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Math 0308 Final Exam Review(answers) Solve the given equations. 1. 3 x + 14 = 8 x − 1 15 = 5 x 2. 3 ( 4 x − 1) = 5 x + 6 12 x − 3 = 5 x + 6 7x = 9 4 = 9x 9 7 4 =x 9 4 9 x= 3= x 3 9 7 4. 5 ( 2 x + 1) − 3 ( x + 4 ) = −21 10 x + 5 − 3 x − 12 = −21 7 x − 7 = −21 7 x = −14 x = −2 −2 5. 3 ( 4 x + 5 ) − 5 ( 2 x − 7 ) = 2 ( 6 x + 1) 12 x + 15 − 10 x + 35 = 12 x + 2 2 x + 50 = 12 x + 2 48 = 10 x 48 =x 10 24 5 6. 2 ( 3 x + 1) + 4 ( 2 x − 2 ) = 2 ( 7 x − 3) 7. 5 ⎡⎣ 2 ( 3 x − 1) − 6 ( x + 2 ) ⎤⎦ = 3 x − 7 5[ 6 x − 2 − 6 x − 12] = 3 x − 7 6 x + 2 + 8 x − 8 = 14 x − 6 5 ⋅ −14 = 3 x − 7 14 x − 6 = 14 x − 6 −70 = 3 x − 7 −6 = −6 −63 = 3 x all real numbers −21 = x −21 8. 3 x + 14 = 3 ( x − 2 ) + 20 3 x + 14 = 3 x − 6 + 20 3. 8 − ( 3 x + 4 ) = 6 x 8 − 3x − 4 = 6 x 9. 3 x + 14 = 5 ( x − 2 ) − 2 ( x + 7 ) 3 x + 14 = 5 x − 10 − 2 x − 14 3 x + 14 = 3 x + 14 3 x + 14 = 3 x − 24 14 = 14 14 = −24 all real numbers no solution Solve the following inequalities. Graph your solution on a number line. 10. x − 7 ≤ −4 11. 4 x + 9 > 3 12. 1 − 9 x ≥ 4 −9 x ≥ 3 4 x > −6 x≤3 x>− 3 2 x≤− 1 3 3 − 13. 2 x + 15 > 7 x − 1 −5 x > −16 x< 3 2 1 3 15. −4 < 4 − 2 x < 6 −8 < −2 x < 2 − 14. −1 < 3 x + 8 ≤ 5 −9 < 3 x < −3 16 5 4 > x > −1 −3 < x < −1 16 5 −3 −1 < x < 4 −1 −1 Use equations and algebraic methods to find solutions of the following problems. 16. Five minus 4 times a number is 3. What is the number? 5 − 4x = 3 −4 x = −2 x= 1 2 1 2 17. Find three consecutive odd integers whose sum is −21 . x + ( x + 2 ) + ( x + 4 ) = −21 3 x + 6 = −21 3 x = −27 x = −9 −9, −7, −5 4 18. A 43 inch board is cut into three pieces. The second piece is one-half as long as the first piece. The third piece is 7 inches longer than the second piece. How long is each piece? x + ( 12 x ) + ( 12 x + 7 ) = 43 2 x + 7 = 43 2 x = 36 x = 18 18",9",16" 19. The perimeter of a rectangle is 36 feet. The length is 2 feet more than 3 times the width. What are the dimensions of the rectangle? L is the length; W is the width. 2 L + 2W = 36 L = 2 + 3W 2 ( 2 + 3W ) + 2W = 36 4 + 6W + 2W = 36 8W + 4 = 36 8W = 32 W = 4' L = 14' 20. Michael has $7.95 in dimes and quarters. He has a total of 39 coins. How many of each type of coin does he have? D is the number of dimes; Q is the number of quarters. 10 D + 25Q = 795 D + Q = 39 10 ( 39 − Q ) + 25Q = 795 390 − 10Q + 25Q = 795 15Q + 390 = 795 15Q = 405 Q = 27 D = 12 21. Jeffrey has $11.75 in nickels and quarters. The number of quarters is 5 more than 4 times the number of nickels. How many of each type of coin does he have? N is the number of nickels; Q is the number of quarters. 