Math 082 Diagnostic Review Material
MATH 082
DIAGNOSTIC
REVIEW MATERIALS
YOU WILL NOT BE GIVEN
THE DIAGNOSTIC TEST UNTIL
YOU RETURN THIS MATERIAL.
DO NOT WRITE IN
THIS MATERIAL.
Fall’09
Directions: The following problems are types of problems that you can expect to see on the diagnostic test. Complete as many problems as you can. You are not expected to know everything in this review packet. Try the problems that you are able to do. The solutions to every problem are worked out for you at the end of the practice test. There are also are extra practice problems that you can do to supplement the practice test.
Practice Test:
1. Add the following and simplify if necessary:
2. Subtract the following and simplify if necessary:
3. Multiply the following and simplify if necessary:
4. Divide the following and simplify:
5. Simplify: 3(4 – 5x) + (12x – 7)
6. Simplify:
7. Simplify:
8. Simplify:
9. Simplify:
10. Solve for x:
11. Solve for x:
12. Solve for x:
13. Solve for x:
14. Use or to make each statement true:
a. _____ –3 b. (-5) _____
15. True or False: -3 is a solution of 4x – 2 3x + 13.
16. Graph the following inequality on a number line: x -5.
17. Solve for x: -6(x + 4) 42
18. Graph the solution set for the following inequality: 7x + 19 > 5
19. Find Z if Z = xy and x = 3 and y =15.
20. In the equation E = 8FG, solve for G.
21. In the equation 14x + 7y = 21, solve for y.
22. True or False: (4, -3) is a solution for -x + 2y = -5.
23. Find the missing value so that ( ____, -9 ) is a solution for 3x – y = 12.
24. Write the ordered pair (x , y) for each of the points represented on the graph by the capital letter.
A
B
C
D
25. Graph 5x – 2y = 10.
26. Graph x – 3y > 6.
27. Compute the slope of the graphed line.
28. Compute the slope of the line that passes through the points (-8, 6) and (2, 4).
29. Write the equation of the line that passes through the points (-2, 7) and (-5,1).
30. Solve the system of equations:
6x + 4y = 14
5x + 2y = 5
31. Solve the system of equations:
x = 2y – 3
3x – 5y = 7
32. Simplify and rewrite your answer without using negative exponents:
33. Simplify and rewrite your answer without using negative exponents:
34. Simplify and rewrite your answer without using negative exponents:
35. Write the following number in scientific notation: 53,000,000
36. Write the following number in scientific notation: 0.0006372
37. Write the following number in decimal notation: 0.0968 x 106
38. Multiply: 2x(x + 7)
39. Multiply: (3x – 4)(x + 5)
40. Multiply: (x – 3)(x2 – 2x +8)
41. Factor completely: 16x2 – 24x
42. Factor completely: x2 – 64
43. Factor completely: x2 – 14x – 15
44. Solve by factoring: x2 – x – 6 = 0
45. Solve by factoring: x2 + 11x = -10
46. Solve by factoring: 6x2 – 10x = 0
Solutions to Practice Test
1. Add the following and simplify if necessary:
2. Subtract the following and simplify if necessary:
3. Multiply the following and simplify if necessary:
4. Divide the following and simplify:
=
Extra Practice Problems
a. Simplify:
b. Simplify:
c. Multiply and simplify if necessary:
d. Multiply and simplify if necessary:
e. Divide and simplify if necessary:
f. Add and simplify if necessary:
g. Add and simplify if necessary:
h. Subtract and simplify if necessary:
i. Subtract and simplify if necessary:
Answers to Extra Practice Problems
a. b. c. d. e. f. g. h. i.
Solutions to Practice Test: Questions 5 - 9
5. Simplify: 3(4 – 5x) + (12x – 7)
3 (4 – 5x) + 1 (12x – 7)
= 12 – 15x + 12x – 7
= -3x + 5 The answer can also be written 5 – 3x.
