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Solution Manual for Beginning and Intermediate Algebra and College Algebra: A Corequisite Solution

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     S OLUTIONS M ANUAL GEX P UBLISHING S ERVICES B EGINNING AND I NTERMEDIATE A LGEBRA S IXTH E DITION Margaret L. Lial American River College John Hornsby University of New Orleans Terry McGinnis

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    Table of Contents R Prealgebra Review.................................................................................................................. 1 R.1 R.2 Fractions •................................................................................................................. Decimals and Percents • ........................................................................................... 1 15 1 The Real Number System ...................................................................................................... 26 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Exponents, Order of Operations, and Inequality •.................................................... Variables, Expressions, and Equations • .................................................................. Real Numbers and the Number Line •...................................................................... Adding and Subtracting Real Numbers • ................................................................. Multiplying and Dividing Real Numbers •............................................................... Summary Exercises Performing Operations with Real Numbers • .......................... Properties of Real Numbers • ................................................................................... Simplifying Expressions • ........................................................................................ Chapter 1 Review Exercises • .................................................................................. Chapter 1 Mixed Review Exercises • ....................................................................... Chapter 1 Test •........................................................................................................ 26 32 40 44 54 63 65 73 80 85 86 2 Linear Equations and Inequalities in One Variable ............................................................ 89 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 The Addition Property of Equality • ....................................................................... The Multiplication Property of Equality •............................................................... More on Solving Linear Equations • ....................................................................... Summary Exercises Applying Methods for Solving Linear Equations •.................. Applications of Linear Equations • ......................................................................... Formulas and Additional Applications from Geometry •........................................ Ratio, Proportion, and Percent • .............................................................................. Further Applications of Linear Equations •............................................................. Solving Linear Inequalities • ................................................................................... Chapter 2 Review Exercises • ................................................................................. Chapter 2 Mixed Review Exercises • ...................................................................... Chapter 2 Test •....................................................................................................... Chapters R–2 Cumulative Review Exercises •........................................................ 89 98 106 119 122 138 149 160 172 187 193 195 197 3 Linear Equations and Inequalities in Two Variables ............................................................ 200 3.1 3.2 3.3 3.4 3.5 Linear Equations and Rectangular Coordinates •.................................................... Graphing Linear Equations in Two Variables • ...................................................... The Slope of a Line •............................................................................................... Slope-Intercept Form of a Linear Equation • .......................................................... Point-Slope Form of a Linear Equation and Modeling • ......................................... Summary Exercises Applying Graphing and Equation-Writing Techniques for Lines • ........................................................................................................ Chapter 3 Review Exercises • ................................................................................. Chapter 3 Mixed Review Exercises • ...................................................................... Chapter 3 Test •....................................................................................................... Chapters R–3 Cumulative Review Exercises •........................................................ 200 212 230 239 251 260 264 269 271 273

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    4 Exponents and Polynomials .................................................................................................... 276 4.1 4.2 4.3 4.4 4.5 4.6 4.7 The Product Rule and Power Rules for Exponents •................................................ Integer Exponents and the Quotient Rule • .............................................................. Summary Exercises Applying the Rules for Exponents •.......................................... Scientific Notation • ................................................................................................. Adding, Subtracting, and Graphing Polynomials • .................................................. Multiplying Polynomials • ....................................................................................... Special Products •..................................................................................................... Dividing Polynomials • ............................................................................................ Chapter 4 Review Exercises • .................................................................................. Chapter 4 Mixed Review Exercises • ....................................................................... Chapter 4 Test •........................................................................................................ Chapters R–4 Cumulative Review Exercises •......................................................... 276 283 291 295 304 317 327 335 350 356 358 361 5 Factoring and Applications.......................................................................................................364 5.1 5.2 5.3 5.4 5.5 5.6 The Greatest Common Factor; Factoring by Grouping • ........................................ Factoring Trinomials • ............................................................................................ More on Factoring Trinomials •.............................................................................. Special Factoring Techniques •............................................................................... Summary Exercises Recognizing and Applying Factoring Strategies • .................. Solving Quadratic Equations Using the Zero-Factor Property • ............................. Applications of Quadratic Equations •.................................................................... Chapter 5 Review Exercises •................................................................................. Chapter 5 Mixed Review Exercises •........................................................................ Chapter 5 Test • ...................................................................................................... Chapters R–5 Cumulative Review Exercises • ....................................................... 364 372 381 393 405 409 420 430 435 437 439 6 Rational Expressions and Applications ...................................................................................442 6.1 6.2 6.3 6.4 6.5 6.6 6.7 The Fundamental Property of Rational Expressions •............................................ Multiplying and Dividing Rational Expressions •.................................................. Least Common Denominators • ............................................................................. Adding and Subtracting Rational Expressions •..................................................... Complex Fractions •............................................................................................... Solving Equations with Rational Expressions • ..................................................... Summary Exercises Simplifying Rational Expressions vs. Solving Rational Equations •......................................................................................... Applications of Rational Expressions •.................................................................. Chapter 6 Review Exercises • ................................................................................ Chapter 6 Mixed Review Exercises •..................................................................... Chapter 6 Test •...................................................................................................... Chapters R–6 Cumulative Review Exercises •....................................................... 442 455 465 475 492 507 529 532 544 550 553 556

