1
00:00:00,000 --> 00:00:00,550


2
00:00:00,550 --> 00:00:04,649
Luis receives a gift card worth $25 to an online

3
00:00:04,650 --> 00:00:07,970
retailer that sells digital music and games.

4
00:00:07,969 --> 00:00:14,489
Each song costs $0.89 and each game costs $1.99.

5
00:00:14,490 --> 00:00:20,560
He wants to buy at least 15 items with this card.

6
00:00:20,559 --> 00:00:23,039
Set up a system of inequalities that represents

7
00:00:23,039 --> 00:00:27,019
this scenario and identify the range of possible purchases

8
00:00:27,019 --> 00:00:28,439
using a graph.

9
00:00:28,440 --> 00:00:31,320
And that's why we have some graph paper over here.

10
00:00:31,320 --> 00:00:32,939
So let's define some variables.

11
00:00:32,939 --> 00:00:36,489
Let's let s equal the number of songs he buys.

12
00:00:36,490 --> 00:00:39,120


13
00:00:39,119 --> 00:00:45,369
And then let's let g equal the number of games that he buys.

14
00:00:45,369 --> 00:00:46,949
Now if we look at this constraint right here, he

15
00:00:46,950 --> 00:00:51,190
wants to buy at least 15 items with this card.

16
00:00:51,189 --> 00:00:53,339
So the total number of items are going to be the number of

17
00:00:53,340 --> 00:00:55,420
songs plus the number of games.

18
00:00:55,420 --> 00:00:57,179
And that has to be at least 15.

19
00:00:57,179 --> 00:01:01,600
So it has to be greater than or equal to 15.

20
00:01:01,600 --> 00:01:04,390
So that's what that constraint tells us right there.

21
00:01:04,390 --> 00:01:08,790
And then the other constraint is the gift card is worth $25.

22
00:01:08,790 --> 00:01:11,310
So the amount that he spends on songs plus the amount that

23
00:01:11,310 --> 00:01:14,810
he spends on games has to be less than or equal to 25.

24
00:01:14,810 --> 00:01:17,010
So the amount that he spends on songs are going to be the

25
00:01:17,010 --> 00:01:22,460
number of songs he buys times the cost per song.

26
00:01:22,459 --> 00:01:27,869
Times $0.89 times-- so I will say 0.89-- times s.

27
00:01:27,870 --> 00:01:31,660
That's how much he spends on songs plus the cost per game,

28
00:01:31,659 --> 00:01:35,679
which is $1.99 times the number of games.

29
00:01:35,680 --> 00:01:37,540
This is going to be the total amount that he spends.

30
00:01:37,540 --> 00:01:44,940
And that has to be less than or equal to 25.

31
00:01:44,939 --> 00:01:47,950
Now if we want to graph these, we first have to define the

32
00:01:47,950 --> 00:01:50,180
axes, so let me do that right here.

33
00:01:50,180 --> 00:01:52,420
And we only care about the first quadrant because we only

34
00:01:52,420 --> 00:01:54,909
care about positive values for the number of songs and the

35
00:01:54,909 --> 00:01:55,549
number of games.

36
00:01:55,549 --> 00:01:59,239
We don't talk about scenarios where he buys a negative

37
00:01:59,239 --> 00:02:00,579
number of songs or games.

38
00:02:00,579 --> 00:02:03,609
So just the positive quadrant right here.

39
00:02:03,609 --> 00:02:06,109
Let me draw the axes.

40
00:02:06,109 --> 00:02:10,210
So let's make the vertical axis that I'm drawing right

41
00:02:10,210 --> 00:02:15,219
here, let's make that the vertical axis and let's call

42
00:02:15,219 --> 00:02:18,139
that the song axis.

43
00:02:18,139 --> 00:02:19,979
So that's the number of songs he buys.

44
00:02:19,979 --> 00:02:21,869
Let me make sure you can see that.

45
00:02:21,870 --> 00:02:23,930
That is the song axis.

