Chapter 7 Practice Test Problems
Write the partial fraction decomposition of the rational
expression.
x - 2
1)
(x - 4)(x - 3)
2)
3)
4)
Solve the system by the addition method.
16) x2 + y2 = 9
x2 - y2 = 9
17) x2 + y2 = 25
25x2 + 16y2 = 400
x
x2 + 3x + 2
18) x2 + y2 = 53
x2 - y2 = 45
3x2 - x - 14
x3 - x
19) x2 - 3y2 = 1
2x2 + 3y2 = 11
15x + 45
x2 + 7x + 10
20) 2x2 + y 2 = 66
5)
5x + 7
(x - 3)2
x2 + y 2 = 41
21) x2 - 3y2 - 1 = 0
6)
7)
x + 5
3
x - 2x2 + x
4x2 + 3y2 - 19 = 0
22) y2 + 2x2 = 13
y2 - x2 = 1
-8x2 - 23x - 18
(x + 2)(x + 1)2
23) x2 + y2 - 8x - 8y + 31 = 0
x2 - y2 - 8x + 8y - 1 = 0
Solve the system by the substitution method.
8) 15x - y = 58
y = x2 - 2
Solve by the method of your choice.
24) x2 + y2 = 41
9) y = x - 2
y2 = -8x
(x - 5)2 + y2 = 16
25) x3 + y = 0
8x2 - y = 0
10) xy = 72
x + y = -17
Graph the inequality.
26) x - y < -4
11) x2 = y 2 + 39
x - y = 3
y
10
12) x - 2y = 3
x2 - xy = 20
13) 3y - x = 10
x2 + y2 - 100 = 0
-10
14) y = x2 - 3
x2 + y2 = 5
10
-10
15) x + y = -8
x2 + y2 = 10y + 60
1
x
31) (x - 5)2 + (y - 4)2 > 4
27) x + 5y ≥ 1
y
y
10
10
5
-10
10
x
-10
-5
5
10
-5
-10
-10
28) 2x + 5y ≤ 10
32) y ≤ x2 + 7
y
10
10
y
5
-10
10
-10
x
-5
5
10 x
5
10 x
-5
-10
-10
33) y ≤ 3 x
29) y ≤ -4x - 4
10
y
y
10
5
5
-10
-10
-5
5
10
x
-5
-5
-5
-10
-10
30) x2 + y2 ≤ 16
10
y
5
-10
-5
5
10 x
-5
-10
2
x
37) y > x2
3x + 6y ≤ 18
Graph the solution set of the system of inequalities or
indicate that the system has no solution.
34) 3x - y ≤ -3
x + 4y ≥ -4
y
10
y
10
5
8
6
4
-10
2
-10 -8 -6 -4 -2
-2
2
4
6
-5
5
10
x
10
x
10
x
-5
8 10 x
-4
-10
-6
-8
-10
38) y ≥ 3x
y ≤ 8
35) x + 2y ≥ 2
x - y ≤ 0
y
10
y
6
4
-10
2
-6
-4
-2
2
6 x
4
-2
-10
-4
39) x2 + y 2 ≤ 25
y - x2 > 0
-6
y
36) -x + 6y < 30
x ≥ 2
10
y
5
10
-10
5
-5
5
-5
-10
-5
5
10
x
-10
-5
-10
3
40) x2 + y 2 ≤ 49
x + y > 1
43) Mrs. White wants to crochet hats and afghans for
a church fundraising bazaar. She needs 8 hours to
make a hat and 3 hours to make an afghan, and
she has no more than 51 hours available. She has
material for no more than 12 items, and she wants
to make at least two afghans. Let x = the number
of hats she makes and y = the number of afghans
she makes. Write a system of inequalities that
describes these constraints.
y
10
5
-10
-5
5
10
x
-5
Find the maximum or minimum value of the given objective
function of a linear programming problem. The figure
illustrates the graph of the feasible points.
44) Objective Function: z = 5x + 8y
Find maximum and minimum.
-10
41) (x + 1)2 + (y + 3)2 > 16
(x + 1)2 + (y + 3)2 < 36
y
y
10
(0, 9)
(9, 9)
5
-10
-5
5
10
x
(9, 3)
(0, 3)
-5
(3, 0)
-10
x
Find the maximum or minimum value of the given objective
function of a linear programming problem. The figure
illustrates the graph of feasible points.
Solve the problem.
42) A steel company produces two types of machine
dies, part A and part B and is bound by the
following constraints:
· Part A requires 1 hour of casting time and 10
hours of firing time.
· Part B requires 4 hours of casting time and 3
hours of firing time.
· The maximum number of hours per week
available for casting and firing are 100 and 70,
respectively.
· The cost to the company is $0.75 per part A and
$3.00 per part B. Total weekly costs cannot exceed
$45.00.
Let x = the number of part A produced in a week
and y = the number of part B produced in a week.
Write a system of three inequalities that describes
these constraints.
45) Objective Function: z = -x - 8y
Find maximum.
46) Objective Function: z = x + 6y + 9
Find minimum.
4
52) An office manager is buying used filing cabinets.
Each small file cabinet costs $7, takes up 5 square
feet of floor space, and holds 10 cubic feet of files.
