Chapter 4 Solving Systems of Linear Equations and Inequalities
76
Mindstretchers
1. The answer is obtained by solving the
system of equations xy+=sum and xy−=
2. To eliminate the x-terms (or the y-terms),
identify the LCM of the coefficients and
multiply one or both equations by the
3. 4330
326 5
5
xy z
xyz
xz
−− =
−−=−
−=
4.4 Solving Systems of
Linear Inequalities
ordered pairs.
4. In the graph of a system of linear
()
Test point is 0,0 .
()
?
030 2
0
<−
<2−
Section 4.4 Solving Systems of Linear Inequalities
77
8. 11
4
The boundary line is dashed.
yx>− −
()
11,
4
xy x xy=− −
()
()
?
Test point is 0,0 .
1
001
4
01
>− −
>−
Shade the half-plane containing
()
0,0 .
3
2
yx≥
Shade the half-plane not containing
()
4,0 .
10. 3912
3912
34
yx
yx
yx
−≤
≤+
≤+
The boundary line is solid.
() ()
?
30 90 12
012
−≤
≤
Shade the half-plane containing
()
0,0 .
5515
5515
3
yx
yx
yx
+≤
≤− +
≤− +
Chapter 4 Solving Systems of Linear Equations and Inequalities
78
12. 2416
4216
14
xy
yx
yx
−− <
−< +
>− −
Shade the-half plane containing
()
0,0 .
28
28
xy
yx
−≥
−≥−+
()
()
?
Test point is 0,0 .
020 8
0
−≥
≥8
Shade the half-plane not containing
()
0,0 .
14. 420
420
15
xy
yx
yx
+≤
≤− +
≤− +
Shade the half-plane containing
()
0,0 .
312 24
12 3 24
xy
yx
+>−
>− −
() ()
?
30 120 24
024
+>−
>−
Shade the half-plane containing
()
0,0 .
Section 4.4 Solving Systems of Linear Inequalities
79
16. 1.2 0.4 0.4
0.4 1.2 0.4
31
xy
yx
yx
+<−
<− −
<− −
The boundary line is dashed.
()
() ()
() ()
The boundary line is dashed.
12 ,
33
12
11 11,1
33
12
22 02,0
33
xyx xy=−
−−−=−−−
−=
18. 3
3
xy
yx
+<
<− +
The boundary line is dashed.
3,
xy x xy=− +
()
()
()
3
The boundary line 3 is solid.
Test point is 0,0 .
03
y
y
≥−
=−
≥−
Shade the half-plane containing
()
0,0 .
Chapter 4 Solving Systems of Linear Equations and Inequalities
80
20. 21
21
yx
yx
+≥−
≥− −
The boundary line is solid.
()
()
()
()
2
The boundary line is solid.
2,
202,0
222,2
Test point is 0,0 .
02
x
xyxy
≤
=
≤
Shade the half-plane containing
()
0,0 .
Shade the half-plane containing
()
0,0 .
22. 48 8
848
11
xy
yx
+>
>− +
() ()
?
40 80 8 0+>⇒>8
Shade the half-plane not containing
()
0,0 .
23
23
23
xy
yx
yx
−>
−>− +
<−
()
The boundary line is dashed.
23 ,
xyx xy=−
()
()
()
The boundary line is dashed.
5,
505,0
525,2
xyxy
=
()
Test point is 0,0 .
05<
Shade the half-plane containing
()
0,0 .
Section 4.4 Solving Systems of Linear Inequalities
81
24. 10 5 5
5105
21
xy
yx
yx
+>−
>− −
>− −
The boundary line is dashed.
Shade the half-plane containing
()
0,0 .
93 0
39
3
xy
yx
yx
−≥
−≥−
≤
The boundary line is solid.
Shade the half-plane containing
()
2,0 .
4
The boundary line is solid.
y≥−
()
()
()
4,
040,4
242,4
xy xy=−
−−
−−
26. 3
3
xy
yx
+≤−
≤− −
The boundary line is solid.
1
1
yx
yx
−≤
≤+
()
() ( )
The boundary line is solid.
1,
00110,1
1112 1,2
xyx xy=+
+=
+=
2
yx
>−
()
() ( )
() ( )
The boundary line is dashed.
11,
2
1
00110,1
2
1
22102,0
xy x xy=−
−=− −
−=
Chapter 4 Solving Systems of Linear Equations and Inequalities
82
28. 32 6
236
33
2
xy
yx
yx
−+ ≤−
≤−
≤−
The boundary line is solid.
Shade the half-plane not containing
()
0,0 .
11
4
11
4
xy
yx
+>
>− +
The boundary line is dashed.
()
() ()
?
Test point is 0,0
1001
4
0
+>
>1
Shade the half-plane not containing
()
0,0 .
30. 1x≥
The boundary line 1x= is solid.
3y>−
The boundary line 3y=− is dashed.
()
() ( )
() ( )
21 ,
020110,1
222132,3
xy x xy=− −
−−=− −
−−=− −
33
33
xy
yx
−+≤
≤+
The boundary line is solid.
Section 4.4 Solving Systems of Linear Inequalities
83
34. a. Let x be the number of part A produced and
y be the number of part B produced.
()
() ( )
() ( )
575 ,
2
5
0 0 75 75 0,75
2
5
30 30 75 0 30,0
2
xy x xy=− +
−+=
−+=
()
() ( )
() ( )
()
The boundary line is solid.
380 ,
03080800,80
25 3 25 80 5 25,5
Test point is 0,0 .
xy x xy
=− +
−+=
−+=
36. a. Let x be the amount of money invested
at 6.5% and y be the amount invested at
5%.
()
() ( )
() ( )
8500 ,
0 0 8500 8500 0,8500
8500 8500 8500 0 8500,0
xyx xy=− +
−+ =
−+=
()
?
Test point is 0,0 .
0 0 8500
+≤
() ()
()
() ()
?
1.3 4000 6000
4000 4000,800
800
Test point is 0,0 .
0.065 0 0.05 0 300
0
−+
=
+≥
≥300
Chapter 4 Solving Systems of Linear Equations and Inequalities
84
36. b. (continued)
38. a. Let x be the number of shares of
pharmaceutical stock and y be the number of
shares of technology stock purchases.
40 25 10,000
xy
+=
()
?
Test point is 0,0 .
38. b. (continued)
200
The boundary line is solid.
x≤
()
()
200 ,
200 0 200,0
xyxy=
()
()
250 50 250,50
Test point is 0,0 .
0
≥50
40 7500
187.5
x
x
≤
≤
Section 4.4 Solving Systems of Linear Inequalities
85
40. a. Let x be the number of donuts and y be the
number of muffins.
0.9 1.5 465
360
240
xy
x
y
+≥
≤
≤
360
The boundary line is solid.
x≤
()
()
()
()
360 ,
360 0 360,0
360 100 360,100
Test point is 0,0 .
xyxy=
40. b. (continued)
Mindstretchers
1. Answers may vary. Possible answer:
Elliot’s school schedule permits him to
work less than 10 hours each week at two
different part time jobs. The pharmacy will
let him work up to 7 hours in one week and
the bakery will let him have at most 8
hours of work each week.