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Algebra for College Students, 8th Edition Solution Manual

Algebra for College Students, 8th Edition Solution Manual breaks down difficult textbook problems into simple solutions, making your study time more effective.

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     ’ S S OLUTIONS M ANUAL GEX P UBLISHING S ERVICES A LGEBRA FOR C OLLEGE S TUDENTS E IGHTH E DITION Margaret L. Lial American River College John Hornsby University of New Orleans Terry McGinnis

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    Table of Contents R Review of the Real Number System.......................................................................................................1 R.1 R.2 R.3 R.4 Basic Concepts • ................................................................................................................ Operations on Real Numbers • .......................................................................................... Exponents, Roots, and Order of Operations • .................................................................... Properties of Real Numbers • ............................................................................................ Chapter R Test •................................................................................................................. 1 6 12 20 23 1 Linear Equations, Inequalities, and Applications ............................................................................. 26 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Linear Equations in One Variable • ................................................................................... Formulas and Percent • ...................................................................................................... Applications of Linear Equations •.................................................................................... Further Applications of Linear Equations • ....................................................................... Summary Exercises Applying Problem-Solving Techniques • ........................................... Linear Inequalities in One Variable • ................................................................................ Set Operations and Compound Inequalities • .................................................................... Absolute Value Equations and Inequalities •..................................................................... Summary Exercises Solving Linear and Absolute Value Equations and Inequalities • .................................................................................................... Chapter 1 Review Exercises •............................................................................................ Chapter 1 Mixed Review Exercises • ................................................................................ Chapter 1 Test • ................................................................................................................. 26 37 46 59 68 72 84 92 106 110 116 119 2 Linear Equations, Graphs, and Functions ....................................................................................... 122 2.1 2.2 2.3 2.4 2.5 2.6 Linear Equations in Two Variables •................................................................................. The Slope of a Line • ......................................................................................................... Writing Equations of Lines • ............................................................................................. Summary Exercises Finding Slopes and Equations of Lines •........................................... Linear Inequalities in Two Variables • .............................................................................. Introduction to Relations and Functions •.......................................................................... Function Notation and Linear Functions •......................................................................... Chapter 2 Review Exercises •............................................................................................ Chapter 2 Mixed Review Exercises • ................................................................................ Chapter 2 Test • ................................................................................................................. Chapters R − 2 Cumulative Review Exercises •.................................................................. 122 138 151 165 168 178 185 193 199 200 203

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    3 Systems of Linear Equations ............................................................................................................. 206 3.1 3.2 3.3 Systems of Linear Equations in Two Variables • .............................................................. Systems of Linear Equations in Three Variables • ............................................................ Applications of Systems of Linear Equations • ................................................................. Chapter 3 Review Exercises •............................................................................................ Chapter 3 Mixed Review Exercises • ................................................................................ Chapter 3 Test • ................................................................................................................. Chapters R − 3 Cumulative Review Exercises •.................................................................. 206 227 246 265 271 273 276 4 Exponents, Polynomials, and Polynomial Functions....................................................................... 281 4.1 4.2 4.3 4.4 4.5 Integer Exponents and Scientific Notation •...................................................................... Adding and Subtracting Polynomials •.............................................................................. Polynomial Functions, Graphs, and Composition • ........................................................... Multiplying Polynomials •................................................................................................. Dividing Polynomials •...................................................................................................... Chapter 4 Review Exercises •............................................................................................ Chapter 4 Mixed Review Exercises • ................................................................................ Chapter 4 Test • ................................................................................................................. Chapters R − 4 Cumulative Review Exercises •.................................................................. 281 295 301 312 323 334 340 342 345 5 Factoring ............................................................................................................................................. 350 5.1 5.2 5.3 5.4 5.5 Greatest Common Factors and Factoring by Grouping •..................................................... Factoring Trinomials •......................................................................................................... Special Factoring • .............................................................................................................. A General Approach to Factoring • ..................................................................................... Solving Equations by the Zero-Factor Property •................................................................ Chapter 5 Review Exercises • ............................................................................................. Chapter 5 Mixed Review Exercises • .................................................................................. Chapter 5 Test •................................................................................................................... Chapters R − 5 Cumulative Review Exercises •.................................................................... 350 356 364 373 380 392 397 398 400