Each large file cabinet costs $8, takes up 10 square
feet of floor space, and holds 14 cubic feet of files.
The total cost cannot exceed $87, and the office
has no more than 75 square feet of floor space
available for the cabinets. How many of each file
cabinet should the manager buy to maximize
storage capacity? What is the maximum storage
capacity?
An objective function and a system of linear inequalities
representing constraints are given. Graph the system of
inequalities representing the constraints. Find the value of
the objective function at each corner of the graphed region.
Use these values to determine the maximum value of the
objective function and the values of x and y for which the
maximum occurs.
47) Objective Function
z = 12x + 5y
Constraints
0 ≤ x ≤ 10
0 ≤ y ≤ 5
3x + 2y ≥ 6
48) Objective Function
Constraints
49) Objective Function
Constraints
z = 3x + 5y
x ≥ 0
y ≥ 0
2x + y ≤ 15
x - 3y ≥ -3
z = 8x + 9y
x ≥ 0
0 ≤ y ≤ 5
2x + 3y ≥ 12
2x + 3y ≤ 20
Solve the problem.
50) A vineyard produces two special wines, a white
and a red. A bottle of the white wine requires 14
pounds of grapes and 1 hour of processing time.
A bottle of red wine requires 25 pounds of grapes
and 2 hours of processing time. The vineyard has
on hand 2,198 pounds of grapes and can allot 160
hours of processing time to the production of
these wines. A bottle of the white wine sells for
$11.00, while a bottle of the red wine sells for
$20.00. How many bottles of each type should the
vineyard produce in order to maximize gross
sales?
51) A doctor has told a sick patient to take vitamin
pills. The patient needs at least 32 units of vitamin
A, at least 8 units of vitamin B, and at least 31
units of vitamin C. The red vitamin pills cost 10¢
each and contain 7 units of A, 1 unit of B, and 2
units of C. The blue vitamin pills cost 20¢ each
and contain 3 units of A, 1 unit of B, and 7 units of
C. How many pills should the patient take each
day to minimize costs?
5
Answer Key
Testname: MTH 112 BTZ PRACTICE PROBLEMS (CH. 7)
1)
2
-1
+ x - 4 x - 3
2)
2
-1
+ x + 2 x + 1
27)
y
y
10
10
5
14
-5
-6
3)
+ + x
x + 1 x - 1
-10
5
10
4)
+ x + 2 x + 5
-10
10
-5
10
x
x
-5
5
22
5)
+ x - 3
(x - 3)2
6)
5
-10
5
6
-5
+ + x x - 1 (x - 1)2
31)
32)
-10
28)
-4
-4
-3
7)
+ + x + 2 x + 1 (x + 1)2
y
10
y
10
5
8) {(8, 62), (7, 47)}
9) {(-2, -4)}
10) {(-8, -9), (-9, -8)}
11) {(8, 5)}
11
12) {(5, 1), (-8, - )}
2
-10
-10
13) {(-10, 0), (8, 6)}
14) {(-2, 1), (-1, -2), (1, -2), (2, 1)}
15) {(-7, -1), (-6, -2)}
16) {(3, 0), (-3, 0)}
17) {(0, 5), (0, -5)}
18) {(7, 2), (-7, 2), (7, -2), (-7, -2)}
19) {(2, 1), (-2, 1), (2, -1), (-2, -1)}
20) {(5, 4), (5, -4), (-5, 4), (-5, -4)}
21) {(2, 1), (2, -1), (-2, 1), (-2, -1)}
22) (2, 5), (2, - 5), (-2, 5), (-2, 5)
23) {(5, 4), (3, 4)}
24) {(5, 4), (5, -4)}
25) {(0, 0), (-8, 512)}
26)
10
-5
x
5
10 x
5
10 x
-5
-10
33)
-10
y
10
29)
y
5
10
5
-10
-5
-5
-10
-5
5
10
-10
-5
34)
y
-10
10
y
10
x
8
30)
6
10
y
4
2
5
-10
10
-4
-10
-10
-10 -8 -6 -4 -2
-2
x
-5
5
-5
-10
6
10 x
-6
-8
-10
2
4
6
8 10 x
Answer Key
Testname: MTH 112 BTZ PRACTICE PROBLEMS (CH. 7)
35)
52) 9 small file cabinets and 3 large
file cabinets; 132 cubic feet
y
10
y
6
5
4
2
-10
-6
-4
-2
2
4
-5
5
10
x
5
10
x
5
10
x
-5
6
-2
-10
-4
39)
y
-6
10
36)
5
y
10
-10
5
-5
-5
-10
-5
5
10
x
-10
40)
-5
y
10
-10
37)
5
y
10
-10
5
-10
-5
-5
5
10
-10
38)
y
10
10 x
-10
-10
x
41)
42)
-5
-10
-5
x + 4y ≤ 100
10x + 3y ≤ 70
0.75x + 3y ≤ 45
43) 8x + 3y ≤ 51
x + y ≤ 12
y ≥ 2
44) maximum value: 117; minimum
value: 15
45) maximum: -20
46) minimum: 25
47) Maximum: 145; at (10, 5)
48) Maximum 33; at (6, 3)
49) Maximum: 80; at (10, 0)
50) 132 bottles of white and 14 bottles
of red
51) 5 red and 3 blue
7