College Algebra 9th txtbk

SECTION 7–7 Systems of Linear Inequalities in Two Variables 19
In Problems 19–24, solve each system of linear inequalities
graphically.
19. x 5
20. x 4 21. 3x y 6
y 6
y 2
x 4
22. 3x 4y 12 23. x 2y 12 24. 2x 5y 20
y 3 2x y 4 x 5y 5
B
In Problems 25–28, match the solution region of each system of
linear inequalities with one of the four regions shown in the figure.
Identify the corner points of each solution region.
(0, 16)
10
(0, 6)
(0, 0)
y
5 (8, 0) 10
5
II
III
(6, 4)
IV
2x y 16
(18, 0)
25. x 3y 18 26. x 3y 18
2x y 16 2x y 16
x 0
x 0
y 0
y 0
I
27. x 3y 18 28. x 3y 18
2x y 16 2x y 16
Region II; corner
x 0
x 0 points: (6, 4), (0, 6),
y 0
y 0 (0, 16)
Region I, corner points:
(0, 16), (6, 4), (18, 0)
In Problems 29–40, solve the systems graphically, and indicate
whether each solution region is bounded or unbounded. Find the
coordinates of each corner point.
29. 2x 3y 6 30. 4x 3y 12 31. 4x 5y 20
x 0
x 0
x 0
y 0
y 0
y 0
32. 5x 6y 30 33. 2x y 8 34. x 2y 10
x 0 x 3y 12 3x y 15
y 0
x 0
x 0
y 0
y 0
35. 4x 3y 24 36. x 2y 8 37. 2x y 12
2x 3y 18 2x y 10 x y 7
x 0
x 0 x 2y 10
y 0
y 0
x 0
y 0
38. 3x y 21 39. x 2y 16 40. 3x y 30
x y 9 x y 12 x y 16
x 3y 21 2x y 14 x 3y 24
x 0
x 0
x 0
y 0
y 0
y 0
x
x 3y 18
Region III; corner
points: (0, 0), (0, 6),
(6, 4), (8, 0)
C
In Problems 41–48, solve the systems graphically, and indicate
whether each solution region is bounded or unbounded. Find the
coordinates of each corner point.
41. x y 11
42. 4x y 32
5x y 15
x 3y 30
x 2y 12
5x 4y 51
43. 3x 2y 24
44. 3x 4y 48
3x y 15
x 2y 24
x 4
y 9
The feasible region is empty. The feasible region is empty.
45. x y 10 46. 3x y 1
3x 5y 15
x 5y 9
3x 2y 15
x y 9
5x 2y 6
y 5
47. 16x 13y 119 48. 8x 4y 41
12x 16y 101
15x 5y 19
4x 3y 11
2x 6y 37
APPLICATIONS
49. MANUFACTURING—RESOURCE ALLOCATION A manufacturing
company makes two types of water skis: a trick ski and a slalom ski.
The trick ski requires 6 labor-hours for fabricating and 1 labor-hour
for finishing. The slalom ski requires 4 labor-hours for fabricating
and 1 labor-hour for finishing. The maximum labor-hours available
per day for fabricating and finishing are 108 and 24, respectively. If
x is the number of trick skis and y is the number of slalom skis
produced per day, write a system of inequalities that indicates appropriate
restraints on x and y. Find the set of feasible solutions
graphically for the number of each type of ski that can be produced.
50. MANUFACTURING—RESOURCE ALLOCATION A furniture manufacturing
company manufactures dining room tables and chairs. A
table requires 8 labor-hours for assembling and 2 labor-hours for
finishing. A chair requires 2 labor-hours for assembling and 1
labor-hour for finishing. The maximum labor-hours available per
day for assembly and finishing are 400 and 120, respectively. If x is
the number of tables and y is the number of chairs produced per day,
write a system of inequalities that indicates appropriate restraints
on x and y. Find the set of feasible solutions graphically for the
number of tables and chairs that can be produced.
51. MANUFACTURING—RESOURCE ALLOCATION Refer to Problem 49.
The company makes a profit of $50 on each trick ski and a profit of
$60 on each slalom ski.
(A) If the company makes 10 trick and 10 slalom skis per day, the
daily profit will be $1,100. Are there other feasible production schedules
that will result in a daily profit of $1,100? How are these schedules
related to the graph of the line 50x 60y 1,100?
(B) Find a feasible production schedule that will produce a daily
profit greater than $1,100 and repeat part A for this schedule.
(C) Discuss methods for using lines like those in parts A and B to
find the largest possible daily profit.
52. MANUFACTURING—RESOURCE ALLOCATION Refer to Problem
50. The company makes a profit of $50 on each table and a
profit of $15 on each chair.
(A) If the company makes 20 tables and 20 chairs per day, the daily
profit will be $1,300. Are there other feasible production schedules
that will result in a daily profit of $1,300? How are these schedules
related to the graph of the line 50x 15y 1,300?