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    6 Rational Expressions and Functions................................................................................................. 404 6.1 6.2 6.3 6.4 6.5 6.6 Rational Expressions and Functions; Multiplying and Dividing • ...................................... Adding and Subtracting Rational Expressions •.................................................................. Complex Fractions • ............................................................................................................ Equations with Rational Expressions and Graphs •............................................................. Summary Exercises Simplifying Rational Expressions vs. Solving Rational Equations •................................................................................................. Applications of Rational Expressions • ............................................................................... Variation • ........................................................................................................................... Chapter 6 Review Exercises • ............................................................................................. Chapter 6 Mixed Review Exercises • .................................................................................. Chapter 6 Test •................................................................................................................... Chapters R − 6 Cumulative Review Exercises •.................................................................... 404 413 426 436 449 454 469 479 485 488 492 7 Roots, Radicals, and Root Functions ................................................................................................ 498 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Radical Expressions and Graphs •....................................................................................... Rational Exponents • ........................................................................................................... Simplifying Radicals, the Distance Formula, and Circles •................................................. Adding and Subtracting Radical Expressions • ................................................................... Multiplying and Dividing Radical Expressions • ................................................................ Summary Exercises Performing Operations with Radicals and Rational Exponents • ................................................................................................ Solving Equations with Radicals • ...................................................................................... Complex Numbers • ............................................................................................................ Chapter 7 Review Exercises • ............................................................................................. Chapter 7 Mixed Review Exercies •.................................................................................... Chapter 7 Test •................................................................................................................... Chapters R − 7 Cumulative Review Exercises •.................................................................... 498 505 513 525 532 543 547 558 565 573 575 578

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    8 Quadratic Equations, Inequalities, and Functions .......................................................................... 583 8.1 8.2 8.3 8.4 8.5 The Square Root Property and Completing the Square • .................................................... The Quadratic Formula • ..................................................................................................... Equations Quadratic in Form • ............................................................................................ Summary Exercises Applying Methods for Solving Quadratic Equations •......................... Formulas and Further Applications •................................................................................... Polynomial and Rational Inequalities • ............................................................................... Chapter 8 Review Exercises • ............................................................................................. Chapter 8 Mixed Review Exercises • .................................................................................. Chapter 8 Test •................................................................................................................... Chapters R − 8 Cumulative Review Exercises •.................................................................... 583 598 609 634 638 652 673 681 684 689 9 Additional Graphs of Functions and Relations ............................................................................... 695 9.1 9.2 9.3 9.4 9.5 Review of Operations and Composition • ......................................................................... Graphs of Quadratic Functions • ....................................................................................... More About Parabolas and Their Applications •............................................................... Symmetry; Increasing and Decreasing Functions • ........................................................... Piecewise Linear Functions • ............................................................................................ Chapter 9 Review Exercises • ........................................................................................... Chapter 9 Mixed Review Exercises • ................................................................................ Chapter 9 Test •................................................................................................................. Chapters R − 9 Cumulative Review Exercises •.................................................................. 695 708 717 733 743 755 761 763 766 10 Inverse, Exponential, and Logarithmic Functions ................................................................................ 772 10.1 10.2 10.3 10.4 10.5 10.6 Inverse Functions • ............................................................................................................ Exponential Functions • .................................................................................................... Logarithmic Functions • .................................................................................................... Properties of Logarithms •................................................................................................. Common and Natural Logarithms •................................................................................... Exponential and Logarithmic Equations; Further Applications •...................................... Chapter 10 Review Exercises • ......................................................................................... Chapter 10 Mixed Review Exercises • .............................................................................. Chapter 10 Test •............................................................................................................... Chapters R − 10 Cumulative Review Exercises •................................................................ 772 782 790 802 808 813 825 832 834 836

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    11 Polynomial and Rational Functions.................................................................................................. 842 11.1 11.2 11.3 11.4 Zeros of Polynomial Functions (I) • .................................................................................. Zeros of Polynomial Functions (II) •................................................................................. Graphs and Applications of Polynomial Functions •......................................................... Summary Exercises Examining Polynomial Functions and Graphs •............................... Graphs and Applications of Rational Functions • ............................................................. Chapter 11 Review Exercises • ......................................................................................... Chapter 11 Mixed Review Exercises • .............................................................................. Chapter 11 Test •............................................................................................................... Chapters R − 11 Cumulative Review Exercises •................................................................ 842 848 860 880 886 902 911 912 915 12 Conic Sections and Nonlinear Systems............................................................................................. 922 12.1 12.2 12.3 12.4 Circles Revisited and Ellipses •......................................................................................... Hyperbolas and Functions Defined by Radicals • ............................................................. Nonlinear Systems of Equations • ..................................................................................... Second-Degree Inequalities, Systems of Inequalities, and Linear Programming • ........... Chapter 12 Review Exercises • ......................................................................................... Chapter 12 Mixed Review Exercises • .............................................................................. Chapter 12 Test •............................................................................................................... Chapters R − 12 Cumulative Review Exercises •................................................................ 922 932 943 955 976 983 984 987 13 Further Topics in Algebra ................................................................................................................. 994 13.1 13.2 13.3 13.4 13.5 13.6 13.7 Sequences and Series • ...................................................................................................... Arithmetic Sequences • ..................................................................................................... Geometric Sequences •...................................................................................................... The Binomial Theorem • ................................................................................................... Mathematical Induction • .................................................................................................. Counting Theory • ............................................................................................................. Basics of Probability • ....................................................................................................... Chapter 13 Review Exercises • ......................................................................................... Chapter 13 Mixed Review Exercises • .............................................................................. Chapter 13 Test •............................................................................................................... Chapters R − 13 Cumulative Review Exercises •................................................................ 994 1000 1009 1020 1025 1037 1044 1050 1057 1059 1063