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    7 Graphs, Linear Equations, and Systems .................................................................................562 7.1 7.2 7.3 7.4 7.5 7.6 7.7 Review of Graphs and Slopes of Lines •................................................................ Review of Equations of Lines; Linear Models • .................................................... Solving Systems of Linear Equations by Graphing • ............................................. Solving Systems of Linear Equations by Substitution •......................................... Solving Systems of Linear Equations by Elimination • ......................................... Summary Exercises Applying Techniques for Solving Systems of Linear Equations • ..................................................................................................... Systems of Linear Equations in Three Variables •................................................. Applications of Systems of Linear Equations •............................ ................ .......... Chapter 7 Review Exercises • ................................................................................ Chapter 7 Mixed Review Exercises •..................................................................... Chapter 7 Test •...................................................................................................... Chapters R–7 Cumulative Review Exercises •....................................................... 562 581 596 607 618 628 634 653 673 684 686 691 8 Inequalities and Absolute Value...............................................................................................696 8.1 8.2 8.3 8.4 Review of Linear Equations in One Variable • ...................................................... Set Operations and Compound Inequalities •......................................................... Absolute Value Equations and Inequalities • ......................................................... Summary Exercises Solving Linear and Absolute Value Equations and Inequalities •................................................................................................... Linear Inequalities in Two Variables •................................................................... Chapter 8 Review Exercises • ................................................................................ Chapter 8 Mixed Review Exercises •..................................................................... Chapter 8 Test •...................................................................................................... Chapters R–8 Cumulative Review Exercises •....................................................... 696 705 712 727 731 744 747 749 751 9 Relations and Functions ............................................................................................................756 9.1 9.2 9.3 9.4 Introduction to Relations and Functions • .............................................................. Function Notation and Linear Functions • ............................................................. Polynomial Functions, Operations, and Composition •.......................................... Variation • .............................................................................................................. Chapter 9 Review Exercises • ................................................................................ Chapter 9 Mixed Review Exercises •..................................................................... Chapter 9 Test •...................................................................................................... Chapters R–9 Cumulative Review Exercises •....................................................... 756 762 770 782 792 795 796 797

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    10 Roots, Radicals, and Root Functions .......................................................................................803 10.1 10.2 10.3 10.4 10.5 10.6 10.7 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 10 Review Exercises • .............................................................................. Chapter 10 Mixed Review Exercises •................................................................... Chapter 10 Test •.................................................................................................... Chapters R − 10 Cumulative Review Exercises • .................................................... 803 813 821 834 840 852 855 867 874 882 884 887 11 Quadratic Equations, Inequalities, and Functions .............................................................. 892 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 Solving Quadratic Equations by the Square Root Property • ................................... Solving Quadratic Equations by Completing the Square •....................................... Solving Quadratic Equations by the Quadratic Formula • ....................................... Solving Equations Quadratic in Form •.................................................................... Summary Exercises Applying Methods for Solving Quadratic Equations • ............. Formulas and Further Applications • ....................................................................... Graphs of Quadratic Functions • .............................................................................. More About Parabolas and Their Applications •...................................................... Polynomial and Rational Inequalities • .................................................................... 892 901 920 931 956 961 975 984 999 Chapter 11 Review Exercises • .............................................................................. Chapter 11 Mixed Review Exercises • ................................................................... Chapter 11 Test •.................................................................................................... Chapters R–11 Cumulative Review Exercises •..................................................... 1021 1032 1036 1041 12 Inverse, Exponential, and Logarithmic Functions................................................................1047 12.1 12.2 12.3 12.4 12.5 12.6 Inverse Functions •.................................................................................................. Exponential Functions • .......................................................................................... Logarithmic Functions •.......................................................................................... Properties of Logarithms • ...................................................................................... Common and Natural Logarithms • ........................................................................ Exponential and Logarithmic Equations; Further Applications ............................. Chapter 12 Review Exercises • ............................................................................... Chapter 12 Mixed Review Exercises • .................................................................... Chapter 12 Test •..................................................................................................... Chapters R–12 Cumulative Review Exercises •...................................................... 1047 1057 1065 1077 1083 1088 1100 1107 1109 1111

