MATH 110
FINAL REVIEW A
Find the domain of the expression
y − 17
1.
2
y − 9 y − 22
a) (− ∞,−2) ∪ (− 2,11) ∪ (11,17 ) ∪ (17, ∞ ) b) (− ∞,17) ∪ (17, ∞ ) c) (− ∞, ∞ )
2.
3.
5
x−4
a) {x x ≠ 4}
b)
3 y 2 − 12
y2 + 4
a) (− ∞,−2) ∪ (− 2,2) ∪ (2, ∞ ) b)
c)
(− ∞, ∞ )
c)
{x x ≠ −4}
(− ∞,−2) ∪ (− 2, ∞ )
d)
d)
{x x ≠ 4, x ≠ −5}
(− ∞,2) ∪ (2, ∞)
g ( y ) = 4 6 y − 24
4.
a)
[4, ∞ )
Reduce the expression
− 15(9 x − 3)( x + 10 )
5.
2
3( x + 10 )
− 5(9 x − 3)
a)
(x + 10)
− 17 w + 40 = 8w 2 + 5 w
5
a) w = 8, w = −
8
d) (− ∞,4]
b)
(4, ∞)
c)
b)
− 15(9 x − 3)
3
c)
− 5(9 x − 3)
d)
4
, t = −5
3
c)
4
t = − ,t = 5
3
5
d) t = , t = −4
3
c)
w = −3, w =
Find all solutions to the equation
6.
3t 2 + 11t − 20 = 0
5
a) t = − , t = 4
b)
3
7.
{x x = 4}
d) (− ∞,−2) ∪ (− 2,11) ∪ (11, ∞ )
b)
t=
w = 20, w =
29
4
Solve the equation by using the square root property
8.
r 2 = 81
a) r = 9
b) r = ±9
(− ∞, ∞ )
c) No Solution
6
5
5(9 x − 3)
(x + 10 )
d) w =
5
, w = −4
4
d) None of the Above
Solve using the quadratic formula
9.
11 y − 6 + 30 y 2 = 0
a)
y = 0 .3 + 3 .7 i , y = 0 .3 − 3 .7 i
1
2
b) y = , y =
3
5
c)
2
3
y =− ,y =
3
10
d) y =
3
10
,y =−
2
3
7 x( x − 2) = 5
10.
a)
x=
14 ± 2i 21
7
b) x =
7 ± 21
2
c) x = 1± 4 21
d) x =
7 ± 2 21
7
11. Find the discriminant, b 2 − 4ac , and determine the number and type of solutions. Choose from two
rational solutions, one rational solution, two irrational solutions, or two imaginary solutions.
6n 2 = 8
a) -192: Two imaginary solutions
b) 192: Two imaginary solutions
c) 192: Two irrational solutions
d) None of the above
Solve the equation
z 2 1 17 z
12.
− =
8 4 24
a) No solution
13.
b) z = 3, z = −
2
3
c) z = 6
1
d) z = − , z = 6
3
5
1
c) x = − , x =
2
3
d) No solution
3
3
3 x − 15
+
=
x x−6
x−6
a)
x =1
b) x = 6, x = 1
14. Multiply:
a)
y − 2x
2x − y
15. Divide:
a)
12
3 y − 6x
•
2x − y
4
b)
16
− 4 x + 4 xy − y 2
c)
− 3(3 y − 6 x )
2x − y
d) -9
b)
x3 − 6x 2 + 9x
5 x 2 + 15 x
c)
5
x+3
d)
x+3
5 x 2 − 3x
c)
7 x 2 + 21x
4 x 2 − 16
d)
4( x − 4 )
x
2
x − 3 x 2 − 3x
÷
x+3
5x
5
3x
7 x − 28 x 2 + 3 x
•
x
4 x − 16
3
7 x − 84 x
7 ( x + 3)
b)
2
4
4 x − 16 x
16. Multiply:
a)
18
4
17. Divide: z
12
z
3
a)
2z 3
18. Subtract:
a)
216
b)
z5
18 + z
c)
z4 + z
3z 3
d)
2
7 n + 14 − 3 + 2n
− 2
n2 − 5
n −5
5n + 17
b)
5n + 17
n2 − 5
c)
9n + 17
n2 − 5
d)
9n + 11
n2 − 5
Perform the indicated operations
y +1 2y − 7
4y
19. 2
+
−
y − 3 y − 10 y − 5 y + 2
a)
20.
− y 2 + 24 y − 33
y 2 − 3 y − 10
8
3
5
−
+ 2
x + 5 x − 5 x − 25
8x + 3
a)
x 2 − 25
b)
− y 2 − 10 y + 37
y 2 − 3 y − 10
b)
c)
25 x + 25
x−5
3y + 8
y − 4 y − 13
2
c)
d)
(
3y + 8
y − 3 y − 10 ( y − 5)( y + 2 )
2
5 x − 50
x 2 − 25
)
d)
3x − 5
x2 − 5
21. If z (t ) = 2t 2 + 7t − 4 , find z (−1) and z ( 4) .
b) z ( −1) = −9 ; z ( 4) = 56
a)
z ( −1) = −13 ; z ( 4) = 56
c)
d) z ( −1) = −7 ; z ( 4) = 88
z ( −1) = 5 ; z ( 4) = 35
22. If f ( x) = 3 x 2 + 7 x − 10 , find and simplify f ( 2 + x )
b) 3 x 2 + 2 x + 16
a) 16 + x .
c) 3 x 2 + 19 x + 16
d) 3 x 2 + 7 x − 12
23. Use the graph of the function f(x) to find the x-value for which f(x) = 3
a) -1
b) 2
24. Which of the following is a linear function?
11
a) f ( x ) = + 7
b) f ( x) = 11 + x + x 2
x
c) 3
d) -2
c)
d)
f ( x) = 11x + 7
f ( x ) = 11 − 7 x
25. What is the range of the relation whose graph is below?
a)
{x − 20 ≤ x ≤ 15}
{
}
b) − 20,−10, 0, 10, 15
c)
{y − 40 ≤ y ≤ 20}
26. Find the x- and y- intercepts of the function: g ( x ) = −8 x − 6
−3 
 −3
a) (0,0)
b) (0, -6) and 
c) (-6, 0) and  0,
,0 

