Download Math 96 Final Exam Review Problems Spring 2014

Math 96 Final Exam Review Problems
For problems 1-6, specify all values of x that
make the expression undefined, then write
the expression in lowest terms. If the
expression is already in lowest terms, write
already simplified.
x+4
1.
x!7
5x + 3
2.
( x + 6 )( x ! 2 )
11.
12.
Spring 2014
6x + 33
5
!
x + 9x + 18 x + 3
2
5x + 17
4
!
x + x ! 12 x ! 3
2
For problems 13-18, perform each
multiplication or division and write the result
in lowest terms.
x 2 + 4x ! 32
3. 2
x + 2x ! 24
13.
x+6 x!8
"
x!8 x+4
x 2 + 9x + 14
4.
x2 ! 4
14.
10x 3 ! 20x 2 4x ! 4
"
2x 2 + 10x 5x 2 ! 10x
5.
3x 3 ! 24x 2 + 45x
x 3 ! 11x 2 + 24x
6.
2x 4 + 10x 3 + 8x 2
4x 5 + 28x 4 + 48x 3
For problems 7-12, perform each addition or
subtraction and write the result in lowest
terms.
7.
8.
9.
10.
5x + 6 x ! 5
+
x+3 x+3
7x + 8 3x ! 4
+
5x + 2 5x + 2
2x ! 3 x ! 5
!
x!4 x!4
x+3
x2
!
x + 5 ( x + 2 )( x + 5 )
x 2 ! 4x ! 21 x 2 + 3x ! 40
÷
15. 2
x ! 5x ! 14 x 2 ! 3x ! 10
x 2 ! 64
3x 2 + 24x
÷
16. 2
x ! 3x ! 40 x 2 ! x ! 30
17.
40x 2 ! 60x 10x 2 ! 15x
÷
3x 2 + 12x 12x 2 + 48x
18.
x 4 ! 36x 2
x 5 ! 6x 4
÷
x 3 + 4x 2 ! 12x x 5 + 3x 4 ! 10x 3
For problems 19-48, solve each equation.
19. x 2 ! 6x = 16
20. 3x 2 + 12x = 2x + 8
21. 6x 2 ! 12 = !x
22. 2x 3 ! 8x 2 = 56x ! 14x 2
23. 5x 2 ! 30x = 24 ! 4x
43.
9−! =!+3
24. 16 = 10x ! x 2
44.
5−!−! =1
45.
7! + 1 − ! = 1
46.
!+4+2=!
25. x ( x ! 6 ) = !4
26. ( x ! 5 ) ( x + 3) = !10
27. ( x ! 6 ) ( x ! 2 ) = !37
47. 2 + 3! − 8 = !
28. ( x ! 8 ) ( x + 6 ) = 1
48. 1 +
12! + 16 = !
29. x 2 + 6 = 2x
30. 2x 3 ! 8x 2 + 12x = 0
For problems 49-54, solve each equation for
the given variable
31. ( x ! 3) = 25
2
!
49.
32. ( 2x + 7 ) = 5
2
33. ( 6x ! 1) ! 121 = 0
50.
34. ( 4x + 3) = 1
51.
2
2
35. ( 5x ! 14 ) = !36
!
!
= !!! for y
!(!!!)
!!!
!
=
!!!
=
!
!
!
for y
for t
!
2
52. 36. ( 2x ! 3) = 12
!
!
= !!!! for y
!!!
2
37.
38.
39.
40.
41.
42.
!
!!!
!!
!
+ !!! = ! ! !!!!!
!!
!
!! !!
!
= !!! − !!!
!!"
! ! !!!!!
!
54.
!
!
!
!
− !!! = ! ! !!!!!
!!!
!!!
!!
!!"
− !!! = ! ! !!
!
!"!
!!!
!!!!
!
!
= ! for n
=
!
!
for x
!
+ !!! = !!!
!
53.
!! !
+ !!! = ! ! !!!!!"
!!!
For problems 55-60, write each radical
expression using rational exponents.
55.
!
!!
56.
!
!!
57.
!
58.
3
59.
!
60. !
!!
x2
!!
!!
For problems 61-66, simplify each radical
expression. Assume all variables represent
positive real numbers.
61.
144! ! ! !
62.
200!!"
63.
!
64!!" !!"
64.
!
162! !" !!"
65.
54! ! ! !
66.
32!! !!"
For problems 67-73, rationalize each
denominator. Assume all variables represent
positive real numbers.
67.
68.
69.
70.
71.
!
!
!
!"
!
!"!
!
!!!
! !
!"!!
72.
73.
!! !
!! !
!!!
!!!
Set up and solve each of the problems 74-79.
74. A boat travels 12 miles with the
current in the same time it travels 8
miles against the current. If the speed
of the current is 2 mph, find the speed
of the boat.
75. It takes Nick 4 hours to paint a house
and Sierra 12 hours to paint the same
house. If they work together, how
long will it take them to paint the
house?