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    Chapter 1 Linear Equations, Inequalities, and Applications 26 Chapter 1 Linear Equations, Inequalities, and Applications 1.1 Linear Equations in One Variable Classroom Examples, Now Try Exercises 1. (a) 9 10 0 x + = is an equation because it contains an equals symbol. (b) 9 10 x + is an expression because it does not contain an equals symbol. N1. (a) 2 17 3 x x + − is an expression because it does not contain an equals symbol. (b) 2 17 3 x x + = is an equation because it contains an equals symbol. 2. 4 8 9 17 1 12 17 10 Combine terms. 12 17 17 10 17 Subtract 17 . 5 10 Combine terms. 5 10 Divide by 5. 5 5 2 x x x x x x x x x x x x x + = − + − = − − = − − − = − − − = − − − = Check by substituting 2 for x in the original equation. ? ? ? 4 8 9 17 1 4(2) 8(2) 9 17(2) 1 Let 2. 8 16 9 34 1 24 25 1 24 24 True x x x x + = − + − + = − + − = + = − + − = − = The solution set is { } 2 . N2. 5 11 2 13 3 5 11 13 Combine terms. 5 11 13 Add . 6 11 13 Combine terms. 6 11 11 13 11 Subtract 11. 6 24 Combine terms. 6 24 Divide by 6. 6 6 4 x x x x x x x x x x x x x x x + = − − + = − − + + = − − + + = − + − = − − = − − = = − Check by substituting 4 − for x in the original equation. ? ? ? 5 11 2 13 3 5( 4) 11 2( 4) 13 3( 4) Let 4. 20 11 8 13 12 9 21 12 9 9 True x x x x + = − − − + = − − − − = − − + = − − + − = − + − = − The solution set is { 4}. − 3. ( ) 3 4 5 3 12 3 5 3 Distributive prop. 12 2 5 3 Combine terms. 12 2 2 5 3 2 Subtract 2 . 12 3 3 Combine terms. 12 3 3 3 3 Add 3. 15 3 Combine terms. 15 3 Divide by 3. 3 3 5 x x x x x x x x x x x x x x x x x x + − = − + − = − + = − + − = − − = − + = − + = = = We will use the following notation to indicate the value of each side of the original equation after we have substituted the proposed solution and simplified. Check 5 : 27 5 25 3 True x = − = − The solution set is { } 5 . N3. ( ) 5 4 12 3 2 x x − − = − 5 20 12 3 2 Distributive prop. 5 32 3 2 Combine terms. 5 32 2 3 2 2 Add 2 . 7 32 3 Combine terms. 7 32 32 3 32 Add 32. 7 35 Combine terms. 7 35 Divide by 7. 7 7 5 x x x x x x x x x x x x x x − − = − − = − − + = − + − = − + = + = = = We will use the following notation to indicate the value of each side of the original equation after we have substituted the proposed solution and simplified. Check 5 : 5 12 3 10 True x = − = − The solution set is { } 5 .

