Chapter 9 Inequalities and Problem Solving
37.
234 10
23 14
x
x
38.
6.5 1
16.51
h
h
39. 34 12xy
First, find the intercepts to the equation
34 12xy
.
Find the x–intercept by setting y = 0.
34 12
340
xy
x
12
This is a false statement. This means that the point,
0, 0 , will not fall in the shaded half-plane.
40. 36xy
First, find the intercepts to the equation 36xy
.
36
6
2
y
x
y
Next, use the origin as a test point.
0, 0 , will fall in the shaded half-plane.
2
1
00
2
yx
2
02
This is a true statement. This means that the point
Chapter 9 Review Exercises
42. 3
5
yx
Replacing the inequality symbol with an equal sign,
we have 3
5
yx. Since the equation is in slope-
intercept form, use the slope and the intercept to
graph the equation. The y–intercept is 0 and the
This is a true statement, so we know the point
1,1
lies in the shaded half-plane.
43. 2x
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
44. 3y
Replacing the inequality symbol with an equal sign,
we have 3.y We know that equations of the
This is a true statement, so we know the point
0, 0
lies in the shaded half-plane.
24
24
yx
yx
y-intercept = –4
slope = 2
Now, use the origin as a test point.
Next consider 5xy. If we solve for y in
5xy, we can graph using the slope and the
y-intercept.
5
5
xy
yx
y -intercept = 5
Chapter 9 Inequalities and Problem Solving
46. 4
4
yx
yx
First consider 4.yx Change the inequality
slope and the y–intercept.
y–intercept = –4
slope = 1
Now, use the origin as a test point.
004
04
This is a true statement. This means that the point
47. 35x
Rewrite the three part inequality as two separate
inequalities. We have 3x and 5.x We
replace the inequality symbols with equal signs and
obtain 3x and 5.x Equations of the form x
48. 26y
Rewrite the three part inequality as two separate
inequalities. We have 2y and 6.y We
49. 3
0
x
y
First consider 3.x Change the inequality symbol
to an equal sign and we obtain the vertical line
3.x Because we have 3,x we know the
shading is to the right of the line 3.x
Next consider 0.y Change the inequality symbol
50. 24
xy
Chapter 9 Review Exercises
Now, use the origin as a test point.
24
20
xy
04
04
This is a true statement. This means that the point
0, 0 will fall in the shaded half-plane.
Next consider 0.x Change the inequality symbol
51. 6
23
xy
yx
First consider 6.xy Replace the inequality
Now, use the origin as a test point.
00 6
06
This is a true statement. This means that the point
0, 0 will fall in the shaded half-plane.
This is a true statement. This means that the point
0, 0 will fall in the shaded half-plane.
Next, graph each of the inequalities. The solution to
the system is the intersection of the shaded half-
planes.
234
32
2
yx
yx
This is a false statement. This means that the point
0, 0 will not fall in the shaded half-plane.
Now consider 3.xy Replace the inequality
symbol with an equal sign and we have 3.xy
Solve for y to obtain slope-intercept form.
Chapter 9 Inequalities and Problem Solving
Now consider the inequalities 0 and 0.xy
The
inequalities mean that both x and y will be positive.
This means that we only need to consider quadrant I.
53. 22
22
xy
xy
First consider 22.xy Replace the inequality
symbol with an equal sign and we have 22.xy
This is a false statement. This means that the point
0, 0 will not fall in the shaded half-plane.
Now consider
22.xy
Replace the inequality
symbol with an equal sign and we have
22.xy
Solve for y to obtain slope-intercept
form.
22
xy
This is a false statement. This means that the point
0, 0 will not fall in the shaded half-plane.
5
y
1.
34512
312512
212 12
xx
xx
x
68 24
()
13
24 24 24 24
68 24
431263
4 3 12 18
83 18
xx
xx
xx
x
+≤−
+≤ −
+≤ −
−+≤−
The solution set is 21 ,.
8
Chapter 9 Test
3. a.
cost fixed costs variable cost
60,000 200Cx x
d.
0
250 60, 000 0
250 60, 000
240
Px
x
x
x
More than 240 computer desks need to be
produced and sold to make a profit.