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    Chapter 1 The Real Number System 26 Chapter 1 The Real Number System 1.1 Exponents, Order of Operations, and Inequality Classroom Examples, Now Try Exercises 1. (a) 2 9 81 9 9 = ⋅ = (b) 4 1 1 1 1 16 1 is us 1 ed as a factor 4 times. 2 1 2 2 2 2 2 ⎛ ⎞ ⋅ ⋅ = ⎠ = ⎜ ⎟ ⎝ ⋅  N1. (a) 2 6 6 6 36 = ⋅ = (b) 3 4 4 4 4 64 5 5 5 5 125 4 is used as a factor 3 times. 5 ⋅ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ⋅ =  2. (a) 10 6 2 10 3 Divide. 7 Subtract. − ÷ = − = (b) ( ) ( ) 18 2 6 3 18 2 3 Subtract inside parentheses. 18 6 Multiply. 24 Add. + − = + = + = (c) ( ) ( ) 7 6 3 8 1 7 6 3 9 Add inside parentheses. 42 27 Multiply. 15 Subtract. ⋅ − + = ⋅ − = − = (d) 2 2 3 2 2 9 5 2 Apply exponents. 2 9 10 Multiply. 11 10 Add. 1 Subtrac 5 t. + ⋅ = + − ⋅ = + − = − = − N2. (a) 15 2 6 15 12 Multiply. 3 Subtract. − ⋅ = − = (b) ( ) ( ) 8 2 5 1 8 2 4 Subtract inside parentheses. 8 8 Multiply. 16 Add. + − = + = + = (c) ( ) ( ) 6 2 4 7 5 6 6 7 5 Add inside parentheses. 36 35 Multiply. 1 Subtract. + − ⋅ = − ⋅ = − = (d) 3 2 8 10 4 2 3 4 8 10 4 8 3 16 Apply exponents. 80 4 8 48 Multiply. 20 8 48 Divide. 12 48 Subtract. 60 Add. ⋅ ÷ − + ⋅ = ⋅ ÷ − + ⋅ = ÷ − + = − + = + = 3. (a) ( ) [ ] [ ] 9 12 3 Add inside parentheses. 9 9 Subtra 9 4 8 3 inside br ct 81 M acket ultip y s. l . = − = = + − ⎡ ⎤ ⎣ ⎦ (b) 2(7 8) 2 3 5 1 2(15) 2 Add inside parentheses. 3 5 1 30 2 Multiply. 15 1 3 16 2 Add. Divide 2 . = + + ⋅ + + = ⋅ + + = = + N3. (a) ( ) [ ] 2 7 3 1 4 7[(9 1) 4] Apply exponents. 7[8 4] Subtract inside parentheses. 7 12 Add inside brackets. 84 Multiply. ⎡ ⎤ − + ⎣ ⎦ = − + = + = =

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    1.1 Exponents, Order of Operations, and Inequality 27 (b) 9(14 4) 2 4 3 6 9(10) 2 Subtract inside parentheses. 4 3 6 90 2 Multiply. 4 18 88 Subtract and add. 22 4 Divide. − − + ⋅ − = + ⋅ − + = = = 4. (a) The statement 12 6 > is true since 12 is greater than 6. Note that the inequality symbol points to the lesser number. (b) The statement 28 4 7 ≠ ⋅ is false because 28 is equal to 4 7. ⋅ (c) The statement 21 21 ≤ is true since 21 21. = (d) Write the fractions with a common denominator. The statement 1 1 3 4 < is equivalent to the statement 4 3 . 12 12 < Since 4 is greater than 3, the original statement is false. N4. (a) The statement 12 10 2 ≠ − is true because 12 is not equal to 8. (b) The statement 5 4 2 > ⋅ is false because 5 is less than 8. (c) The statement 7 7 ≤ is true since 7 . 7 = (d) Write the fractions with a common denominator. The statement 5 7 9 11 > is equivalent to the statement 55 63 . 99 99 > Since 55 is less than 63, the original statement is false. 5. (a) “Nine is equal to eleven minus two” is written as 9 11 2. = − (b) “Fourteen is greater than twelve” is written as 14 12. > (c) “Two is greater than or equal to two” is written as 2 2. ≥ N5. (a) “Ten is not equal to eight minus two” is written as 10 8 2. ≠ − (b) “Fifty is greater than fifteen” is written as 50 . 15 > (c) “Eleven is less than or equal to twenty” is written as 11 . 20 ≤ 6. 9 15 ≤ may be written as 15 . 9 ≥ N6. 8 9 < may be written as 9 . 8 > Exercises 1. False; 2 3 3 3 9. = ⋅ = 2. False; 1 raised to any power is 1. Here, 3 1 1 1 1 1. = ⋅ ⋅ = 3. False; a number raised to the first power is that number, so 1 3 3. = 4. False; 2 6 means that 6 is used as a factor 2 times, so 2 6 6 6 36. = ⋅ = 5. False; the common error leading to 42 is adding 4 to 3 and then multiplying by 6. One must follow the rules for order of operations. ( ) 4 3(8 2) 4 3 6 4 18 22 + − = + = + = 6. False; multiplications and divisions are performed in order from left to right. 12 2 3 6 3 18 ÷ ⋅ = ⋅ = 7. Additions and subtractions are performed in order from left to right. N N 1 2 18 2 3 − + 8. Multiplications and divisions are performed in order from left to right, and then additions and subtractions are performed in order from left to right. N N 2 1 28 6 2 − ÷ 9. Multiplications and divisions are performed in order from left to right, and then additions and subtractions are performed in order from left to right. N N N 3 2 1 2 8 6 3 ⋅ − ÷

