1
00:00:00,813 --> 00:00:01,930
Solve for z

2
00:00:01,930 --> 00:00:09,732
5z + 7 is less than 27 OR negative 3z is less than or equal to 18.

3
00:00:09,732 --> 00:00:14,283
So this is a compound inequality. We have two conditions here.

4
00:00:14,283 --> 00:00:19,529
So z can satisfy this OR z can satisfy this over here.

5
00:00:19,529 --> 00:00:24,790
So lets just solve each of these inequalities and just know that z can satisfy either of them.

6
00:00:24,790 --> 00:00:26,327
So lets just look at this.

7
00:00:26,327 --> 00:00:28,354
So if we look at just this one over here:

8
00:00:28,354 --> 00:00:33,071
We have 5z + 7 is less than 27.

9
00:00:33,071 --> 00:00:35,801
Let's isolate the z's on the left hand side,

10
00:00:35,801 --> 00:00:40,008
so lets subtract 7 from both sides to get rid of this 7 on the left hand side.

11
00:00:40,008 --> 00:00:46,510
And so our left hand side is just going to be 5z plus 7, minus 7, those cancel out

12
00:00:46,510 --> 00:00:52,733
5z is less than 27 minus 20, sorry, 27 minus 7, which is 20

13
00:00:52,733 --> 00:00:55,554
so we have 5z less than 20.

14
00:00:55,554 --> 00:00:59,824
Now we can divide both sides of this inequality by 5

15
00:00:59,824 --> 00:01:01,498
and we don't have to swap the inequality

16
00:01:01,498 --> 00:01:04,110
because we're dividing by a positive number

17
00:01:04,110 --> 00:01:08,348
and so we get z is less than 20 over 5.

18
00:01:08,348 --> 00:01:11,034
Z is less than 4.

19
00:01:11,034 --> 00:01:12,534
Now this is only one of the conditions.

20
00:01:12,534 --> 00:01:15,024
Lets go to the other one over here:

21
00:01:15,024 --> 00:01:20,564
We have negative 3z is less than or equal to 18.

22
00:01:20,564 --> 00:01:23,476
Now to isolate the z we can just divide both sides

23
00:01:23,476 --> 00:01:25,831
of this inequality by negative 3

24
00:01:25,831 --> 00:01:29,200
but remember when you divide or multiply both sides

25
00:01:29,200 --> 00:01:31,138
of an inequality by a negative number,

26
00:01:31,138 --> 00:01:33,200
you have to swap the inequality.

27
00:01:33,200 --> 00:01:36,200
So we can write negative 3z

28
00:01:36,200 --> 00:01:38,522
We're going to divide by negative 3

29
00:01:38,522 --> 00:01:41,402
and then you go 18 divide by negative 3

30
00:01:41,402 --> 00:01:43,533
but we're going to swap the inequality!

31
00:01:43,533 --> 00:01:47,332
So the left center equals greater than or equal to

32
00:01:47,332 --> 00:01:48,621
and so these guys cancel out

33
00:01:48,621 --> 00:01:51,080
negative 3 divided by negative 3 equals 1,

34
00:01:51,080 --> 00:01:57,304
so we have z is greater than or equal to 18 over negative 3 is negative 6

35
00:01:57,304 --> 00:02:00,877
and remember, it's this constraint OR this constraint

36
00:02:00,877 --> 00:02:04,492
and this constraint right over here boils down to this

37
00:02:04,492 --> 00:02:06,543
and this one boils down to this

38
00:02:06,543 --> 00:02:07,601
so our solution set

39
00:02:07,601 --> 00:02:14,377
z is less than 4 OR z is greater than or equal to negative -6.

40
00:02:14,377 --> 00:02:15,553
So let me make this clear.

41
00:02:15,553 --> 00:02:16,406
Let me rewrite it.

42
00:02:16,406 --> 00:02:23,494
So z is less than 4 OR z is greater than or equal to negative 6.

43
00:02:23,494 --> 00:02:25,120
It can satisfy either one of these.

44
00:02:25,120 --> 00:02:28,352
This is kind of interesting here.

45
00:02:28,352 --> 00:02:32,476
So, so let's plot these. Let's plot these.

46
00:02:32,476 --> 00:02:34,802
So there's the number line.

47
00:02:34,802 --> 00:02:35,838
Right over there.