6. 242and 35
22 2
1
xx
xx
x
7. 64 and2 3 2
225
5
2
xx
xx
x
8.
235or364
28 310
xx
xx
9. 31or435
448
2
xx
xx
x
10.
25
36
3
25
33 3 36
3
x
x
The solution set is 13
7, 2
.
11.
537
x
Chapter 9 Inequalities and Problem Solving
12.
61415xx
61415
2115
xx
x
The solutions are
8 and 7
5
and the solution set is
8,7 .
5
13.
217
72 17
x
x
14.
235x
235or235
xx
15.
98.6 8b
98.6 8 or 98.6 8
bb
320
x
6
36
2
x
x
Find the y–intercept by setting x = 0.
32 6
30
xy
26
y
will fall in the shaded half-plane.
Chapter 9 Test
17.
11
2
yx
Replacing the inequality symbol with an equal sign,
we have
11
2
yx
. The equation is in slope-
This is a true statement. This means that the point
will fall in the shaded half-plane.
18. 1y
Replacing the inequality symbol with an equal sign,
we have 1.y Equations of the form y = b are
horizontal lines with y –intercept = b, so this is a
horizontal line at 1.y
Next, use the origin as a test point.
19.
2
4
xy
xy
First consider 2.xy If we solve for y in
2,xy we can graph the line using the slope
will not fall in the shaded half-plane.
Next consider 4xy. If we solve for y in
4xy, we can graph using the slope and the y–
intercept.
Now, use the origin as a test point.
4
00 4
04
xy
Chapter 9 Inequalities and Problem Solving
20.
39
23 6
0, 0
xy
xy
xy
Now, use the origin as a test point.
39
30
xy
09
09
y-intercept = 2 slope =
2
3
Now, use the origin as a test point.
23 6
20
xy
306
06
Graph each of the inequalities. The solution to the
system is the intersection of the shaded half-planes.
21.
24x
Rewrite the three part inequality as two separate
inequalities. We have
2x
and
4.x
We
replace the inequality symbols with equal signs and
obtain
2x
and
4.x
Equations of the form x
Cumulative Review Exercises (Chapters 1 – 9)
1.
512 321
552 63
57 53
xxx
xxx
xx
Cumulative Review
2.
26 47
1
33
xx
3.
24 43
27
73
7
57
5
10 10
15
15
22
33
xy xy
xy
y
xy x
5.
2
341fx x x
2
51gx x x
6. Since the line we are concerned with is
perpendicular to the line, 23,yx
we know the
notation.
1
32
2
1
31
2
yx
yx
14
Find the x–intercept by setting y = 0, and the y–
intercept by setting x = 0.
21 21
yx yx
Chapter 9 Inequalities and Problem Solving
8. 2yx
Consider the line 2yx. Since the line is in slope–
intercept form, we know that the slope is 2 and the
y–intercept is 0. Use this information to graph the
line.
9.
26xy
Graph the equation using the intercepts.
26
xy
10.
1
1
fx
y
Equations of the form y = b are horizontal lines with
y–intercept = b. This is the horizontal line at y = –1.
11. 3 15
2 1
232 0
xyz
xyz
xyz
Add the first two equations to eliminate z.
3 15
xyz
The system of two equations in two variables
becomes as follows.
414
830
xy
xy
830
4 16
4
xy
x
x
Back-substitute –4 for x to find y.
1
z
The solution is
4, 2, 1 and the solution set is
4, 2, 1 .
12.
3
() 4
x
fx
Use y notation:
3
4
x
y
Interchang x and y and solve for y.
3
4
y
x
Cumulative Review
13.
2
() 3 1fx x and () 2gx x.
2
() 3( 2) 1
fgx x
14. Let x = the number of rooms with a kitchen.
Let y = the number of rooms without a kitchen.
60
90 80 5260
xy
xy
Solve the first equation for y.
60
xy
15. Using the vertical line test, we see that graphs a. and
b. are functions.
16.
31
44 2
3
44414
44 2
xx
xx
17.
2 5 11 and 3 18
26 6
xx
xx
18.
41or 3 1 5
5215
26
3
xxx
xx
x
x
The solution set is
10,7 .
20.
387
38 7or387
x
xx