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    1.1 Linear Equations in One Variable 27 4. 6 (4 ) 8 2(3 5) 6 4 8 6 10 Distributive prop. 2 2 10 Combine terms. 2 2 10 Add . 2 3 10 Combine terms. 2 10 3 10 10 Add 10. 12 3 Combine terms. 12 3 Divide by 3. 3 3 4 x x x x x x x x x x x x x x x x x x − + = − + − − = − − − = − − + = − + = − + = − + = = = Check 4 : 2 32 34 True x = − = − The solution set is { } 4 . N4. ( ) ( ) 2 3 2 6 4 1 36 2 6 18 4 4 36 Dist. prop. 4 18 4 40 Combine terms. 4 18 18 4 40 18 Add 18 . 4 22 40 Combine terms. 4 40 22 40 40 Subtract 40. 44 22 Combine terms. 44 22 Divide by 22. 22 22 2 x x x x x x x x x x x x x x x x − + = + + − − = + + − − = + − − + = + + − = + − − = + − − = − = − = Check 2 : 2 30 4 36 True x = − + = − + The solution set is { } 2 . − 5. Multiply each side by the LCD, 4, and use the distributive property 1 3 1 2 4 2 1 3 1 4 4 4 2 4 2 2( 1) 1( 3) 2 2 2 3 2 3 5 2 3 3 Subtract 5. 1 Divide by 3. x x x x x x x x x x x + + + = + + ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ + = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ + + + = + + + = + = = − = − Check 2 1 1: 0 True 4 2 x = − + = The solution set is { 1}. − N5. Multiply each side by the LCD, 8, and use the distributive property. 4 2 4 5 4 8 4 2 4 8 8 8(5) 4 8 2( 4) 1(2 4) 40 2 8 2 4 40 4 4 40 4 44 Add 4. 11 Divide by 4. x x x x x x x x x x x − + + = − + ⎛ ⎞ ⎛ ⎞ + = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ − + + = − + + = − = = = Check 7 13 11: 5 True 4 4 x = + = The solution set is { } 11 . 6. Multiply each term by 100. 0.02(60) 0.04 0.03(50 ) 2(60) 4 3(50 ) 120 4 150 3 4 30 3 30 x x x x x x x x x + = + + = + + = + = + = Check 30 : 1.2 1.2 2.4 True x = + = The solution set is { } 30 . N6. Multiply each term by 100. 0.08 0.12( 4) 0.03( 5) 8 12( 4) 3( 5) 8 12 48 3 15 4 48 3 15 4 63 3 63 7 9 x x x x x x x x x x x x x x x − − = − − − = − − + = − − + = − − + = = = Check 9 : 0.72 0.60 0.12 True x = − = The solution set is { } 9 . 7. (a) 5( 2) 2( 1) 3 1 5 10 2 2 3 1 3 8 3 1 3 8 3 3 1 3 8 1 False x x x x x x x x x x x x + − + = + + − − = + + = + + − = + − = Since the result, 8 1, = is false , the equation has no solution and is called a contradiction. The solution set is . ∅

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    Chapter 1 Linear Equations, Inequalities, and Applications 28 (b) Multiply each side by the LCD, 3, and use the distributive property. 1 2 1 3 3 3 1 2 1 3 3 3 3 3 3 1 2 3 1 3 1 3 1 x x x x x x x x x x x + + = + + ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ + = + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ + + = + + = + This is an identity. Any real number will make the equation true. The solution set is {all real numbers}. (c) 5(3 1) 5 15 5 5 14 5 5 Subtract . 14 0 Subtract 5. 0 Divide by 14. x x x x x x x x + = + + = + + = = = This is a conditional equation. Check 0 : 5(1) 0 5 True x = = + The solution set is { } 0 . N7. (a) 9 3( 4) 6( 2) 9 3 12 6 12 6 12 6 12 x x x x x x x x − + = − − − = − − = − This is an identity. Any real number will make the equation true. The solution set is { } all real numbers . (b) 3(2 1) 2 3 6 3 2 3 8 3 3 9 3 3 Subtract . 9 0 Subtract 3. 0 Divide by 9. x x x x x x x x x x x x − − − = + − + − = + − + = + − + = − = = − This is a conditional equation. Check 0 : 3( 1) 3 True x = − − = The solution set is { } 0 . (c) 10 21 2( 5) 8 10 21 2 10 8 10 21 10 10 10 21 10 10 10 10 21 10 False x x x x x x x x x x x x − = − + − = − + − = − − − = − − − = − Since the result, 21 10, − = − is false , the equation has no solution and is called a contradiction. The solution set is . ∅ Exercises 1. A collection of numbers, variables, operation symbols, and grouping symbols, such as ( ) 2 8 15 , x − is an algebraic expression. While an equation does include an equality symbol, there is not an equality symbol in an algebraic expression. 2. A linear equation in one variable (here x ) can be written in the form , A x B C + = with 0. A ≠ Another name for a linear equation is a first-degree equation, because the greatest power on the variable is one . 3. If we let 2 x = in the linear equation 2 5 9, x + = a true statement results. The number 2 is a solution of the equation, and { } 2 is the solution set. 4. A linear equation with one solution in its solution set, such as the equation in Exercise 3, is a conditional equation. 5. A linear equation with an infinite number of solutions is an identity. Its solution set is { } all real numbers . 6. A linear equation with no solution is a contradiction. Its solution set is the empty set . ∅ 7. 3 2 0 x x + − = can be written as 4 2, x = so it is linear. 9 4 9 x − = is in linear form. Choices A and C are linear. 8. 2 12 x = is not a linear equation because the variable is squared. 1 1 0 8 x x − = is not a linear equation because there is a variable in the denominator of the second term. Choices B and D are nonlinear. 9. ? ? 3( 4) 5 Original equation 3(6 4) 5 6 Let 6. 3(10) 30 Add. 30 30 True x x x + = + = ⋅ = = = Since a true statement is obtained, 6 is a solution.