46
00:02:23,930 --> 00:02:28,069
And then let's make this, this horizontal, that's going to be

47
00:02:28,069 --> 00:02:29,905
the number of games he buys.

48
00:02:29,905 --> 00:02:31,155
Let's bold it in.

49
00:02:31,155 --> 00:02:39,319


50
00:02:39,319 --> 00:02:43,719
And just to make sure that we can fit on this page-- because

51
00:02:43,719 --> 00:02:45,580
I have a feeling we're going to get to reasonably large

52
00:02:45,580 --> 00:02:48,469
numbers-- let's make each of these boxes equal to 2.

53
00:02:48,469 --> 00:02:56,710
So this would be 4, 8, 12, 16, 20, so on and so forth.

54
00:02:56,710 --> 00:03:02,960
And this would be 4-- this obviously would be 0-- 4, 8,

55
00:03:02,960 --> 00:03:08,860
12, 16, 20, and so on.

56
00:03:08,860 --> 00:03:11,930
So let's see if we can graph these two constraints.

57
00:03:11,930 --> 00:03:15,010
Well, this first constraint, s plus g is going to be greater

58
00:03:15,009 --> 00:03:16,560
than or equal to 15.

59
00:03:16,560 --> 00:03:18,469
The easiest way to think about this-- or the easiest way to

60
00:03:18,469 --> 00:03:20,859
graph this is to really think about the intercepts.

61
00:03:20,860 --> 00:03:25,160
If g is 0, what is s?

62
00:03:25,159 --> 00:03:29,340
Well, s plus 0 has to be greater than or equal to 15.

63
00:03:29,340 --> 00:03:35,289
So if g is 0, s is going to be greater than or equal to 15.

64
00:03:35,289 --> 00:03:36,370
Let me put it this way.

65
00:03:36,370 --> 00:03:39,700
So if I'm going to graph this one right here.

66
00:03:39,699 --> 00:03:43,709
If g is 0, s is greater than or equal to 15.

67
00:03:43,710 --> 00:03:49,390
So g is 0, s, 15, let's see, this is 12, 14, 15 is right

68
00:03:49,389 --> 00:03:50,379
over there.

69
00:03:50,379 --> 00:03:53,090
And s is going to be all of the values equivalent to that

70
00:03:53,090 --> 00:03:56,170
or greater than for g equal to 0.

71
00:03:56,169 --> 00:04:00,459
If s is equal to 0, g is greater than or equal to 15.

72
00:04:00,460 --> 00:04:05,700
So if s is equal to 0, g is greater than or equal to 15.

73
00:04:05,699 --> 00:04:08,810
So g is greater than or equal to 15.

74
00:04:08,810 --> 00:04:12,740
So the boundary line, s plus g is equal to 15, we would just

75
00:04:12,740 --> 00:04:14,900
have to connect these two dots.

76
00:04:14,900 --> 00:04:17,649
Let me try my best to connect these dots.

77
00:04:17,649 --> 00:04:19,720
So it would look something like this.

78
00:04:19,720 --> 00:04:22,400


79
00:04:22,399 --> 00:04:24,329
This is always the hardest part.

80
00:04:24,329 --> 00:04:28,000
Let me see how well I can connect these two dots.

81
00:04:28,000 --> 00:04:29,540
Nope.

82
00:04:29,540 --> 00:04:30,480
Let me see.

83
00:04:30,480 --> 00:04:32,960
I should get a line tool for this.

84
00:04:32,959 --> 00:04:35,399
So that's pretty good.

85
00:04:35,399 --> 00:04:38,049
So that's the line s plus g is equal to 15.

86
00:04:38,050 --> 00:04:41,160
And we talk about the values greater than 15, we're going

87
00:04:41,160 --> 00:04:41,920
to go above the line.

88
00:04:41,920 --> 00:04:45,280
And you saw that when g is equal to 0, s is greater than

89
00:04:45,279 --> 00:04:47,009
or equal to 15.