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    Chapter 1 The Real Number System 28 10. Multiplications and divisions are performed in order from left to right, and then additions and subtractions are performed in order from left to right. If grouping symbols are present, work within them first, starting with the innermost. N N N 3 1 2 40 6( 3 1) + − 11. Multiplications and divisions are performed in order from left to right, and then additions and subtractions are performed in order from left to right. If grouping symbols are present, work within them first, starting with the innermost. N N N N 4 1 2 3 3 5 2( 4 2) ⋅ − + 12. Apply all exponents. Then, multiplications and divisions are performed in order from left to right, and additions and subtractions are performed in order from left to right. N N N N 3 3 1 4 2 9 2 3 4 − + ⋅ 13. 2 3 3 3 9 = ⋅ = 14. 2 8 64 8 8 = ⋅ = 15. 2 7 7 7 49 = ⋅ = 16. 2 4 16 4 4 = ⋅ = 17. 2 12 12 12 144 = ⋅ = 18. 2 14 1 4 9 1 14 6 = ⋅ = 19. 3 4 4 4 4 64 = ⋅ ⋅ = 20. 3 5 5 125 5 5 = ⋅ ⋅ = 21. 3 10 10 10 10 1000 = ⋅ ⋅ = 22. 3 11 11 11 11 1331 = ⋅ ⋅ = 23. 4 3 3 3 3 3 81 = ⋅ ⋅ ⋅ = 24. 4 6 6 6 6 1 9 6 2 6 = ⋅ ⋅ ⋅ = 25. 5 4 4 4 4 4 4 1024 = ⋅ ⋅ ⋅ ⋅ = 26. 5 3 3 3 3 3 24 3 3 = ⋅ ⋅ ⋅ ⋅ = 27. 2 1 1 1 1 6 6 6 36 = ⎟ ⋅ ⎞ ⎠ = ⎛ ⎜ ⎝ 28. 2 1 1 1 1 3 3 3 9 = ⎟ ⎠ ⋅ ⎞ ⎜ = ⎛ ⎝ 29. 4 2 2 2 2 2 16 3 3 3 3 3 81 = ⋅ ⋅ ⋅ ⎛ ⎜ = ⎞ ⎟ ⎝ ⎠ 30. 3 3 4 4 3 3 4 3 4 2 6 7 4 = ⋅ ⋅ ⎟ ⎠ = ⎛ ⎞ ⎜ ⎝ 31. ( ) ( )( )( ) 3 0.4 0.4 0.4 0.4 0.064 = = 32. ( ) ( )( ) ( )( ) 4 0.5 0.5 0.5 0. 0.0625 5 0.5 = = 33. Divide. 64 4 2 16 2 32 Multiply. ÷ ⋅ = ⋅ = 34. Divide. 250 5 2 50 2 100 Multiply. ÷ ⋅ = ⋅ = 35. 13 9 5 13 45 Multiply. 58 Add. + ⋅ = + = 36. 11 7 6 11 42 Multiply. 53 Add. + ⋅ = + = 37. 25.2 12.6 4.2 25.2 3 Divide. 22.2 Subtract. = − ÷ = − 38. 12.4 9.3 3.1 12.4 3 D 9. ivide. Subtract 4 . − ÷ = − = 39. 1 2 2 11 1 22 4 3 5 3 6 15 5 44 30 30 30 49 19 , or 1 3 Multiply. LC 0 3 d. 0 D Ad ⋅ + ⋅ = + = = = + 40. 2 4 5 3 4 4 3 5 3 9 8 6 17 5 , or 2 9 Multiply. 2 3 LCD 6 6 Add. 6 6 ⋅ + ⋅ = = = + + = 41. 9 4 8 3 36 24 Multiply. 12 Subtract. = ⋅ − ⋅ = − 42. 11 4 10 3 44 30 Multipl 7 y. Add. 4 = ⋅ + ⋅ = + 43. 20 4 3 5 20 12 5 Multiply. 8 5 Subtract. 13 Add. − ⋅ + = − + = + =