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    1.1 Linear Equations in One Variable 29 10. ? ? ? 5( 4) 3( 6) 9( 1) 5( 2 4) 3( 2 6) 9( 2 1) Let 2. 5(2) 3(4) 9( 1) Add. 10 12 9 Multiply. 2 9 False x x x x + − + = + − + − − + = − + = − − = − − = − − = − Since a false statement is obtained, 2 − is not a solution. 11. 3 2 4 x x − + − = is an equation because it contains an equals symbol. 12. 3 2 4 4 x x − + − − = is an equation because it contains an equals symbol. 13. 4( 3) 2( 1) 10 x x + − + − is an expression because it does not contain an equals symbol. 14. 4( 3) 2( 1) 10 x x + − + + is an expression because it does not contain an equals symbol. 15. 10 12 4 3 x x − + − = − is an equation because it contains an equals symbol. 16. 10 12 4 3 0 x x − + − + = is an equation because it contains an equals symbol. 17. A sign error was made when the distributive property was applied. The left side of the second line should be 8 4 6. x x − + ( ) 8 2 2 3 3 7 8 4 6 3 7 Distributive property 4 6 3 7 Combine like terms. 1 Subtract 3 and 6. x x x x x x x x x x − − = + − + = + + = + = The correct solution is 1. 18. The negative sign − represents 1. − ( ) 5 2 4 5 5 2 4 5 Distributive property 7 9 Combine terms. x x x x x − − − + = − − + + = − + 19. 7 8 1 7 8 8 1 8 Subtract 8. 7 7 7 7 Divide by 7. 7 7 1 x x x x x + = + − = − = − − = = − We will use the following notation to indicate the value of each side of the original equation after we have substituted the proposed solution and simplified. Check 1: 7 8 1 True x = − − + = The solution set is { 1}. − 20. 5 4 21 5 4 4 21 4 Add 4. 5 25 5 25 Divide by 5. 5 5 5 x x x x x − = − + = + = = = Check 5 : 25 4 21 True x = − = The solution set is { } 5 . 21. 5 2 3 6 5 2 3 3 6 3 Subtract 3 . 2 2 6 2 2 2 6 2 Subtract 2. 2 8 2 8 Divide by 2. 2 2 4 x x x x x x x x x x x x + = − + − = − − + = − + − = − − = − − = = − Check 4 : 20 2 12 6 True x = − − + = − − The solution set is { 4}. − 22. 9 1 7 9 9 1 7 7 9 7 Subtract 7 . 2 1 9 2 1 1 9 1 Subtract 1. 2 10 2 10 Divide by 2. 2 2 5 x x x x x x x x x x x x + = − + − = − − + = − + − = − − = − − = = − Check 5 : 45 1 35 9 True x = − − + = − − The solution set is { 5}. − 23. 7 5 15 8 2 15 8 Combine terms. 2 7 Subtract 15. 7 Subtract . x x x x x x x x x − + = + + = + = − = − Check 7 : 49 35 15 7 8 True x = − − + + = − + The solution set is { 7}. −

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    Chapter 1 Linear Equations, Inequalities, and Applications 30 24. 2 4 4 5 4 4 5 Combine terms. 3 4 5 Subtract 4 . 3 9 Subtract 4. 3 Divide by 3. x x x x x x x x x + − = − + = − − + = − − = − = − Check 3 : 6 4 3 12 5 True x = + − = − The solution set is { } 3 . 25. 12 15 9 5 3 5 9 27 4 3 4 Combine terms. 30 4 4. Add 3 . 30 0 Add 4. 0 Divide by 30. w w w w w w w w w + − + = − + − − = − − − = − = = Check 0 : 9 5 5 9 True w = − + = − The solution set is { } 0 . 26. 4 5 8 4 6 4 4 6 4 Combine terms. 5 4 4 Subtract 6 . 5 0 Add 4. 0 Divide by 5. x x x x x x x x x − + − + = − − = − − − = − − = = − Check 0 : 8 4 4 True x = − + = − The solution set is {0}. 27. 3(2 4) 20 2 6 12 20 2 Distributive property 8 12 20 Add 2 . 8 32 Add 12. 4 Divide by 8. t t t t t t t t − = − − = − − = = = Check 4 : 3(4) 20 8 True t = = − The solution set is { } 4 . 28. 2(3 2 ) 4 6 4 4 Distributive property 6 5 4 Subtract . 5 10 Subtract 6. 2 Divide by 5. x x x x x x x x − = − − = − − = − − = − = − Check 2 : 2( 1) 2 4 True x = − = − The solution set is { } 2 . 29. 5( 1) 3 2 6 4 5 5 3 2 6 4 Distributive prop. 2 3 6 4 Combine terms. 3 8 4 Add 2 . 7 8 Subtract 4. 7 Divide by 8. 8 x x x x x x x x x x x x − + + + = + − − + + = + − − = + − = + − = − = Check : Substitute 7 8 − for x and show that both sides equal 1.25. − The screen shows a typical Check on a calculator. The solution set is 7 . 8 ⎧ ⎫ − ⎨ ⎬ ⎩ ⎭ 30. 5( 3) 4 5 4 2 5 15 4 5 4 2 Distributive prop. 9 10 4 2 Combine terms. 11 10 4 Add 2 . 11 6 Subtract 10. 6 Divide by 11. 11 x x x x x x x x x x x x + + − = − + + − = − + = − + = = − = − Check 6 : 11 135 24 55 44 12 True 11 11 11 11 11 x = − − − = + The solution set is 6 . 11 ⎧ ⎫ − ⎨ ⎬ ⎩ ⎭ 31. 2 5 9 3( 4) 5 3 9 3 12 5 3 9 3 17 9 17 False x x x x x x x − + − = − − − = − − − = − − = − The equation is a contradiction. The solution set is . ∅ 32. 6 2 11 2(2 3) 4 4 11 4 6 4 4 11 4 10 11 10 False x x x x x x x − + − = − − + − − = − + + − − = − + − = The equation is a contradiction. The solution set is . ∅