90
00:04:47,009 --> 00:04:49,069
It's all of these values up here.

91
00:04:49,069 --> 00:04:53,329
And when s was 0, g was greater than or equal to 15.

92
00:04:53,329 --> 00:04:57,649
So this constraint right here is all of this.

93
00:04:57,649 --> 00:05:00,439
All of this area satisfies this.

94
00:05:00,439 --> 00:05:03,310
All of this area-- if you pick any coordinate here, it

95
00:05:03,310 --> 00:05:05,560
represents-- and really you should think about the integer

96
00:05:05,560 --> 00:05:08,230
coordinates, because we're not going to buy parts of games.

97
00:05:08,230 --> 00:05:10,689
But if you think about all of the integer coordinates here,

98
00:05:10,689 --> 00:05:13,709
they represent combinations of s and g, where you're buying

99
00:05:13,709 --> 00:05:15,269
at least 15 games.

100
00:05:15,269 --> 00:05:18,609
For example here, you're buying 8 games and 16 songs.

101
00:05:18,610 --> 00:05:19,319
That's 24.

102
00:05:19,319 --> 00:05:21,730
So you're definitely meeting the first constraint.

103
00:05:21,730 --> 00:05:23,569
Now the second constraint.

104
00:05:23,569 --> 00:05:28,120
0.89s plus 1.99g is less than or equal to 25.

105
00:05:28,120 --> 00:05:29,569
This is a starting point.

106
00:05:29,569 --> 00:05:38,269
Let's just draw the line 0.89s plus 1.99 is equal to 25.

107
00:05:38,269 --> 00:05:41,659
And then we could think about what region the less than

108
00:05:41,660 --> 00:05:43,000
would represent.

109
00:05:43,000 --> 00:05:44,889
Oh, 1.99g.

110
00:05:44,889 --> 00:05:46,789
And the easiest way to do this, once again, we could do

111
00:05:46,790 --> 00:05:48,900
slope y-intercept all that type of thing.

112
00:05:48,899 --> 00:05:50,370
But the easiest way is to just find the s- and the

113
00:05:50,370 --> 00:05:51,610
g-intercepts.

114
00:05:51,610 --> 00:06:04,879
So if s is equal to 0 then we have 1.99g is equal to 25 or g

115
00:06:04,879 --> 00:06:08,469
is equal to-- let's get a calculator out for this.

116
00:06:08,470 --> 00:06:17,900
So if we take 25 divided by 1.99, it is 12.56.

117
00:06:17,899 --> 00:06:26,209
g is equal to 12.56.

118
00:06:26,209 --> 00:06:27,969
So when s is 0, let me plot this.

119
00:06:27,970 --> 00:06:32,370
When s is 0, g is 12.56.

120
00:06:32,370 --> 00:06:34,439
This is 12, this is 14.

121
00:06:34,439 --> 00:06:37,449
12.56 is going to be right there, a little

122
00:06:37,449 --> 00:06:38,949
bit more than 12.

123
00:06:38,949 --> 00:06:40,370
That's that value there.

124
00:06:40,370 --> 00:06:43,500
And then let's do the same thing if g is 0.

125
00:06:43,500 --> 00:06:47,699
So if g is equal to 0, then we have-- so this term goes

126
00:06:47,699 --> 00:06:51,769
away-- we have 0.89s.

127
00:06:51,769 --> 00:06:54,639
If we use just the equality here, the equation-- is equal

128
00:06:54,639 --> 00:07:00,219
to 25 or s is equal to-- get the calculator out again.

129
00:07:00,220 --> 00:07:06,410
So if we take 25 divided by 0.89, we get--

130
00:07:06,410 --> 00:07:09,110
it's equal to 28.08.

131
00:07:09,110 --> 00:07:11,180
Just a little over 28.

132
00:07:11,180 --> 00:07:14,670
So 28.08.

133
00:07:14,670 --> 00:07:19,160
So that is, g is 0, s is 28.