 4 
 4 
d)
{0,
d) (0,-6) and (14,0)
Find and write the equation of the line
1
 9
27. Slope = − and y-intercept  0, 
5
 8
9
1
b) y = − 15 x + 89
a) y = − 5 (x − 8 )
c) x = − 15 y − 98
d) y = − 15 (x + 98 )
28. Slope = 12, goes through the point (-1, 4)
a) y = 12 x + 4
b) y = 12 x − 1
c)
d) y = −3 x + 12
y = 12 x + 16
29. Through the point (1, 10) that is parallel to the line 2 y − 4 x = 12
a) y = 4 x + 6
b) y = 2 x + 8
c) 2 y − 4 x = 10
d) y = 2 x + 6
30. Solve the system using the substitution method:
y = −29 − 3 x
4 x + 7 y = −50
a) (9, -9)
b) (-9, -2)
d) (-2, 10)
c) (-2, -9)
Solve the system using the addition method:
11x + y = 69
31.
13 x − y = 99
a)
(7, -8)
b) (4, -7)
c) (-8, 7)
}
−20, −40, 20
d) There is no solution
32.
− 11x − 2 y = 8
11x + 2 y = 121
a)
There is no solution
b)
1 3
c)  ,− 
2 4
{(x, y )11x + 2 y = 121}
d) (1,8)
33. At one store, 5 pairs of jeans and 2 sweatshirts costs $230, while 3 pairs of jeans and 4 sweatshirts costs
$208. Find the cost of one sweatshirt.
a) $25
b) $36
c) $22
d) $38
Solve the inequality. Write the answer in interval notation
34. 8 > 3 x and − 9 + 2 x ≥ −14
5  8 

a)  − ∞,−  ∪  , ∞ 
b) No solution
2  3 

35. − 11 < 9 − 11z ≤ 10
 1 20 
a) − , 
 11 11 
c)
 20 1 
b)  ,− 
 11 11 
 8 5
− 3 , 2 