5 N + 25Q = 1175 Q = 5 + 4N 5 N + 25 ( 5 + 4 N ) = 1175 5 N + 125 + 100 N = 1175 105 N + 125 = 1175 105 N = 1050 N = 10 Q = 45 22. Melissa has $220 in $1 bills, $5 bills, and $10 bills. The number of $1 bills is 3 times the number of $5 bills. The number of $10 bills is 4 more than the number of $5 bills. How many of each type of bill does she have? O is the number of ones; F is the number of fives; T is the number of tens. O + 5 F + 10T = 220 O = 3F T =4+ F 3F + 5 F + 10 ( 4 + F ) = 220 8 F + 40 + 10 F = 220 18 F + 40 = 220 18 F = 180 F = 10 O = 30 T = 14 Using the laws of exponents, simplify the following expressions. Write your answers with positive exponents. Assume that all variables represent non-zero numbers. 36 23. 4 3 32 9 24. ( x ) 4 6 x 24 ⎛ 3x 4 ⎞ 25. ⎜ 8 ⎟ ⎝ y ⎠ 33 x12 y 24 27 x12 y 24 3 26. 39 x5 y12 3x 4 y 6 13xy 6 ⎛ 8 x5 y 6 ⎞ 27. ⎜ 2 2 ⎟ ⎝ 2x y ⎠ 3 28. 5 + 5 0 ( 4x y ) 1+ 64 x9 y12 6 5 4 3 3 14 x −2 y 6 31. 21x −10 y −3 32. ( 4 x −2 y −3 −1 29. x ⋅ x 4 1 5 x −3 1 x3 ) (6 −7 −3 −1 −6 x3 y 4 x y ⋅6 x y 10 −2 6 2x x y y 3 6 21 2 62 y 25 43 36 y 25 64 3 2 x8 y 9 3 x −5 30. 2 x 1 x5 ⋅ x 2 −7 4 ) −2 −2 1 x7 ⎛ 4 x −4 y −3 ⎞ 33. ⎜ 2 −6 ⎟ ⎝ 7x y ⎠ −2 ⎛ 4 y −3 y 6 ⎞ ⎜ 2 4 ⎟ ⎝ 7x x ⎠ −2 ( 3x 34. 38. ( 3 x + 8 )( 6 x − 5 ) 18 x + 33 x − 40 2 −2 −2 3 ⋅ 82 ⋅ x −2 y 23 x 4 y −2 49 x12 16 y 6 24 y 3 x6 4x2 + 7 x + 8 39. ( 5 x + 6 ) ( 3 x 2 − 2 x + 3) 15 x 3 − 10 x 2 + 15 x + 18 x 2 − 12 x + 18 15 x3 + 8 x 2 + 3x + 18 (8x y ) ⎛ 4 y3 ⎞ ⎜ 6⎟ ⎝ 7x ⎠ 4−2 y −6 7 −2 x −12 3 ⋅ 64 ⋅ yy 2 8x4 x2 Perform the indicated operations, and simplify all answers. 35. ( x 2 − 5 x − 12 ) + ( 7 x 2 − 3 x + 4 ) 36. ( 5 x 2 − x + 6 ) − ( x 2 − 8 x − 2 ) 8x2 − 8x − 8 y −2 )( 2 x −2 y −1 ) 3 x −8 y −2 ⋅ 2−3 x 6 y 3 8−2 x 4 y −2 −2 9 y 25 16 −8 37. ( −5 x3 )( 8 x 7 ) −40x10 40. ( x − 11 y )( x + 11 y ) x 2 − 121 y 2 −3 41. ( 3 x + 5 ) 42. ( 4 x − 3 y ) 2 9 x 2 + 30 x + 25 12 x 2 − 40 x + 28 43. 4x 2 12 x 40 x 28 − + 4x 4x 4x 2 16 x 2 − 24 xy + 9 y 2 3 x − 10 + 44. 24 x5 yz + 6 x 4 y 2 z 3 − 4 x 3 y 3 z 5 3 x 4 yz 2 24 x5 yz 6 x 4 y 2 z 3 4 x 3 y 3 z 5 + − 4 2 3 x 4 yz 2 3 x 4 yz 2 3 x yz 2 3 45. 6 x 2 + 22 x − 5 2x + 8 3x − 1 + 3 2x + 8 2 x + 8 6 x 2 + 22 x − 5 − ( 6 x 2 + 24 x ) 8x 4y z + 2 yz − 3x z − 2x − 5 − ( −2 x − 8 ) 3 46. 