134
00:07:19,160 --> 00:07:22,890
So that is 2, 4, 24, 6, 8.

135
00:07:22,889 --> 00:07:23,800
A little over 28.

136
00:07:23,800 --> 00:07:25,600
So it's right over there.

137
00:07:25,600 --> 00:07:32,180
So this line, 0.89s plus 1.99g is equal to 25 is going to go

138
00:07:32,180 --> 00:07:35,810
from this coordinate, which is 0, 28.

139
00:07:35,810 --> 00:07:37,629
So that point right there.

140
00:07:37,629 --> 00:07:41,319
All the way down to the point 12.56,0.

141
00:07:41,319 --> 00:07:43,420
So let me see if I can draw that.

142
00:07:43,420 --> 00:07:47,129
It's going to go-- I'll draw up one more attempt.

143
00:07:47,129 --> 00:07:50,560
Maybe if I start from the bottom it'll be easier.

144
00:07:50,560 --> 00:07:52,220
That was a better attempt.

145
00:07:52,220 --> 00:07:55,100
Let me bold that in a little bit, so you can make sure you

146
00:07:55,100 --> 00:07:57,510
can see it.

147
00:07:57,509 --> 00:08:01,389
So that line represents this right over here.

148
00:08:01,389 --> 00:08:05,709
Now if we're talking about the less than area, what would

149
00:08:05,709 --> 00:08:06,699
that imply?

150
00:08:06,699 --> 00:08:13,920
So if we think about it, when g is equal to 0, 0.89s

151
00:08:13,920 --> 00:08:15,550
is less than 25.

152
00:08:15,550 --> 00:08:18,610
So when g is equal to 0, if we really wanted the less than

153
00:08:18,610 --> 00:08:20,500
there, we could think of it this way.

154
00:08:20,500 --> 00:08:23,379
It's less than instead of just doing less than or equal to.

155
00:08:23,379 --> 00:08:26,629
So s is less than 28.08.

156
00:08:26,629 --> 00:08:30,399
So it'll be the region below.

157
00:08:30,399 --> 00:08:34,610
When s is 0, g-- so if we think s is 0, if we use this

158
00:08:34,610 --> 00:08:38,820
original equation, 1.99g will be less than or equal to.

159
00:08:38,820 --> 00:08:41,350
I use this just to plot the graph, but if we actually care

160
00:08:41,350 --> 00:08:46,670
about the actual inequality, we get 1.99g is less than 25.

161
00:08:46,669 --> 00:08:49,620
g would be less than or equal to 12.56.

162
00:08:49,620 --> 00:08:55,389
So when s is equal to 0, g is less than 12.56.

163
00:08:55,389 --> 00:08:59,949
So the area that satisfies this second constraint is

164
00:08:59,950 --> 00:09:01,800
everything below this graph.

165
00:09:01,799 --> 00:09:06,039


166
00:09:06,039 --> 00:09:09,750
Now we want the region that satisfies both constraints.

167
00:09:09,750 --> 00:09:12,059
So it's going to be the overlap of the regions that

168
00:09:12,059 --> 00:09:13,929
satisfy one of the two.

169
00:09:13,929 --> 00:09:17,059
So the overlap is going to be this region right here.

170
00:09:17,059 --> 00:09:20,919
Below the orange graph and above the blue graph,

171
00:09:20,919 --> 00:09:23,519
including both of them.

172
00:09:23,519 --> 00:09:29,220
So if you pick any combination-- so if he buys 4

173
00:09:29,220 --> 00:09:33,300
games and 14 songs, that would work.

174
00:09:33,299 --> 00:09:37,089
Or if he bought 2 games and 16 songs, that would work.

175
00:09:37,090 --> 00:09:38,170
So you can kind of get the idea.

176
00:09:38,169 --> 00:09:40,299
Anything in that region-- and he can only buy integer

177
00:09:40,299 --> 00:09:42,779
values-- would satisfy

178
00:09:42,779 --> 00:09:44,799