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    1.1 Exponents, Order of Operations, and Inequality 29 44. 18 7 2 6 18 14 6 Multiply. 4 6 Subtract. 10 Add. − ⋅ + = − + = + = 45. 10 40 5 2 10 8 2 Divide. 10 16 Multiply. 26 Add. + ÷ ⋅ = = + = ⋅ + 46. 12 64 8 4 12 8 4 Divide. 20 4 Add. 16 Subtract. + ÷ − = + − = − = 47. 18 2(3 4) 18 2(7) Add inside parentheses. 18 14 Multiply. 4 Subtract. − = + = − − = 48. ( ) ( ) 30 3 4 2 30 3 6 Add inside parentheses. 18 Multiply. 12 Subtrac 0 . 3 t − + = − − = = 49. 3(4 2) 8 3 3 6 8 3 Add. 18 24 Multiply. 42 Add. = + + ⋅ = ⋅ + + = ⋅ 50. ( ) Add. 7 9 1 2 10 Multiply. 82 7 2 5 9 8 2 Ad . 5 d = + + = + ⋅ = ⋅ + ⋅ 51. 2 18 4 3 18 16 3 Apply exponents. 2 3 Subtract. 5 Add. − + = − + = + = 52. 3 22 2 9 22 8 9 Apply exponents. 14 9 Subtract. 23 Add. − + = − + = + = 53. 2 3[5 4(2)] 2 3[5 8] Multiply. 2 3[13] Add. 2 39 Multiply. 41 Add. + + = + + = + = + = 54. ( ) [ ] [ ] 5 4 1 7 3 5 4 1 21 Multiply. 5 4 22 Add. 5 88 Multiply. 93 Add. + + = + + ⎡ ⎤ ⎣ ⎦ = + = + = 55. ( ) 2 5 3 4 2 5[3 4(4)] Apply exponents. 5(3 16) Multiply. 5(19) Add. 95 Multiply. ⎡ ⎤ + = + ⎣ ⎦ = + = = 56. ( ) [ ] 3 6 2 8 3 6 2 8 27 Apply exponents. 6(2 216) Multiply. 6 218 Add. 1308 Multiply. ⎡ ⎤ + ⎣ ⎦ = + ⋅ = + = ⋅ = 57. 2 2 2 3 [(11 3) 4] 3 [14 4] Add inside parentheses. 3 [10] Subtract. 9[10] Apply exponents. 90 Multiply. + − = − = = = 58. [ ] [ ] [ ] 2 2 2 4 9 [(13 4) 8] 4 17 8 Add inside parentheses. 4 Subtract. 16 9 Apply exponents. 144 Multiply. + − = − = = = 59. Simplify the numerator and denominator separately, and then divide. ( ) 2 2 6 3 1 8 6(9 1) 8 8 4 8 2 6(8) 8 4 48 8 4 56 14 4 − + − + = − − + = + = = = 60. Simplify the numerator and denominator separately, and then divide. ( ) ( ) ( ) 2 3 2 4 8 2 64 4 8 29 27 29 3 2 60 8 120 8 8 2 6 2 12 2 4 8 − + − + = − − + = + = = =

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    Chapter 1 The Real Number System 30 61. Simplify the numerator and denominator separately, and then divide. 2 2 4(6 2) 8(8 3) 4(8) 8(5) 6(4 2) 2 6(2) 2 4(8) 8(5) 6(2) 4 32 40 12 4 72 9 8 + + − + = − − − + = − + = − = = 62. Simplify the numerator and denominator separately, and then divide. 3 3 6(5 1) 9(1 1) 6(6) 9(2) 5(8 6) 2 5(2) 2 36 18 10 8 18 2 9 + − + − = − − − − = − = = 63. 3 6 4 2 60 ⋅ + ⋅ = Listed below are some possibilities. Use trial and error until you get the desired result. (3 6) 4 2 18 8 26 60 (3 6 4) 2 22 2 44 60 3 (6 4 2) 3 14 42 60 3 (6 4) 2 3 10 2 30 2 60 ⋅ + ⋅ = + = ≠ ⋅ + ⋅ = ⋅ = ≠ ⋅ + ⋅ = ⋅ = ≠ ⋅ + ⋅ = ⋅ ⋅ = ⋅ = 64. 2 8 1 3 42 ⋅ − ⋅ = ( ) 2 8 1 3 2 7 3 14 3 42 ⋅ − ⋅ = ⋅ ⋅ = ⋅ = 65. 10 7 3 6 − − = 10 (7 3) 10 4 6 − − = − = 66. 15 10 2 7 − − = 15 (10 2) 15 8 7 − − = − = 67. 2 8 2 100 + = ( ) 2 2 8 2 10 10 10 100 + = = ⋅ = 68. 2 4 2 36 + = ( ) 2 2 4 2 6 6 6 36 + = = ⋅ = 69. 9 3 11 16 27 11 16 16 16 ⋅ − ≤ − ≤ ≤ The statement is true since 16 16. = 70. 6 5 12 18 30 12 18 18 18 ⋅ − ≤ − ≤ ≤ The statement is true since 18 18. = 71. 5 11 2 3 60 55 6 60 61 60 ⋅ + ⋅ ≤ + ≤ ≤ The statement is false since 61 is greater than 60. 72. 9 3 4 5 48 27 20 48 47 48 ⋅ + ⋅ ≥ + ≥ ≥ The statement is false since 47 is less than 48. 73. 0 12 3 6 6 0 36 36 0 0 ≥ ⋅ − ⋅ ≥ − ≥ The statement is true since 0 0. = 74. 10 13 2 15 1 10 26 15 10 11 ≤ ⋅ − ⋅ ≤ − ≤ The statement is true since 10 11. < 75. 45 2[2 3(2 5)] 45 2[2 3(7)] 45 2[2 21] 45 2[23] 45 46 ≥ + + ≥ + ≥ + ≥ ≥ The statement is false since 45 is less than 46. 76. 55 3[4 3(4 1)] 55 3[4 3(5)] 55 3[4 15] 55 3[19] 55 57 ≥ + + ≥ + ≥ + ≥ ≥ The statement is false since 55 is less than 57. 77. [3 4 5(2)] 3 72 [12 10] 3 72 [22] 3 72 66 72 ⋅ + ⋅ > + ⋅ > ⋅ > > The statement is false since 66 is less than 72.