20 

c) − ∞, 
11 

 5 8
d) − , 
 2 3
d)
(− 20,1)
Solve the inequality, graph the solution, and write the answer in interval notation
36. 12 y − 6 ≥ 18 or y < −2
a)
(− ∞,−2) ∪ [2, ∞ )
b)
(− ∞,−2) ∪ (2, ∞)
c)
(− ∞,−2] ∪ [2, ∞)
d) None of the above
c) w = 6, w = 2
d) No solution
Solve the equation
37. 2 − 3w − 18 = 8
a)
w = 8, w = 4
11 2
+ 3 y − 6 = −2
4 3
13
19
a) y = , y =
8
8
b) w = 4
38. −
b) y = 2, y = −
5
2
c)
1
11
y= , y=−
3
2
d) No solution
Solve the inequality. Write the answer in interval notation
39. 4w − 7 ≥ 5
a)
1 
 2 ,3


b)
(− ∞,−3] ∪  1 , ∞ 
2

1

c)  − ∞,  ∪ [3, ∞ )
2

1

d) − 3,− 
2

c) All real numbers
d) No solution
Solve the inequality
40. − 11 ≥ 2b − 23
a) 6 ≤ b ≤ 17
b) b ≤ 6 or b ≥ 17
Solve the inequality. Write the answer in interval notation
41. − 2 z + 4 ≤ 4
[0,4]
a)
b)
(− ∞,−2)
c) All real numbers
d) No solution
Write the expression by using rational exponents rather than radical notation
42. 137 x 3
3
7
3
a) 13x 7
b)
(13x )7
c) 13x 3
d)
13
x
3
7
Simplify the expression by using the properties of rational exponents. Write the final answer using positive
exponents only.
43. h
10
3
a) h
•h
20
2
3
9
 81s 12 r − 4 

44. 
−4 4 
 16 s r 
27 s 6
a)
8
b) h
3
20
3
c) h 2
4
b)
3s 16 r 8
2
c)
3s 12
2r 6
Simplify the radical. Assume that all variables represent positive real numbers
27 z
45.
3z
c) 9
a) 3
b) 3z
d)
27 s 12
8r 6
d)
3z
52x 5
46.
a) 2 x 2 13 x
47.
d) h 4
3
a)
b) 2 13x 5
c) 2 x 13x 3
d) 4 x 4 13 x
b) 2 y 3 9 y 5
c) 2 y 2 3 9 y 2
d) 8 y 6 3 9 y 2
c) 2 b10
d)
72 y 8
y 2 3 72 y 2
8b 2
2b12
48.
a)
2
b5
b)
4
b10
2
b b8
Add, if possible
49. 7 2 + 98
a) 28
b) 70
c) 14 2
d) Cannot simplify
Multiply
50. 3 • 24
a)
27
51. − 43 4 • 63 6
a) − 483 3
(
52.
2 8− 3
( 12 − 1)(
3 +5
4
d) 36 2
b) − 483 6
c) 23 10
d) − 1923 3
b) 4 − 3
c) 2 8 − 6
d) 4 + 6
b) 1 + 9 3
c) 31 + 5 12 − 3
d)
)
a) 1
54.
c) 2 6
)
a) 4 − 6
53.
b) 6 2
33 − 5
p 2q3 • 3 p 5q 2
a) 12 p 10 q 6
Rationalize the denominator
− 14
55. 3
10
− 7 3 100
a)
5
q6
c)
p2q
53 100
−7
c)
− 73 10
5
b) p
b)
7
12
p2q5
d) pq
7
p3q
d) Cannot simplify
Answer Key
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
D
A
B
A
A
B
D
B
C
D
C
D
A
D
C
B
A
B
A
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
C
B
C
D
D
C
B
B
C
B
B
A
A
A
D
A
A
D
A
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
C
D
A
A
D
D
A
A
C
A
C
B
A
A
B
C
A