Download Math 0308 Final Exam Review(answers)

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