6. Simplify:
7. Simplify:
8. Simplify:
9. Simplify:
=
=
= – 10x + 6 + 16x + – 40
=
Extra Practice Problems
a. Simplify: 2(5 – 3x) – (6x – 3)
b. Simplify:
c. Simplify:
Answers to Extra Practice Problems
a. -12x + 13 b. 4x – 12 c. 15x – 9
Solutions to Practice Test: Questions 10 - 13
10. Solve for x:
+ 9x + 9x
22 + 2x = 40
-22 - 22
2x = 18
2 2
x = 9
11. Solve for x:
- 18 -18
30x = -13
30 30
x =
12. Solve for x:
6x – 12 + 8 = – 2x – 6 – 9
6x – 4 = – 2x – 15
+ 4 + 4
6x = – 2x – 11
+ 2x + 2x
8x = –11
8 8
x =
13. Solve for x:
18x – 4 = 48 – 21x
+21x +21x
39x – 4 = 48
+ 4 + 4
39x = 52
39 39
Extra Practice Problems
a. Solve for x: 3(2x + 2) = 2(x – 6)
b. Solve for x:
c. Solve for x:
Answers to Extra Practice Problems
a. b. c.
Solutions to Practice Test: Questions 14 - 18
14. Use or to make each statement true:
a. > –3 b. (-5) _____
_____
3 >
15. True or False: -3 is a solution of 4x – 2 3x + 13.
4(-3) – 2 3(-3) + 13
-12 – 2 < -9 + 13
-14 < 4
True
16. Graph the following inequality on a number line: x £ -5 for x >#, use ( arrow going to right of #
for x < #, use ) arrow going to left of #
for x ≥ #, use [ arrow going to right of #
-5 0 for x £ #, use ] arrow going to left of #
17. Solve for x: -6(x + 4) 42
-6x – 24 ³ 42
+ 24 + 24
-6x ³ 66
-6 -6
x £ -11
18. Graph the solution set for the following inequality: 7x + 19 > 5
– 19 –19
7x > –14
7 7
-2 0 x > -2
Extra Practice Problems
a. Graph: x 3
b. Graph: x £ 3
c. Solve and graph the solution: 10 + 2x £ 18
d. Solve and graph the solution: 5x + 3 > 7x + 15
e. True or False: 5 is a solution of 10 + 2x < 18.
Answers to Extra Practice Problems
a. b.
0 3 0 3
c. x 4 d. x < -6
0 4 -6 0
e. False
Solutions to Practice Test: Questions 19 - 21
19. Find Z if Z = xy and x = 3 and y =15.
Z = (3)(15)
Z = 45
Z = – 60
20. In the equation E = 8FG, solve for G.
E = 8FG
8F 8F
or
21. In the equation 14x + 7y = 21, solve for y.
14x + 7y = 21
– 14x – 14x
7y = 21 – 14 x
7 7 7
y = 3 – 2x or y = -2x + 3
Extra Practice Problems
a. Find A if and q = 75 and t = 15.
b. Find y if and x = -2.
c. In the equation , solve for L.
d. In the equation , solve for y.
e. In the equation T = 3(R + S), solve for S.
Answers to Extra Practice Problems
a. A = 5 b. y = 9 c. L = EA d.
e.
Solutions to Practice Test: Questions 22 - 26
22. True or False: (4, -3) is a solution for -x + 2y = -5.
-(4) + 2(-3) = -5
-4 – 6 = -5
-10 ¹ -5
False
23. Find the missing value so that ( ____, -9 ) is a solution for 3x – y = 12.
3x –(-9) = 12
3x + 9 = 12
– 9 – 9
3x = 3
3 3
x = 1
24. Write the ordered pair (x , y) for each of the points represented on the graph by the capital letter.
A
B
C
D
25. Graph 5x – 2y = 10.
x y
0 -5
2 0
26. Graph x – 3y > 6.
x y
0 -2
6 0
A dotted line is used because the inequality symbol is >.
To find where to shade, use a test point.
(0, 0) à 0 – 3(0) > 6
0 > 6
This statement is not true, so the shading is below the line since the set of the solutions are below the dotted line.
Extra Practice Problems
a. True or False: (0 , 5) is a solution for 2x – y = -5.
b. True or False: (3 , -2) is a solution for 3x – 2y = 5.
c. Write the ordered pair (x , y) for each of the points
represented on the graph by the capital letter.
d. Graph the line 2x – y = 4.