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    1.1 Exponents, Order of Operations, and Inequality 31 78. 2 [7 5 3(2)] 58 2 [35 6] 58 2[29] 58 58 58 ⋅ ⋅ − ≤ ⋅ − ≤ ≤ ≤ The statement is true since 58 58. = 79. 3 5(4 1) 3 2 4 1 3 5(3) 3 8 1 3 15 3 9 18 3 9 2 3 + − ≥ ⋅ + + ≥ + + ≥ ≥ ≥ The statement is false since 2 is less than 3. 80. ( ) ( ) 7 3 1 2 2 3 5 2 7 4 2 2 3 10 28 2 2 13 26 2 13 2 2 + − ≤ + ⋅ − ≤ + − ≤ ≤ ≤ The statement is true since 2 2. = 81. 2(5 1) 3(1 1) 3 5(8 6) 4 2 2(6) 3(2) 3 5(2) 8 12 6 3 10 8 6 3 2 3 3 + − + ≥ − − ⋅ − ≥ − − ≥ − ≥ ≥ The statement is true since 3 3. = 82. ( ) 3(8 3) 2(4 1) 7 9(6 2) 11 5 2 3(5) 2(3) 7 9(4) 11(3) 15 6 7 36 33 21 7 3 7 7 − + − ≤ − − − + ≤ − + ≤ − ≤ ≤ The statement is true since 7 7. = 83. “5 ” 17 < means “five is less than seventeen.” The statement is true. 84. “8 ” 12 < means “eight is less than twelve.” The statement is true. 85. “5 ” 8 ≠ means “five is not equal to eight.” The statement is true. 86. “6 ” 9 ≠ means “six is not equal to nine.” The statement is true. 87. “7 ” 14 ≥ means “seven is greater than or equal to fourteen.” The statement is false. 88. “6 ” 12 ≥ means “six is greater than or equal to twelve.” The statement is false. 89. “15 ” 15 ≤ means “fifteen is less than or equal to fifteen.” The statement is true. 90. “21 ” 21 ≤ means “twenty-one is less than or equal to twenty-one.” The statement is true. 91. “ 1 3 ” 3 10 = means “one-third is equal to three- tenths.” The statement is false. 92. “ 10 6 ” 3 2 = means “ten-sixths is equal to three- halves.” The statement is false. 93. 2.5 2 “ ” .50 > means “two and five-tenths is greater than two and fifty-hundredths.” The statement is false. 94. 1.80 “ ” 1.8 > means “one and eighty-hundredths is greater than one and eight-tenths.” The statement is false. 95. “Fifteen is equal to five plus ten” is written as 15 5 10. = + 96. “Twelve is equal to twenty minus eight” is written as 12 20 8. = − 97. “Nine is greater than five minus four” is written as 9 5 4. > − 98. “Ten is greater than six plus one” is written as 10 6 1. > + 99. “Sixteen is not equal to nineteen” is written as 16 19. ≠ 100. “Three is not equal to four” is written as 3 4. ≠ 101. “One-half is less than or equal to two-fourths” is written as 1 2 . 2 4 ≤

