Download Advanced Algebra Review Packet 2004-2005

AATH Review Packet #3
Final Exam Review – Semester One
Chapter One
Evaluate each of the following.
1.
8  4  2 1
3.
Name _______________
2.
-2b2 + 4ab when a = 3 and b = -1
1
1
x  y when x = 3 and y = 2
4
3
Simplify.
4.
3(x – 8) – 2(x + 5)
5.
 2 x 
6.
53  2 2
55
7.
x2 y5 x3
y3
8.
(2a3b2)4
9.
 2x3 
 5 
 y 
11.
187,000
Convert to scientific notation.
10.
0.00000000183
3 2
3
Simplify.
1.8  1015  2  10 4
12.
3.0  1011
Solve.
13. Mark paid $27.50 to rent a rug shampooer at Ace Rentals because he returned the equipment
late. The late fee is 20% of the rental price. What would Mark have paid if he had returned the
equipment on time?
Chapter Two
Solve the following equations.
14.
2x + 19 = 5(x + 2)
15.
1.2n = 2.3n – 2.20
16.
2x  7  3
17.
2
1
x 2
3
4
Solve the following inequalities and then graph each on a number line.
18.
19.
5x  2  7
1  2x  3  5
20.
3x – 9 < -5 or x – 9 > 6
22.
 x3  4
21.
2x  1  3
Solve.
23. The second angle of a triangle is three times the first, and the third angle is twice the second.
Find the measures of the angles.
24. You can earn $6 an hour raking leaves and $4 an hour selling ice cream. What is the most time
you can spend selling ice cream if you can work only 10 hours per week and you want to make $52 a
week?
25.
Solve 5xy + 2x = 3 for y.
27.
Solve a2b + 2b = 12 for b.
26.
Solve a2b + 2b = 12 for a.
Chapter Three
28.
Find the slope of the line containing the points (-1,3) and (-4,5).
29.
Sketch the graph of the line -6x + 2y = 4
30.
What is the slope of 2x + 4y = 15?
Decide whether the two lines are parallel, perpendicular, or neither.
31.
-3x – 5y = 10
32.
-3x - 5y = 10
-3x + 5y = 10
-5x + 3y = 10
Write an equation for the following lines in standard form:
33. (5,-2) and (7,-3)
34. (3, 4) and (3, -7)
35. (4, 10) and (-1, 10)
36. Write an equation in slope-intercept form of a line that passes through (1,4) and is perpendicular
to the line y = -3x + 1.
37. Determine the x and y intercepts of the line 2x – 5y = 20.
Write the following equations in slope-intercept form.
38. x – y = 9
39. 2x – 10y = 15
40. -x + 9y = 30
Write the following equations in standard form.
41. y = 3x – 21
42. x = -4y + 7
43. y = 
4
x6
5
44. Find f(-3) when f(x) = 2x2 + 5x + 2
45. Is (2,1), (1,2), (1,3), (0,4), (3,5)
a function?
Domain:
Range:
Find the domain of the functions listed.
x5
46. f ( x) 
47. h( x)  4 x 2
x2
Consider f(x) = x2 and g(x) = 4x – 6 for the following.
49. Find f(x) - g(x)
50. Find g ( f (2))
48. k ( x)  x  7
51. Find f ( g ( x))
Solve.
52. When a tree was 2 years old it was 6 feet tall. When it was 5 years old it was 10 feet tall. How
tall will the tree be when it is 11 years old?
Chapter Four
Decide if the ordered pair is a solution of the linear system.
53. -x + 2y = 5
(-9,2)
54. -6x + 9y = 3
2x – 4y = 8
2x + 2y = -16
Solve the linear systems listed by the substitution method.
55. x – 4y = 20
56. -x + y = -14
2x + 5y = 1
2x – 3y = 33
Solve the linear systems listed by the elimination method (linear combinations).
57.
2x + 3y = -7
58.
2x – y + z = 7
-4x – 5y = 13
4x + 2y – 3z = -2
x – 3y + 2z = 7
(-2,-1)
Graph the systems of inequalities
Find the maximum and minimum
and determine the vertices (corner points). value of C = 5x + 6y subject to constraints;
y  1
x 1
59.
60.
x  3
y0
y  x  1
3x  2 y  24
Chapter Five
Perform the indicated operations.
61.
3xy2 + 2xy – 4xy2 – 8xy + 2
62.
(x2 – 2x + 9) – (5x2 – 3x + 4)
64.
(5x + 9)(3x – 7)
Factor completely.
66.
7a2x – 63b2x
67.
64y3 – 1
68.
25 + y2
69.
3my + 7x + 7m + 3xy
70.
x3 + 1
71.
3x2 -46x + 15
72.
3y2 – 147
63.
3a2b(5a + 2)
65.
(2a + b)3
Find all solutions of the following equations.
73.
2x3 +4x2 – 10x = 20
74.
75.
x3 + 6x2 – 25x = 150
76.
x3 + 6x2 = -8x
32x3 – 12x2 – 32x + 12 = 0
Solve.
77.
The sum of the squares of two positive, consecutive odd integers is 74. Find the integers.
Chapter Six
Simplify.
x 4  5x 3  4 x 2
78.
x3  7x 2  6x
80.
30 x
3