2 x 4 + 8 x3 + 7 x 2 − 7 2x2 + 2x − 5 x 2 + 3x + 3 + 2x2 + 2 x − 5 9x + 8 2x + 2x − 5 2 2 x 4 + 8 x3 + 7 x 2 − 7 − ( 2 x 4 + 2 x3 − 5 x 2 ) 6 x3 + 12 x 2 − 7 − ( 6 x3 + 6 x 2 − 15 x ) 6 x 2 + 15 x − 7 − ( 6 x 2 + 6 x − 15 ) 9x + 8 7 x Completely factor the following polynomials. 48. x 2 − 81 47. 6 x 3 + 12 xy 6x ( x2 + 2 y ) ( x − 9 )( x + 9 ) 49. 3 x 3 − 27 x 50. x 2 + 6 x + 8 3x ( x 2 − 9 ) ( x + 2 )( x + 4 ) 3 x ( x − 3)( x + 3) 51. x 2 − 9 x + 14 ( x − 2 )( x − 7 ) 52. x 2 − 6 x − 27 ( x − 9 )( x + 3) 55. 6 x 3 − 7 x 2 − 10 x 56. 6 xy + 21x + 10 y + 35 57. 6 xy + x − 30 y − 5 ( 6 xy + 21x ) + (10 y + 35) ( 6 xy + x ) − ( 30 y + 5) x ( 6 x 2 − 7 x − 10 ) x ( 6 x + 5 )( x − 2 ) 58. 12 x3 − 28 x 2 − 3 x + 7 (12 x3 − 28 x2 ) − ( 3x − 7 ) 4 x 2 ( 3x − 7 ) − ( 3x − 7 ) ( 3x − 7 ) ( 4 x 2 − 1) 53. 6 x 2 − 11x − 2 ( 6 x + 1)( x − 2 ) 54. 5 x 2 + 7 x − 6 ( 5 x − 3)( x + 2 ) 3x ( 2 y + 7 ) + 5 ( 2 y + 7 ) x ( 6 y + 1) − 5 ( 6 y + 1) ( 2 y + 7 )( 3x + 5) ( 6 y + 1)( x − 5) 59. x3 − 125 ( x − 5) ( x 2 + 5 x + 25) 60. 8 x 3 + 27 ( 2 x + 3) ( 4 x 2 − 6 x + 9 ) ( 3x − 7 )( 2 x − 1)( 2 x + 1) 61. 4 y 3 − 4 4 ( y 3 − 1) 4 ( y − 1) ( y 2 + y + 1) Solve the given equations. 62. ( x − 2 )( x + 5 ) = 0 x − 2 = 0, x + 5 = 0 2, −5 63. 4 x 2 − 49 = 0 ( 2 x − 7 )( 2 x + 7 ) = 0 2 x − 7 = 0,2 x + 7 = 0 7 7 ,− 2 2 64. 2 x 2 + 5 x − 10 = x 2 + 2 x + 8 x 2 + 3 x − 18 = 0 ( x + 6 )( x − 3) = 0 x + 6 = 0, x − 3 = 0 −6,3 65. 3 x 2 − 11x + 14 = ( 2 x + 1)( x − 2 ) 66. ( 3 x + 10 )( 3 x − 2 ) = ( x + 3)( 3 x + 10 ) ( 3x + 10 )( 3x − 2 ) − ( x + 3)( 3x + 10 ) = 0 ( 3x + 10 ) ⎡⎣( 3x − 2 ) − ( x + 3)⎤⎦ = 0 ( 3x + 10 )( 2 x − 5) = 0 3 x 2 − 11x + 14 = 2 x 2 − 3 x − 2 x 2 − 8 x + 16 = 0 ( x − 4) 2 =0 3 x + 10 = 0, 2 x − 5 = 0 x−4=0 4 − 10 5 , 3 2 67. 4 x3 + 10 x 2 − 50 x = 0 2 x ( 2 x 2 + 5 x − 25 ) = 0 2 x ( 2 x − 5 )( x + 5 ) = 0 2 x = 0, 2 x − 5 = 0, x + 5 = 0 5 0, , −5 2 Reduce the following rational expressions to lowest terms. 27 x3 y x2 + 4 x + 3 68. 69. x2 − 1 18 x 2 y 4 3x 2 y3 ( x + 1)( x + 3) ( x − 1)( x + 1) x+3 x −1 3− x 3x − 5 x − 12 3− x ( 3x + 4 )( x − 3) 70. 2 − ( x − 3) ( 3x + 4 )( x − 3) − 1 ( 3x + 4 ) Perform the indicated operations, and reduce the answers to lowest terms. 8 x 3 10 x 3 2x2 + x 2x2 + 6x 3 x 2 − 9 xy x3 − 3 x 2 y 72. 73. 71. 2 ⋅ ÷ ⋅ 5 x 12 x 4 4 x + 8 x3 + 3x 2 6 x2 − 6 y 2 xy + y 2 x ( 2 x + 1) 2 x ( x + 3) 3x ( x − 3 y ) y( x + y) ⋅ 2 ⋅ 2 4 ( x + 2 ) x ( x + 3) 6 ( x − y )( x + y ) x ( x − 3 y ) 4 3 2x + 1 2( x + 2) y 2x ( x − y ) 2 x2 + 8x 3x 2 − 2 x ÷ 2 x 2 − 4 x − 30 3 x 2 + 7 x − 6 2 x ( x + 4) ( 3x − 2 )( x + 3) ⋅ 2 ( x − 5 )( x + 3) x ( 3x − 2 ) 74. 