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    Chapter 1 The Real Number System 32 102. “One-third is less than or equal to three-ninths” is written as 1 3 . 3 9 ≤ 103. 5 20 < becomes 20 5 > when the inequality symbol is reversed. 104. 30 9 > becomes 9 30 < when the inequality symbol is reversed. 105. 4 3 5 4 > becomes 3 4 4 5 < when the inequality symbol is reversed. 106. 5 3 4 2 < becomes 3 5 2 4 > when the inequality symbol is reversed. 107. 2.5 1.3 ≥ becomes 1.3 2.5 ≤ when the inequality symbol is reversed. 108. 4.1 5.3 ≤ becomes 5.3 4.1 ≥ when the inequality symbol is reversed. 109. (a) Substitute “40” for “age” in the expression for women. 14.7 40 0.13 − ⋅ (b) 14.7 40 0.13 14.7 5.2 Multiply 9. . Subtract 5 . = − ⋅ = − (c) 85% of 9.5 is 0.85(9.5) 8.075. = Walking at 5 mph is associated with 8.0 METs, which is the table value closest to 8.075. (d) Substitute “55” for “age” in the expression for men. 14.7 55 0.11 − ⋅ 14.7 55 0.11 14.7 6.05 M 8 ultiply. Subtract. .65 − ⋅ = − = 85% of 8.65 is 0.85(8.65) 7.3525. = Swimming is associated with 7.0 METs, which is the table value closest to 7.3525. 110. Answers will vary. 111. The states that had a number greater than 13.8 are Alaska ( ) 16.2 , Texas ( ) 14.7 , California ( ) 24.1 , and Idaho ( ) 17.6 . 112. The states that had a number that was at most 14.7 are Texas ( ) 14.7 , Wyoming ( ) 12.5 , Maine ( ) 12.3 , and Missouri ( ) 13.8 . 113. The states that had a number not less than 13.8, which is the same as greater than or equal to 13.8, are Alaska ( ) 16.2 , Texas ( ) 14.7 , California ( ) 24.1 , Idaho ( ) 17.6 , and Missouri ( ) 13.8 . 114. The states that had a number less than 13.0 are Wyoming ( ) 12.5 and Maine ( ) 12.3 . 1.2 Variables, Expressions, and Equations Classroom Examples, Now Try Exercises 1. (a) 16 3 Repla 1 ce with 3. 48 Multiply. 6 16 p p p ⋅ = ⋅ = = (b) 3 3 3 2 2 2 3 Replace with 3. 2 27 Cube 3. 54 Multiply. p p p = ⋅ = ⋅ = ⋅ = N1. (a) 9 9 6 Replace with 6. 54 Multipl 9 y. x x x = ⋅ = ⋅ = (b) 2 2 2 4 Replace with 6. 4 6 Square 6. 4 36 Multiply. 144 4 x x x = ⋅ = ⋅ = ⋅ = 2. (a) 4 5 4(6) 5(9) 24 45 Multipl 69 Add. y. x y + = + + = = (b) 4 2 4(6) 2(9) 1 6 1 24 18 Multiply. 6 1 Sub 6 tract a dd 7 nd a x y x . = + − + = − − = + (c) 2 2 2 2 2 6 36 81 Use exponents. Multiply. 153 A 2 9 2 7 81 d . 2 d x y + = + = = + = ⋅ ⋅ +

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    1.2 Variables, Expressions, and Equations 33 N2. (a) 3 4 3(4) 4(7) 12 28 Multipl 40 Add. y. x y + = + + = = (b) 6 2 6(4) 2(7) 2 9 2(7) 9 24 14 Multiply. 14 9 10 2 Subtract; reduce. 5 x y y − − = − − − − = = = (c) 2 2 2 2 4 4 4 7 4 16 49 Use exponents. 64 49 Multiply. 15 Subtract. x y = = = = − ⋅ − ⋅ − − 3. (a) Since a number is subtracted from 48, write this as 48 x − when using x as the variable to represent the number. (b) “Product” indicates multiplication. Using x as the variable to represent the number, “the product of 6 and a number” translates as 6 x ⋅ or 6 x . (c) “The sum of a number and 5” suggests a number plus 5. Using x as the variable to represent the number, “9 multiplied by the sum of a number and 5” translates as ( ) 9 5 . x + N3. (a) Using x as the variable to represent the number, “the sum of a number and 10” translates as 0, 1 x + or 10 . x + (b) “A number divided by 7” translates as 7, x ÷ or . 7 x (c) “The difference between 9 and a number” translates as 9 . x − Thus, “the product of 3 and the difference between 9 and a number” translates as 3(9 . ) x − 4. ? ? 8 11 5 8 2 11 Replace with 2. 16 11 M 5 5 5 5 True ultiply. p p − = ⋅ − = − = = The number 2 is a solution of the equation. N4. ? ? 8 5 61 8 7 5 Replace with 61 61 7. 56 5 Multiply. 61 61 True k k = = + = ⋅ + = + The number 7 is a solution of the equation. 5. Using x as the variable to represent the number, “three times a number is subtracted from 21, giving 15” translates as 21 3 15. x − = Now try each number from the set {0, 2, 4, 6, 8, 10}. ( ) ? 15 0 : 21 3 0 21 15 False x − = = = ( ) ? 15 2 : 21 3 2 15 15 True x − = = = ( ) ? 15 4 : 21 3 4 9 15 False x = − = = Similarly, 6, 8, x = or 10 result in false statements. Thus, 2 is the only solution. N5. Using x as the variable to represent the number, “the sum of a number and nine is equal to the difference between 25 and the number” translates as 9 25 . x x + = − Now try each number from the set {0, 2, 4, 6, 8, 10}. ? 25 4 : 4 9 13 21 Fals 4 e x − = + = = ? 25 6 : 6 9 15 19 Fals 6 e x − = + = = ? 2 8 : 8 9 17 17 True 5 8 x − = + = = Similarly, 0, 2, x = or 10 result in false statements. Thus, 8 is the only solution. 6. (a) 3 1 5 x − has no equals symbol, so this is an expression. (b) 3 1 5 x = has an equals symbol, so this is an equation. N6. (a) 2 5 6 x + = has an equals symbol, so this is an equation. (b) 2 5 6 x + − has no equals symbol, so this is an expression .