76. The length of a rectangular flower
garden is 5 feet more than twice the
width of the garden. If the area
encompassed by the garden is 133
square feet, what is the length and the
width?
77. A picture that measures 8 inches by 7
inches is placed in a wood frame of
uniform width. If the area of the
frame and picture together is 306
square inches, what is the uniform
width of wood frame?
78. Sal can wax a car twice as fast as
Louis can wax the same car. When Sal
and Louis work together, they can wax
the car in 1 1/3 hours. How long
would it take each of them to wax the
car alone?
79. A traffic island is the shape of a
triangle whose base is three feet less
than its height. Find the base and the
height, if the island has an area of 65
square meters.
84.
For problems 80-87, state the domain and
range of the following relations. Determine
whether the relation is a function or not.
Explain your reasoning.
80.
{(3,4), (5, "2), (0,6), (1,8), ("7,4), ("4, "1)}
81.
{(2,5), (3,7), (4,9), (2,11)}
85.
!
!
82.
86.
83.
87.
Math 96 Final Exam Review
Answers
Spring 2014
1. The value that makes the expression
undefined is 7. The expression is already
in lowest terms.
2. The values that make the expression
undefined are –6 and 2. The expression is
already in lowest terms.
3. The values that make the expression
undefined are –6 and 4. In lowest terms,
x+8
the expression is
.
x+6
4. The values that make the expression
undefined are –2 and 2. In lowest terms,
x+7
the expression is
.
x!2
10.
5x + 6
( x + 2 )( x + 5 )
11.
1
x+6
12.
x +1
( x + 4 )( x ! 3)
13.
x+6
x+4
14.
4 ( x ! 1)
x+5
15.
x+3
x+8
16.
x!6
3x
17. 16
5. The values that make the expression
undefined are 0, 3 and 8. In lowest
3( x + 5 )
terms, the expression is
.
x!8
6. The values that make the expression
undefined are –4, –3 and 0. In lowest
x +1
terms, the expression is
.
2x ( x + 3)
7.
6x + 1
x+3
8. 2
9.
x+2
x!4
18. x + 5
19. x = !2, 8
20. x = !4,
2
3
3 4
21. x = ! ,
2 3
22. x = !7, 0, 4
23. x =
!4
, 6
5
24. x = 2, 8
25. x = 3 ± 5
26. x = 1 ± 6
27. x = 4 ± i 33
28. x = 1 ± 5 2
29. x = 1 ± i 5
30. x = 0, 2 ± i 2
31. x = !2, 8
32. x =
!7 ± 5
2
5
33. x = ! , 2
3
1
34. x = ! , ! 1
2
14 6
35. x =
± i
5 5
36. x =
3
3± 2 3
± 3 or
2
2
37. x = !
12
7
45. ! = 0, 5
46. ! = 5
47. ! = 3, 4
48.
x = 15
49. ! =
50.
y=
52. ! =
!!!!"
53. ! =
!!!!
54. ! =
!!!!
!
!
!
!
!
55. ! !
!
56. ! !
4
57. x 3
2
39. ! = 3
59. !!
42. ! =
10x + 24
10
or y = x + 8
3
3
!!!!
38. ! = −3
x3
!
60. !
41. no solution
!
51. ! =
58.
40. x = 2
! !!!
!
!
61. 12!! ! !"
!"
!
43. ! = 0
44. ! = 1
62. 10!! 2!
!
63. 4! ! ! ! !
!
64. 3! ! ! ! 2! ! ! !
! !
65. 3! !
82. domain = {0,1,2,3,4} , range =
6!
{0,1,4,9,16} , it is a function since every x-
value goes to exactly one y-value
!
66. 4! ! ! ! 2!"
67.
68.
69.
70.
71.
72.
!
! !
!
{t, u,v, w} ,
!
!
! !"!
84. domain = [ "5, # ) , range = ( "#, # ) , not
!!
a function since the graph fails the vertical
line test
! !!!"
!
!
5− 2
!
73.
!−3
74.
10 mph
!
!
86. domain = [ "3,3] , range = [ "6,6] , not a
function since it fails the vertical line test
87.
76. width = 7 feet; length = 19 feet
77. 5 inches
!
!
domain = ( "#, # ) , range = [0, ") , it is a
function since it passes the vertical line
test
!
78. Louis = 2 hrs; Sal = 4 hrs
79. base = 10 m; height = 13 m
80. domain = {"7, "4,0,1,3,5} , range =
{"2, "1,4,6,8},
!
85. domain = ( "#, # ) , range = ( "#, # ) , it is
a function since the graph passes the
vertical line test
!"! !! !"!!
it is a function since every
x-value goes to exactly one y-value
!
81. domain = {2,3,4} , range = {5,7,9,11} ,
not a function since 2 corresponds to two
values in the range
!
it is a function since every
domain value goes to exactly one range
value
!
!"
75. 3 hours
!
83. domain = { A, B,C, D, E} , range =
!
!