48
00:02:35,838 --> 00:02:37,751
Let's say 0 is over here

49
00:02:37,751 --> 00:02:41,653
and we have 1, 2, 3, 4 is right over there

50
00:02:41,653 --> 00:02:46,716
and negative 6, we have 1, 2 ,3, 4, 5, 6.

51
00:02:46,716 --> 00:02:48,944
That's negative 6 over there.

52
00:02:48,944 --> 00:02:52,334
Now let's think about z being less than 4

53
00:02:52,334 --> 00:02:54,577
Z being less than 4,

54
00:02:54,577 --> 00:02:55,385
we put a circle around 4,

55
00:02:55,385 --> 00:02:57,287
since we're not including 4

56
00:02:57,287 --> 00:03:03,656
and it'd be everything, everything less than 4. Everything less than 4.

57
00:03:03,656 --> 00:03:07,861
Now lets think about z being greater than or equal to negative 6 would be

58
00:03:07,861 --> 00:03:10,104
That means we can include negative 6

59
00:03:10,104 --> 00:03:12,504
and it's everything... let's do this in a more different color...

60
00:03:12,504 --> 00:03:16,278
It means you can include negative 6...

61
00:03:16,278 --> 00:03:18,118
I'm going to do that....

62
00:03:18,118 --> 00:03:19,208
It means you can include negative 6...

63
00:03:19,208 --> 00:03:21,383
Let me do that in a more different color...

64
00:03:21,383 --> 00:03:23,051
I'm going to do it in orange...

65
00:03:23,051 --> 00:03:25,015
so z is greater than equal to negative 6

66
00:03:25,015 --> 00:03:26,812
and this means you include negative 6

67
00:03:26,812 --> 00:03:28,995
and its everything greater than that

68
00:03:28,995 --> 00:03:31,031
greater than that

69
00:03:31,031 --> 00:03:32,801
including 4.

70
00:03:32,801 --> 00:03:35,539
It's everything greater than that.

71
00:03:35,539 --> 00:03:37,134
So we see we've essentially shaded in the

72
00:03:37,134 --> 00:03:38,898
entire number line!

73
00:03:38,898 --> 00:03:42,195
Every number will meet either one of these constraints

74
00:03:42,195 --> 00:03:43,404
or both of them.

75
00:03:43,404 --> 00:03:45,693
If we're over here, we're going to meet both of the constraints,

76
00:03:45,693 --> 00:03:48,277
if we're one of these numbers, if we're a number out here, we're going

77
00:03:48,277 --> 00:03:49,287
to meet this constraint

78
00:03:49,287 --> 00:03:50,667
if we're a number down here,

79
00:03:50,667 --> 00:03:51,874
we're going to meet this constraint.

80
00:03:51,874 --> 00:03:53,441
You can just try it out with a bunch of numbers.

81
00:03:53,441 --> 00:03:55,106
0 will work.

82
00:03:55,106 --> 00:03:58,186
0 plus 7 is 7

83
00:03:58,186 --> 00:03:59,579
which is less than 27.

84
00:03:59,579 --> 00:04:01,037
And three times 0 is less than 18

85
00:04:01,037 --> 00:04:03,016
so it meets both constraints.

86
00:04:03,016 --> 00:04:04,224
If we put 4 here,

87
00:04:04,224 --> 00:04:06,235
it should only meet one of the constraints.

88
00:04:06,235 --> 00:04:08,588
Negative 3 times 4 is negative 12,

89
00:04:08,588 --> 00:04:10,275
which is less than 18.

90
00:04:10,275 --> 00:04:12,736
So it meets this constraint but won't meet this constraint.

91
00:04:12,736 --> 00:04:14,205
Because you do 5 times 4

92
00:04:14,205 --> 00:04:15,987
plus 7 is 27

93
00:04:15,987 --> 00:04:18,795
which is not LESS than 27, it's equal to 27.

94
00:04:18,795 --> 00:04:20,663
Remember this is an OR, so you just have to

95
00:04:20,663 --> 00:04:22,195
meet one of the constraints.

96
00:04:22,195 --> 00:04:23,975
So 4 meets this constraint

97
00:04:23,975 --> 00:04:25,401
so even 4 works.

98
00:04:25,401 --> 00:04:27,211
So it's really the entire the number line

99
00:04:27,211 --> 99:59:59,999
will satisfy either one or both of these constraints.