-e Logarithms......Page 836
5 Evaluate Natural Logarithmic Expressions......Page 837
6 Graph the Natural Logarithmic Function......Page 838
7 Use Base-e Logarithmic Formulas and Functions in Applications......Page 840
SECTION 9.5 STUDY SET......Page 841
1 Use the Four Basic Properties of Logarithms......Page 846
2 Use the Product Rule for Logarithms......Page 847
3 Use the Quotient Rule for Logarithms......Page 848
4 Use the Power Rule for Logarithms......Page 849
5 Write Logarithmic Expressions as a Single Logarithm......Page 851
6 Use the Change-of-Base Formula......Page 853
7 Use Properties of Logarithms to Solve Application Problems......Page 854
SECTION 9.6 STUDY SET......Page 855
9.7 Exponential and Logarithmic Equations......Page 857
1 Solve Exponential Equations......Page 858
2 Solve Logarithmic Equations......Page 862
3 Use Exponential and Logarithmic Equations to Solve Application Problems......Page 865
SECTION 9.7 STUDY SET......Page 867
CHAPTER SUMMARY AND REVIEW......Page 870
CHAPTER TEST......Page 881
CUMULATIVE REVIEW......Page 883
CHAPTER 10 Conic Sections; More Graphing......Page 887
1 Identify Conic Sections and Some of Their Applications......Page 888
2 Graph Equations of Circles Written in Standard Form......Page 890
4 Convert the General Form of the Equation of a Circle to Standard Form......Page 892
6 Convert the General Form of the Equation of a Parabola to Standard Form to Graph It......Page 894
SECTION 10.1 STUDY SET......Page 898
1 Define an Ellipse......Page 901
2 Graph Ellipses Centered at the Origin......Page 902
3 Graph Ellipses Centered at (h, k)......Page 904
4 Solve Application Problems Involving Ellipses......Page 906
SECTION 10.2 STUDY SET......Page 907
2 Graph Hyperbolas Centered at the Origin......Page 910
3 Graph Hyperbolas Centered at (h, k)......Page 913
4 Graph Equations of the Form xy= k......Page 914
5 Solve Application Problems Involving Hyperbolas......Page 915
SECTION 10.3 STUDY SET......Page 916
1 Solve Systems by Graphing......Page 919
2 Solve Systems by Substitution......Page 920
3 Solve Systems by Elimination (Addition)......Page 922
SECTION 10.4 STUDY SET......Page 923
CHAPTER SUMMARY AND REVIEW......Page 926
CHAPTER TEST......Page 931
GROUP PROJECT......Page 932
CHAPTER 11 Miscellaneous Topics......Page 933
11.1 The Binomial Theorem......Page 934
1 Use Pascal’s Triangle to Expand Binomials......Page 935
2 Use Factorial Notation......Page 937
3 Use the Binomial Theorem to Expand Binomials......Page 938
4 Find a Specific Term of a Binomial Expansion......Page 940
SECTION 11.1 STUDY SET......Page 941
1 Find Terms of a Sequence Given the General Term......Page 943
2 Find Terms of an Arithmetic Sequence by Identifying the First Term and the Common Difference......Page 944
3 Find Arithmetic Means......Page 946
4 Find the Sum of the First n Terms of an Arithmetic Sequence......Page 947
5 Solve Application Problems Involving Arithmetic Sequences......Page 948
6 Use Summation Notation......Page 949
SECTION 11.2 STUDY SET......Page 950
1 Find Terms of a Geometric Sequence by Identifying the First Term and the Common Ratio......Page 953
2 Find Geometric Means......Page 955
3 Find the Sum of the First n Terms of a Geometric Sequence......Page 957
4 Define and Use Infinite Geometric Series......Page 958
5 Solve Application Problems Involving Geometric Sequences......Page 960
SECTION 11.3 STUDY SET......Page 962
CHAPTER SUMMARY AND REVIEW......Page 965
CHAPTER TEST......Page 969
GROUP PROJECT......Page 970
CUMULATIVE REVIEW......Page 971
1 Roots and Powers......Page 975
2 Answers to Selected Problems......Page 977
INDEX......Page 1035