 6x 2 
15 x 2  3x
4 x 2  4 x  24
79.
x 2  7 x  8 4x3

3x 2  24 x x 2  1
81.
x 1
x2  x  2 x 1


3
3x
x 2  5x  6
State the least common denominator of the following expressions.
5
6
13
4
5
, 2
82.
83.
,
,
2
2x  1 4x  1
x  8 x  12 x  6 x( x  2)
Perform the indicated operations and simplify.
5
4 2
4
6
 2 
84.
85.
x3
x x
x3
86.
x
3

x  9 x( x  3)
2
Simplify the complex fractions.
2 
 4

 2

x  25 x  5 

87.
1 
 1



 x 5 x 5
88.
Solve the following equations. Check each solution.
x
2 x  10

89.
90.
x  1 x  11
91.
x
4
1 
x4
x4
93.
Solve
3  x 1
x 2  5
2x  4 x  2
3x


x
x2 x2
92.
1 
1
 

 x x 1  2
 1 


 x 1
96.
 3 2 0 1
8 5 3 3 6
 9 4 2 0
98.
 6 1 2
1 5 

7 4  3  6 8

 4 9 5


1 1 1
for f.
 
p q f
Chapter Thirteen
 2 7  9 7
94.
What are the dimensions of  6 5 2 9 ?


11 6  1 6
Perform the indicated matrix operations.
95.
0 1  5  10 3 11
4 1 6    2 8 3 

 

97.
 6 4 5 10 
 2 

 
 0 3 1 3  
99.
1 0    1 1 
 4 9    3 2

 

100.
1 
0
10 2 1 5   
  2
 
3
101.
4  1 3   0 1 
7 10  3   9 2

 

1 2  5 12 0
102.
2 8 1  0 1  1
0 5  2  8 2 4 

 

Evaluate the determinants.
103.
2 3
6 3
2
104.
1
4
0
1
2
1
2
3
Solve by using Cramer’s Rule.
105. 2x + 5y = 13
-3x + 4y = 15
Find the inverse of each matrix.
 2  2
106. 
107.

 4  3
2 3 
7 11


Solve the following matrix equation.
 5  13
 3 1
109. 
X 


5 
2
  4 0
Write the linear system as a matrix equation.
110. 2x – 4y = 7
-3x + y = 12
Use an inverse matrix to solve the linear system.
111. 2x + 3y + z + m = 3
-x + 2y + 3z + m = 8
2x - 4y + z - m = 9
x+y+z+m=6
108.
6 10
2 4 


Answer Key to Review Packet Sem I – AATH
1. 5
2. -14
3. 17/12
4. x – 34
5. 4/x6
6. 4/25
7. x5y2
8. 16a12b8
9. 8x9/y15
10. 1.83 X 10-9
11. 1.87 X 105
12. 1.2
13. $22.92
14. x = 3
15. n = 2
16. x = -2,-5
17. x = 21/8
18. x  9/5
19. 2 < x < 4
20. x < 4/3 or x > 15
21. -1  x  2
22. x < -1 or x > 7
23. 108, 18, 54
24. 6 hours of raking leaves
4 hours of selling ice cream
 2x  3
5x
12  2b
26. a  
b
12
27. b  2
a 2
25. y 
36. y =
1
2
x3
3
3
37. x-int (10,0) y-int (0,-4)
38. y = x-9
1
3
x
5
2
1
10
40. y = x 
9
3
39. y =
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
-3x + y + 21 = 0
x + 4y – 7 = 0
4x + 5y – 30 = 0
5
no, D=-2,-1,0,3 R=1,2,3,4,5
All real numbers except -2
All real numbers
x 7
x2 – 4x + 6
10
16x2 -48x + 36
18 ft.
no
no
(8,-3)
(9,-5)
(-2,-1)
(2,1,4)
Vertices (-3, 4), (-3,-1), (2,-1)
28. m = -2/3
29. y = 3x + 2
60. Max(1, 10.5)=68, (8,0)=40
Min(1,0)=5
30.
31.
32.
33.
34.
35.
61.
62.
63.
64.
65.
66.
m = -1/2
neither
perpendicular
x + 2y – 1 = 0
x–3=0
y -10 = 0
–xy2 – 6xy + 2
-4x2 + x + 5
15a3b + 6a2b
15x2 – 8x – 63
8a3 + 12a2b + 6ab2 + b3
7x(a-3b)(a+3b)
 24 16 0 8 
 40 24 24 48
96.


 72 32 16 0 
 2 12
97. 

 2 0
98. Not possible
67.
68.
69.
70.
71.
72.
(4y-1)(16y2 + 4y + 1)
does not factor
(3y + 7)(m+x)
(x + 1)(x2 – x + 1)
(3x – 1)(x –15)
3(y – 7)(y + 7)
73.
74.
75.
76.
77.
-2, 
0,-4,-2
5,-5,-6
5
 1, 3 8
5,7
x ( x  4)
x6
4x 2
79.
3( x  1)
78.
80. 8x(x+3)(x-2)
81.
x( x  1)
x3
82. (2x+1)(2x-1)
83. x(x-6)(x-2)
6 x  13
x3
2(4 x 2  5 x  3)
85.
x 2 ( x  3)
84.
x 2  3x  9
x( x  3)( x  3)
x3
87.
x
x (3 x  1)
88.
1  5x 2
86.
89.
90.
91.
92.
x = 5,-2
x=4
no solution
x= ½
93. f =
qp
q p
94. 3 x 4
95.
10 4 6
 2 9 9


 1 1 
 23 22


100. 23
2
 27


101.
 54 27
 42 5 
99.
102. Not possible
103. 12
104. 4
105. (-1,3)
 3 
1
106.  2
  2 1


 11  3
107. 

 7 2 

1
108. 
1

2
 37
109. 
 14
 5
2 
3 

2 
5
 2
 2  4  x   7 
    
 3 1   y  12
110. 
If solved x = -5.5 and y = -4.5
111. (1,-2,3,4)