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    Chapter 1 The Real Number System 34 Exercises 1. The expression 2 8 x means 8 . x x ⋅ ⋅ The correct choice is B. 2. If 2 x = and 1, y = then the value of xy is 2 1 2. ⋅ = The correct choice is C. 3. The sum of 15 and a number x is represented by the expression 15 . x + The correct choice is A. 4. There is no equals symbol in 6 7 x + or 6 7, x − so those are expressions. 6 7 x = and 6 7 0 x − = have equals symbols, so those are equations. 5. 3 2 2 , x x x x = ⋅ ⋅ ⋅ while 3 2 2 2 (2 ) . x x x x ⋅ ⋅ = The last expression is equal to 3 8 . x 6. “7 less than a number” is an expression indicating subtraction, 7, x − while “7 is less than a number” is a statement relating 7 and x , 7 . x < 7. The exponent 2 applies only to its base, which is x . (The expression 2 (5 ) x would require multiplying 5 by 4 x = first.) 8. An expression cannot be solved—it indicates a series of operations to perform. An expression is simplified; an equation is solved. 9. (a) 7 4 7 11 x + = + = (b) 7 6 7 13 x + = + = 10. (a) 3 4 3 1 x − = − = (b) 3 6 3 3 x − = − = 11. (a) 4 4(4) 16 x = = (b) 4 4(6) 24 x = = 12. (a) ( ) 6 6 4 24 x = = (b) ( ) 6 6 6 36 x = = 13. (a) 2 2 4 4 4 4 16 64 x = ⋅ = ⋅ = (b) 2 2 4 4 6 4 36 144 x = ⋅ = ⋅ = 14. (a) 2 2 5 5 4 5 16 80 x = ⋅ = ⋅ = (b) 2 2 5 5 6 5 36 180 x = ⋅ = ⋅ = 15. (a) 1 4 1 3 3 5 3 x + + = = (b) 1 6 1 3 3 7 3 x + + = = 16. (a) 2 4 2 5 2 5 5 x − − = = (b) 2 6 2 4 5 5 5 x − − = = 17. (a) 3 5 3 4 5 2 2 4 12 5 8 7 8 x x − ⋅ − = ⋅ − = = (b) 3 5 3 6 5 2 2 6 18 5 12 13 12 x x − ⋅ − = ⋅ − = =

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    1.2 Variables, Expressions, and Equations 35 18. (a) ( ) ( ) 4 4 1 4 1 3 3 4 16 1 12 15 5 12 4 x x − − = − = = = (b) ( ) ( ) 4 6 1 4 1 3 3 6 24 1 18 23 18 x x − − = − = = 19. (a) 2 2 3 3 4 4 3 16 4 48 4 52 x x + = ⋅ + = ⋅ + = + = (b) 2 2 3 3 6 6 3 36 6 108 6 114 x x + = ⋅ + = ⋅ + = + = 20. (a) ( ) 2 2 2 2 4 4 8 16 24 x x + = + = + = (b) ( ) 2 2 2 2 6 6 12 36 48 x x + = + = + = 21. (a) 6.459 6.459 4 25.836 x = ⋅ = (b) 6.459 6.459 6 38.754 x = ⋅ = 22. (a) 3.275 3.275 4 13.1 x = ⋅ = (b) 3.275 3.275 6 19.65 x = ⋅ = 23. (a) 8 3 5 8(2) 3(1) 5 16 3 5 19 5 24 x y + + = + + = + + = + = (b) 8 3 5 8(1) 3(5) 5 8 15 5 23 5 28 x y + + = + + = + + = + = 24. (a) 4 2 7 4(2) 2(1) 7 8 2 7 17 x y + + = + + = + + = (b) 4 2 7 4(1) 2(5) 7 4 10 7 21 x y + + = + + = + + = 25. (a) 3( 2 ) 3(2 2 1) 3(2 2) 3(4) 12 x y + = + ⋅ = + = = (b) 3( 2 ) 3(1 2 5) 3(1 10) 3(11) 33 x y + = + ⋅ = + = = 26. (a) ( ) ( ) ( ) 2(2 ) 2 2 2 1 2 4 1 2 5 10 x y + = + ⎡ ⎤ ⎣ ⎦ = + = = (b) ( ) ( ) ( ) 2(2 ) 2 2 1 5 2 2 5 2 7 14 x y + = + ⎡ ⎤ ⎣ ⎦ = + = = 27. (a) 4 4 2 1 2 4 6 x y + = + = + = (b) 4 4 1 5 5 4 5 5 9 5 x y + = + = + = 28. (a) 8 8 2 1 1 4 5 y x + = + = + = (b) 1 8 8 5 5 8 13 y x + = + = + =

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