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    1.1 Linear Equations in One Variable 31 33. 2( 3) 4( 1) 2 6 4 4 Remove parentheses. 6 6 4 Add 4x. 6 10 Subtract 6. 10 5 Divide by 6. 6 3 x x x x x x x + = − + + = − − + = − = − − = = − − Check 5 4 2 : 2 4 True 3 3 3 x ⎛ ⎞ ⎛ ⎞ = − = − − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ The solution set is 5 . 3 ⎧ ⎫ − ⎨ ⎬ ⎩ ⎭ 34. 4( 9) 8( 3) 4 36 8 24 Remove parentheses. 4 36 24 Subtract 8 . 4 60 Add 36. 15 Divide by 4. x x x x x x x x − = + − = + − − = − = = − − Check 15 : 4( 24) 8( 12) True x = − − = − The solution set is { 15}. − 35. 3(2 1) 2( 2) 5 6 3 2 4 5 4 7 5 4 2 2 1 4 2 x x x x x x x + − − = + − + = + = = − − = = − Check 1 5 : 3(0) 2 5 True 2 2 x ⎛ ⎞ = − − − = ⎜ ⎟ ⎝ ⎠ The solution set is 1 . 2 ⎧ ⎫ − ⎨ ⎬ ⎩ ⎭ 36. 4( 2) 2( 3) 6 4 8 2 6 6 6 2 6 6 8 8 4 6 3 x x x x x x x − + + = − + + = − = = = = Check 4 2 13 : 4 2 6 True 3 3 3 x ⎛ ⎞ ⎛ ⎞ = − + = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ The solution set is 4 . 3 ⎧ ⎫ ⎨ ⎬ ⎩ ⎭ 37. 2 3( 4) 2( 3) 2 3 12 2 6 5 12 2 6 3 6 6 2 3 x x x x x x x x x x + − = − + − = − − = − = = = Check 2 : 4 3( 2) 2( 1) True x = + − = − The solution set is {2}. 38. 6 3(5 2) 4(1 ) 6 15 6 4 4 9 6 4 4 5 10 10 2 5 x x x x x x x x x x − + = − − − = − − − = − − = = = − − Check 2 : 12 3( 8) 4(3) True x = − − − − = The solution set is { 2}. − 39. 6 4(3 2 ) 5( 4) 10 6 12 8 5 20 10 14 12 5 30 9 18 2 x x x x x x x x x x − − = − − − + = − − − = − = − = − Check 2 : 12 4(7) 5( 6) 10 True x = − − − = − − The solution set is { 2}. − 40. 2 3(4 2 ) 2( 3) 2 2 12 6 2 6 2 4 12 2 4 2 8 4 x x x x x x x x x x − − − = − + − − + = − + − = − = = Check 4 : 8 3( 4) 2(1) 2 True x = − − − = + The solution set is {4}. 41. 2( 3) 4 3( 4) 2 2 6 4 3 12 2 3 10 3 10 x x x x x x x x − + − − = − + + − − − − = − − + − − = − − The equation is an identity. The solution set is {all real numbers}. 42. 4(2 7) 2 25 3(2 1) 8 28 2 25 6 3 8 28 8 28 x x x x x x x x + = + + + + = + + + + = + The equation is an identity. The solution set is {all real numbers}.

