Section 9.4 Llnear Inequalities In Two Variables
11.
1
3
yx
Replacing the inequality symbol with an equal sign,
we have
1.
3
yx
Since the equation is in slope-
This is a true statement, so we know the point
1,1
lies in the shaded half-plane.
12.
13. 32yx
First, graph the equation 32yx
. Since the
This is a true statement. This means that the point
0,0
will fall in the shaded half-plane.
15.
1
4
yx
Replacing the inequality symbol with an equal sign,
we have
1.
4
yx
Since the equation is in slope-
1,1
as a test point.
1
11
4
1
14
This is a false statement, so we know the point
1,1
does not lie in the shaded half-plane.
Chapter 9 Inequalities and Problem Solving
17.
2
x
Replacing the inequality symbol with an equal sign,
we have
2.
x
We know that equations of the
form x = a are vertical lines with x -intercept = a.
Next, use the origin as a test point.
2
x
18.
19. 4y
Replacing the inequality symbol with an equal sign,
we have 4.y We know that equations of the
21. 0y
Replacing the inequality symbol with an equal
sign, we have 0y. We know that equations of
the form y = b are horizontal lines with
y-intercept = b. In this case, we have 0y, the
equation of the x-axis.
22.
intercept 1 intercept 8
yy
Use the origin as a test point to determine shading.
Section 9.4 Llnear Inequalities In Two Variables
24.
25.
2510
32 6
xy
xy
Graph the equations using the intercepts.
2510 32 6
intercept 5 intercept 2
xy xy
xx
26.
27.
23
6
yx
yx
Graph the equations using the intercepts.
23 6
yx yx
28.
29.
24
3
xy
yx
Graph the equations using the intercepts.
24 3
xy yx
half-planes.
30.
31.
2
1
x
y
Chapter 9 Inequalities and Problem Solving
32.
33.
25
x
Since x lies between –2 and 5, graph the two
vertical lines, x = –2 and x = 5. Since x lies between
34.
35.
1
2
xy
x
Graph the equations.
1 2
intercept 1 intercept 2
xy x
xx
36.
37.
4
1
xy
xy
or .
38.
3
2
xy
xy
Section 9.4 Llnear Inequalities In Two Variables
39.
4
1
xy
xy
Graph the equations using the intercepts.
4 1
xy xy
40.
41.
2
2
3
xy
x
y
Use the origin as a test point to determine shading.
42.
2510 34 12
intercept 5 intercept 4
intercept 2 intercept 3
xy xy
xx
yy
Use the origin as a test point to determine shading.
Chapter 9 Inequalities and Problem Solving
45.
36
21
2
4
xy
xy
x
y
Graph the equations using the intercepts.
1750 550 2300
Use the origin as a test point to determine shading.
included in the solution set.
46.
47.
48.
50.
53. Find the union of solutions of
32
2
yx
and
Section 9.4 Llnear Inequalities In Two Variables
54. Find the union of solutions of 1xy
and
52 10xy
.
55. The system
33 9
33 9
xy
xy
has no solution. The
number 33xy cannot both be less than 9 and
greater than 9 at the same time.
39xy.
58. The system
624
624
xy
xy
has infinitely many
solutions. The solutions are all points on the line
624xy .
59. a. The coordinates of point A are
20,150 .
This
b.
10 70
10 20 70, true
a
0.7 220
Ha
60. a. The coordinates of point B are
40,130 .
This
means that a 40-year-old person with a heart rate
of 130 beats per minute falls within the target
zone.
b.
10 70
a
130 0.7 220 40
130 126, true
0.8 220
Ha
62.
10 70
0.5 220
0.6 220
a
Ha
Ha
63. a. 0
5
1
y
xy
x
Chapter 9 Inequalities and Problem Solving
64. a. 0
0
80 160 2000
x
y
xy
65. – 74. Answers will vary.
75. 44yx
76. 22
3
yx
77. 26
26
xy
yx
78. 32 6
236
33
xy
yx
yx
79–81. Answers will vary.
82. does not make sense; Explanations will vary.
Sample explanation: If (0, 0) is on the line, it can
not be selected as the test point.
85. makes sense
86. false; Changes to make the statement true will vary.
A sample change is: The graph of 35 10xy
consists of a dashed line and a shaded half-plane
above the line.
A sample change is: The ordered pair (–2, 40) does
not satisfy 13 14.xy
89. true
90. The related line has a y-intercept of –2 and a slope
of 1 giving 2.yx The shading is above a
Section 9.4 Llnear Inequalities In Two Variables
92. The dashed line has a y-intercept of –3 and a slope
of 1 giving 3.yx The solid line has a y–
93. The slope of the linear equation is
21
21
6 ( 8) 14 2.
4(3) 7
yy
mxx
Use the point slope form to find the of the linear
equation.
94. An answer is not possible. A system implies an
intersection and it is impossible to intersect two
95.
96. 3 8
52
xy
xy
12
318
152
RR
Since we know y = 1, we can use back-substitution
to find x.
The solution is
3,1 .
97. 32
28
yx
yx
The solution is
2, 4 .
42 2
42
(2 20 50 )
(5)
xx xy y
xx y
3
100.
() 3 12
(1) 3(8) 12
36
6
fx x
f
Chapter 9 Inequalities and Problem Solving
Chapter 9 Review
1.
6315
612
x
x
2. 69 43
10 9 3
10 6
xx
x
x
3. 31
34 2
xx
3
12 12 12 1 12
34 2
433126
xx
xx
The solution set is
21
2
xx
or
21
,.
2
4.
65 2 325
65 2625
65 219
xx
xx
xx
5.
32 1 2 4 7 23 4
6328768
45138
xx x
xx x
xx
125 40 357,000
125 40 357,000
85 357,000
xx
xx
x
b.
0
Px
1150Rx x
9.
profit revenue cost
Px Rx Cx
Chapter 9 Review Exercises
10.
0
300 360,000 0
Px
x
11. Let x = the number of checks written per month.
The cost using the first method is
1
11 0.06 .cx
The cost using the second method is
2
40.20.cx
The first method is a better deal if it costs less than
12. Let x = the amount of sales per month in dollars.
The salesperson’s commission is
500 0.20 .cx
We are looking for the amount of sales, x, the
13.
,
AB ac
14.
AC a
17.
3 and 6
xx
The solution set is
3
xx
or
,3 .
18.
3 or 6
xx
31
xx
Chapter 9 Inequalities and Problem Solving
21.
25 1and3 3
24 1
2
xx
xx
x
22.
251or33
24 1
2
xx
xx
x
The solution set is
1 or 2
xx x
or
,1 2,
.
23.
13or435
448
2
xx
xx
x
24.
52 22or324
520 36
42
xx
xx
xx
25.
54 11or14 9
515 48
32
xx
xx
xx
is a real number
xx .
26.
324
32 22 42
52
x
x
x
27. 14 26
12 4 22 62
34 4
34 4
444
31
4
x
x
x
x
x
5
351
80 90
5
351
580 5 590
x
x
Chapter 9 Review Exercises
29. 217
x
30. 32 5x
There are no values of x for which the absolute
31. 23710
2317
38.5
x
x
x
32. 4379xx
4379
xx
or
4379
xx
33.
2315
15 2 3 15
x
x
26
x
60
xx
The solution set is
6 or 0xx x
or
12 51
15 2 5515
62 4
x
x
x
36.
4257
4212
x
x