75. x2 + 4 x + 4 2 ( x − 2 )( x + 2 ) ( x + 2) 2 ( x − 2) 2 76. 12 x 9 x+4 x−5 x 2 + 3 x + 5 x 2 − 3 x − 13 77. 2 + x + 4 x + 4 x2 + 4 x + 4 2x2 − 8 x2 + 4 x + 4 2 ( x2 − 4) 7 x 5x + 9 9 2x − y 2x − y 4x 3 78. 6x − 5 y 4x − 4 y − 2x − y 2x − y 1 6 x + 1 3 x − 13 − 3x − 4 4 − 3x 6 x + 1 3 x − 13 + 3x − 4 3x − 4 9 x − 12 3x − 4 3( 3x − 4 ) 3x − 4 79. 11x + 2 2 x + 15 3 x − 1 − − 2x − 4 2x − 4 2x − 4 6 x − 12 2x − 4 6 ( x − 2) 2 ( x − 2) 3 x+2 Given the following linear equations, complete the table of solution pairs. 80. 6 x − 5 y = 10 x y 0 −2 5 0 3 10 3 2 x −3 y 0 0 81. 3 x − 4 y + 9 = 0 −3 5 6 82. x + y = 0 x 1 −2 −5 y −1 2 5 3 Graph the solutions of the following linear equations. Indicate the intercept(s). 83. x + y = −3 84. 3 x − y = 9 85. 5 x − 4 y = −20 86. 4 x + 3 y = 24 87. x = 2 89. − x + y = −5 88. 3 y + 6 = 0 90. 2 x + y = 0 91. − x − 2 y = 4 Determine the solutions of the following linear systems of equations. If there are no solutions, write inconsistent. If there are infinitely many solutions, write dependent. 92. 3x − 7 y = 8 x = −2 93. x = 3y +1 2x − 5 y = 3 2 ( 3 y + 1) − 5 y = 3 94. 3x + 8 y = 1 x+ y=2 3( 2 − y ) + 8 y = 1 −6 − 7 y = 8 6y + 2 − 5y = 3 −7 y = 14 y+2=3 6 − 3y + 8y = 1 5y + 6 =1 y = −2 y =1 5 y = −5 x = −2 x=4 y = −1 x=3 95. 2x − 6 y = 4 − x + 3 y = −2 2x − 6 y = 4 + 2 ( − x + 3 y = −2 ) 0=0 infinitely many solutions dependent 96. 6 x − y = −1 3 x + 2 y = 12 2 ( 6 x − y = −1) + ( 3 x + 2 y = 12 ) 9 x = 10 10 x= 9 13 y= 3 97. 5 x + y = −14 2 x − y = −7 5 x + y = −14 + ( 2 x − y = −7 ) 7 x = −21 x = −3 y =1 98. −3 x + 7 y = −7 99. 3 x + 2 y = 25 6 x − 4 y = 10 −3 x + 2 y = −4 −3 x + 7 y = −7 6 x − 4 y = 10 + ( 3 x + 2 y = 25 ) +2 ( −3 x + 2 y = −4 ) 9 y = 18 0=2 y=2 no solution x=7 inconsistent Simplify the following radicals, if possible. numbers. 100 101. 49 102. 9 10 7 3 104. 107. 63 105. 13x10 3 7 x5 13 275x11 y 25 5 12 5x y 108. x=7 7 +11( 4 x + 11 y = 31) 81 y = 189 7 3 4 x= 3 y= Assume all variables represent positive 103. x 2 + x − 12 = 6 x − 6 x − 5x − 6 = 0 2 ( x − 6 )( x + 1) = 0 6, −1 3 −27 125 3 − 5 80x 4 y 7 106. 4 x2 y3 5 y 11 144 Solve the following equations. 3x x 7 x+4 6 110. − = 111. = 5 2 10 x −1 x − 3 11x + 10 y = 38 4 x + 11 y = 31 −4 (11x + 10 y = 38 ) 99 x16 16 109. 3 x8 11 4 11 12 11xy 6 x − 5x = 7 100. 112. x 5 −20 − = 2 x + 1 x − 3 x − 2x − 3 x ( x − 3) − 5 ( x + 1) = −20 x 2 − 8 x − 5 = −20 x 2 − 8 x + 15 = 0 ( x − 3)( x − 5) = 0 5