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    Chapter 1 Linear Equations, Inequalities, and Applications 32 43. 2[ (2 4) 3] 2( 1) 2[ 2 4 3] 2( 1) 2[ 1] 2( 1) 1 1 Divide by 2. 1 2 1 Add . 2 2 Subtract 1. 1 Divide by 2. x x x x x x x x x x x x x x − + + = + − − + = + − − = + − − = + − = + − = − = Check 1: 2[ 1 2 3] 0 True x = − − − + = The solution set is { 1}. − 44. 4[2 (3 ) 5] (2 7 ) 4[2 3 5] (2 7 ) 4[3 2] (2 7 ) 12 8 2 7 19 8 2 Add 7 . 19 10 Subtract 8. 10 Divide by 19. 19 x x x x x x x x x x x x x x − − + = − + − + + = − + + = − + + = − − + = − = − = − Check 10 : 19 20 67 95 32 4 True 19 19 19 19 x = − ⎡ ⎤ ⎛ ⎞ − − + = − − ⎜ ⎟ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ The solution set is 10 . 19 ⎧ ⎫ − ⎨ ⎬ ⎩ ⎭ 45. [2 (5 2)] 2 (2 7) [2 5 2] 2 2 7 [ 3 2] 2 2 7 3 2 2 9 7 x x x x x x x x x x x − − + = + + − − − = + + − − − = + + + = + = Check 7 : [14 37] 2 21 True x = − − = + The solution set is {7}. 46. [6 (4 8)] 9 (6 3) [6 4 8] 9 6 3 (2 8) 6 12 2 8 6 12 8 4 4 1 8 2 x x x x x x x x x x x x − − + = + + − − − = + + − − = + − + = + − = = = − − Check 1 : [ 3 6] 9 0 True 2 x = − − − − = + The solution set is 1 . 2 ⎧ ⎫ − ⎨ ⎬ ⎩ ⎭ 47. 3 6 5( 1) 5 (2 4) 5 3 6 5 5 5 2 4 5 8 11 7 9 2 2 x x x x x x x x x x x x − + − − = − − − + − + − + = − − + + − + = − + − = − = Check 2 : 6 6 5 10 0 5 True x = − + − = − − + The solution set is {2}. 48. 4( 2) 8 5 3 9 2( 6) 4 8 8 5 3 9 2 12 4 3 5 3 6 x x x x x x x x x x x + − − = − + − + + − − = − + − − − + = − − = − Check 6 : 16 48 5 18 9 0 True x = − − + − = + − The solution set is { 6}. − 49. 7[2 (3 4 )] 2 9 2(1 15 ) 7[2 3 4 ] 2 9 2 30 7[ 1 4 ] 2 7 30 7 28 2 7 30 7 30 7 30 x x x x x x x x x x x x x x − + − = − + − − − − = − + − − − − = − − − − − = − − − − = − − The equation is an identity. The solution set is {all real numbers}. 50. 4[6 (1 2 )] 10 2(10 3 ) 8 4[6 1 2 ] 10 20 6 8 4(5 2 ) 10 20 2 20 8 10 20 2 20 2 20 2 x x x x x x x x x x x x x x x x − + + = − + − − + = − + − + = + − + = + + = + The equation is an identity. The solution set is {all real numbers}. 51. [3 (2 5)] 4 [3(2 4) 3 ] [3 2 5] 4 [6 12 3 ] [ 5] 4 [3 12] 5 4 3 12 5 3 8 2 3 3 2 x x x x x x x x x x x x x x x x − − + = − − − − − − − = − − − − − − = − − − − + = − − + − + = − + = =