Answers to Extra Practice Problems
a. True b. False c. A (-2 , 1) B (6 , 8) C (-6 , -4) D (5 , 0)
d. e.
Solutions to Practice Test: Questions 27 - 29
27. Compute the slope of the graphed line.
·
Slope = -2
28. Compute the slope of the line that passes through the points (-8 , 6) and (2 , 4).
=
29. Write the equation of the line that passes through the points (-2 , 7) and (-5 , 1).
=
y = mx + b
(7) = 2(-2) + b
7 = -4 + b
+4 +4
11 = b
y = 2x + 11
Extra Practice Problems
a. Compute the slope of the line that passes through the points (3 , 4) and (5 , 7).
b. Compute the slope of the line that passes through the points (0 , 10) and (6 , -2).
c. Write the equation of the line that passes through the points (2, 6) and (6, 8).
d. Write the equation of the line that passes through the points (2, 0) and (-1, 9).
Answers to Extra Practice Problems
a. b. m = -2 c. d. y = 3x – 6
Solutions to Practice Test: Questions 30 - 31
30. Solve the system of equations.
6x + 4y = 14 6x + 4y = 14 6x + 4y = 14
5x + 2y = 5 Þ-2(5x + 2y) = (-2)5 -10x – 4y = -10
-4x = 4
-4 -4
x = -1
6x + 4y = 14
6(-1) + 4y = 14
-6 + 4y = 14
+6 +6
4y = 20
4 4
y = 5
(-1, 5)
31. Solve the system of equations.
x = 2y -3
3x – 5y = 7
3(2y – 3) – 5y = 7
6y – 9 – 5y = 7
y – 9 = 7
+9 +9
y = 16
x = 2y – 3
x = 2(16) – 3
x = 32 – 3
x = 29
(29, 16)
Extra Practice Problems
a. b.
c. d.
Answers to Extra Practice Problems
a. (5 , 2) b. (-3 , 5) c. (2 , 1) d. No Solution
Solutions to Practice Test: Questions 32 - 37
32. Simplify:
= (-3)2 (x-3)2 (y14)2
=9x-6y28
=
33. Simplify:
= -36a4+0b7+-6
= -36a4b1 = -36a4b
34. Simplify:
=
35. Write the following number in scientific notation: 53,000,000
5.3 x 107
36. Write the following number in scientific notation: 0.0006372
6.372 x 10-4
37. Write the following number in decimal notation: 0.0968 x 106
96,800
Extra Practice Problems
a. Simplify b. Simplify
c. Simplify d. Simplify
e. Simplify
f. Write the following number in scientific notation: 0.00000000034
g. Write the following number in scientific notation: 547000
h. Write the following number in decimal notation: 6.8725 x 106
Answers to Extra Practice Problems
a. b. c. d. e. f. 3.4 x 10-10
g. 5.47 x 105 h. 6,872,500
Solutions to Practice Test: Questions 38 - 40
38. Multiply: 2x(x + 7)
2x(x + 7)
= 2x2 + 14x
39. Multiply: (3x – 4)(x + 5)
(3x – 4)(x + 5) = 3x2 + 15x – 4x – 20
= 3x2 + 11x – 20
40. Multiply: (x – 3)(x2 – 2x +8) = x3 – 2x2 + 8x – 3x2 + 6x – 24
= x3 – 5x2 + 14x – 24
Extra Practice Problems: Multiply.
a. -5x(2x + 3) b. (3x + 2)(x – 3) c. (x + 2)(x – 2) d. (x + 2)(x2 – 3x + 5)
Answers to Extra Practice Problems: Multiply
a. -10x2 – 15x b. 3x2 – 7x – 6 c. x2 – 4 d. x3 – x2 – x – 10
Solutions to Practice Test: Questions 41 - 43
41. Factor completely: 16x2 – 24x
= 8x(2x – 3)
42. Factor completely: x2 – 64
= (x + 8)(x – 8)
43. Factor completely: x2 – 14x – 15
= x2 – 15x + 1x – 15
= x(x – 15) + 1(x – 15)