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    1.1 Linear Equations in One Variable 33 Check 3 : 2 9 9 8 4 3 True 2 2 x = ⎡ ⎤ ⎡ ⎤ − − = − − − − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ The solution set is 3 . 2 ⎧ ⎫ ⎨ ⎬ ⎩ ⎭ 52. 2[ ( 1) 4] 5 [ (6 7) 9 ] 2[ 1 4] 5 [ 6 7 9 ] 2[ 5] 5 [3 7] 2 10 3 12 5 2 2 5 x x x x x x x x x x x x − − + = + − − + − + + = + − + + − + = + + − + = + − = = − Check 2 : 5 7 47 18 2 4 5 True 5 5 5 x = − ⎡ ⎤ ⎡ ⎤ + = + − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ The solution set is 2 . 5 ⎧ ⎫ − ⎨ ⎬ ⎩ ⎭ 53. The denominators of the fractions are 3, 4, 6, and 1. The least common denominator is 12 since it is the smallest number into which each denominator can divide without a remainder. 54. Yes, the coefficients will be larger, but you will get the correct solution. As long as you multiply both sides of the equation by the same nonzero number, the resulting equation is equivalent and the solution does not change. 55. (a) We need to make 0.04, 0.06, 0.03, and 1.46 integers. These numbers can be written as 4 , 100 6 , 100 3 , 100 and 146 , 100 respectively. Multiplying by 2 10 , or 100, will eliminate the decimal points (the denominators) so that all the coefficients are integers. (b) We need to make 0.006, 0.02, 0.03, 0.008, and 0.25 integers. These numbers can be written as 6 , 1000 2 , 100 3 , 100 8 , 1000 and 25 , 100 respectively. Multiplying by 3 10 , or 1000, will eliminate the decimal points (the denominators) so that all the coefficients are integers. 56. 0.06(10 )(100) 0.06(100)(10 ) 6(10 ) 60 6 Choice B is correct. x x x x − = − = − = − 57. 5 2 9 5 18 Multiply by 9. 18 18 Divide by 5. 5 5 x x x − = − = = = − − − Check 18 5 18 : 2 True 5 9 5 x ⎛ ⎞ ⎛ ⎞ = − − − = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ The solution set is 18 . 5 ⎧ ⎫ − ⎨ ⎬ ⎩ ⎭ 58. 3 5 11 3 55 Multiply by 11. 55 55 Divide by 3. 3 3 x x x = − = − − = = − Check 55 3 55 : 5 True 3 11 3 x ⎛ ⎞⎛ ⎞ = − − = − ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ The solution set is 55 . 3 ⎧ ⎫ − ⎨ ⎬ ⎩ ⎭ 59. 6 1 5 6 5 Multiply by 5. 5 5 Divide by 6. 6 6 x x x = − = − − = = − Check 5 6 5 : 1 True 6 5 6 x ⎛ ⎞⎛ ⎞ = − − = − ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ The solution set is 5 . 6 ⎧ ⎫ − ⎨ ⎬ ⎩ ⎭ 60. 7 6 8 7 48 Multiply by 8. 48 48 Divide by 7. 7 7 x x x − = − = = = − − − Check 48 7 48 : 6 True 7 8 7 x ⎛ ⎞⎛ ⎞ = − − − = ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ The solution set is 48 . 7 ⎧ ⎫ − ⎨ ⎬ ⎩ ⎭

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    Chapter 1 Linear Equations, Inequalities, and Applications 34 61. Multiply both sides by the LCD, 6. 5 2 3 6 6(5) 2 3 6 6 30 Distributive property 2 3 3 2 30 5 30 Add. 6 Divide by 5. x x x x x x x x x x + = ⎛ ⎞ + = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ + = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ + = = = Check 6 : 3 2 5 True x = + = The solution set is {6}. 62. Multiply both sides by the LCD, 20. 1 5 4 20 20(1) 5 4 20 20 20 Distributive property 5 4 4 5 20 20 Subtract. 20 Multiply by 1. x x x x x x x x x x − = ⎛ ⎞ − = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ − = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ − = − = = − − Check 20 : 4 5 1 True x = − − + = The solution set is { 20}. − 63. Multiply both sides by the LCD, 4. 3 5 13 4 2 3 5 4 4(13) 4 2 3 5 4 4 4(13) Distributive prop. 4 2 3 10 52 13 52 Combine terms. 4 Divide by 13. x x x x x x x x x x + = ⎛ ⎞ + = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ + = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ + = = = Check 4 : 3 10 13 True x = + = The solution set is {4}. 64. Multiply both sides by the LCD, 6. 8 13 3 2 8 6 6( 13) 3 2 8 6 6 6( 13) Distributive prop. 3 2 16 3 78 13 78 6 Divide by 26. x x x x x x x x x x − = − ⎛ ⎞ − = − ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ − = − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ − = − = − = − Check 6 : 16 3 13 True x = − − + = − The solution set is { 6}. − 65. Multiply both sides by the LCD, 6. 10 2 5 5 3 10 2 15 15 5 5 3 3( 10) 3(2) 5 3 30 6 5 8 24 24 3 8 x x x x x x x x x x − + = − − ⎛ ⎞ ⎛ ⎞ + = − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ − + = − − + = − = = = Check 7 2 3 : 1 True 5 5 x = − + = − The solution set is {3}. 66. Multiply both sides by the LCD, 54. ( ) ( ) 5 5 6 6 54 5 5 54 54 6 6 54 9 5 9 5 45 9 45 9 90 90 10 90 9 10 x x x x x x x x x x x x − + = − ⎛ ⎞ ⎛ ⎞ + = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ − + = − + = − + = = = = Check 36 45 9 9 : True 54 54 54 x − = + = The solution set is {9}.

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