1
00:00:00,000 --> 00:00:02,799
You go for a job interview.

2
00:00:03,282 --> 00:00:05,719
And the first thing your interviewer says is,

3
00:00:05,719 --> 00:00:09,701
"Look you have great work experience, you seem like a nice young person,

4
00:00:10,167 --> 00:00:14,449
but what I really care about is your logical reasoning capabilities."

5
00:00:14,449 --> 00:00:19,088
So what she says is, "Just sit down, I'm going to ask you a question about some math expressions."

6
00:00:19,088 --> 00:00:23,384
And you say, "Sure shot away, give me some questions about math expressions."

7
00:00:23,399 --> 00:00:30,387
And she says, "Alright so you have two integers, integer 'a' and integer 'b.'"

8
00:00:30,387 --> 00:00:37,661
And she tells you that integer 'a' is greater than zero and integer 'b' is less than zero.

9
00:00:37,661 --> 00:00:48,601
Then, she says that we also know that 'a' over 'b' is greater than 'a' times 'b.'

10
00:00:48,601 --> 00:00:57,390
So then she says tell me some interesting things about 'a' and 'b' and 'a' over 'b' and 'a' times 'b.'

11
00:00:57,390 --> 00:00:59,544
And you say, "Okay, I'll give my best shot"

12
00:00:59,544 --> 00:01:04,423
And I encourage you to actually pause now and try to think about how these things might relate

13
00:01:04,423 --> 00:01:09,177
or how 'b' might be constrained or 'a' might be constrained and then unpause it.

14
00:01:09,177 --> 00:01:12,691
So, I've assumed you've unpaused it so let's think about this a little bit.

15
00:01:13,045 --> 00:01:17,418
So the first thing, we know 'a' is positive and 'b' is negative.

16
00:01:17,418 --> 00:01:21,487
So if I take a positive divided by a negative what am I going to get?

17
00:01:21,487 --> 00:01:25,806
Well this right over here, a positive divided by a negative is going to be a negative.

18
00:01:25,806 --> 00:01:28,745
What happens if I take a positive times a negative?

19
00:01:28,745 --> 00:01:30,930
Well 'ab' is also going to be a negative.

20
00:01:30,930 --> 00:01:33,245
A postive times a negative is also a negative.

21
00:01:33,245 --> 00:01:37,136
So really what we're saying is that we have two negative quantities and

22
00:01:37,136 --> 00:01:40,275
this one over here is greater than that one.

23
00:01:40,275 --> 00:01:43,056
So let's visualize that on a number line.

24
00:01:43,056 --> 00:01:44,971
Let's visualize that on a number line.

25
00:01:44,971 --> 00:01:47,684
So let's say that this right over here is zero

26
00:01:47,684 --> 00:01:51,802
this right over here is the positive direction, this is the negative direction.

27
00:01:51,802 --> 00:01:55,146
We know both of these are negative, but 'a' over 'b' is larger.

28
00:01:55,146 --> 00:01:58,753
So 'a' over 'b' is going to be to the right of 'a' times 'b.'

29
00:01:58,753 --> 00:02:02,575
So, we know that 'a' over 'b'

30
00:02:02,575 --> 00:02:05,041
'a' over 'b,' it has to be negative.

31
00:02:05,041 --> 00:02:07,279
It has to be to the left of zero.

32
00:02:07,279 --> 00:02:11,670
'a' over 'b' is going to be to the right of 'a' times 'b.'

33
00:02:11,670 --> 00:02:16,633
To the right of 'a' times 'b.'

34
00:02:16,633 --> 00:02:19,105
So one way to think about it is that they're both negative, but

35
00:02:19,105 --> 00:02:21,645
'a' over 'b' is going to be less negative

36
00:02:21,645 --> 00:02:23,052
or another way of thinking about it,

37
00:02:23,052 --> 00:02:25,345
it's going to have a smaller absolute value.

38
00:02:25,345 --> 00:02:28,981
it's distance from zero, which is another way of thinking about absolute value,

39
00:02:28,981 --> 00:02:31,096
it's distance to the left of zero

40
00:02:31,096 --> 00:02:34,592
is less than 'a' times 'b's distance to the left of zero

41
00:02:34,669 --> 00:02:36,399
but because they're both negative

42
00:02:36,399 --> 00:02:38,576
and we'll talking about distances from the left of zero

43
00:02:38,576 --> 00:02:41,957
the one that has the smaller distance to the left of zero is less negative

44
00:02:41,957 --> 00:02:44,830
and therefore is a greater number.

45
00:02:44,830 --> 00:02:46,495
So, so far your interviewer seems impressed

46
00:02:46,495 --> 00:02:47,734
so that was pretty good,

47
00:02:47,734 --> 00:02:52,210
you were able to get a lot of clues just out of this little bit of information that I gave you.

48
00:02:52,210 --> 00:02:55,922
But tell me more about what 'b' has to be and

49
00:02:55,922 --> 00:02:58,956
whether we can constrain it in some way.

50
00:02:58,956 --> 00:03:01,774
And you say, well okay, well we got some clues here

51
00:03:01,774 --> 00:03:04,208
the fact that this absolute value right over here

52
00:03:04,208 --> 00:03:06,557
is going to be less than this absolute value

53
00:03:06,557 --> 00:03:12,332
we know that the absolute value of 'a' over 'b' is going to be less than 'a' times 'b,'

54
00:03:12,332 --> 00:03:14,126
the absolute value of 'a' times 'b,'

55
00:03:14,126 --> 00:03:20,371
Once again, it's going to be not as far left of zero as 'a' times 'b.'

56
00:03:20,371 --> 00:03:22,264
So that is how we can make that statement, but

57
00:03:22,310 --> 00:03:26,371
let's manipulate, algebraically manipulate this inequality here.

58
00:03:26,371 --> 00:03:29,289
We could multiply both sides times 'b.'

59
00:03:29,289 --> 00:03:33,522
So let's do that, let's multiply both sides by 'b',

60
00:03:33,522 --> 00:03:37,864
'b' is less than zero, 'b' is less than zero.

61
00:03:37,864 --> 00:03:43,505
So if your multiplying both sides of an inequality by something less than zero, it swaps the inequality.

62
00:03:43,505 --> 00:03:45,558
So let me, that's why I got rid of it, I'm going to rewrite it.

63
00:03:45,558 --> 00:03:50,485
So 'b' times 'a' over 'b' is going to be negative.

64
00:03:50,638 --> 00:03:55,567
We know it's going to be negative, so its going to be less than 'ab' times 'b.'

65
00:03:55,567 --> 00:03:57,605
If we just multiply it out, these cancel out.

66
00:03:57,605 --> 00:04:02,942
We get 'a' is less than 'a' times 'b' squared.

67
00:04:02,942 --> 00:04:06,343
Well now if we want to simplify, we can divide both sides by 'a'

68
00:04:06,343 --> 00:04:10,013
and since 'a' is greater than zero it does not change the inequality

69
00:04:10,013 --> 00:04:12,343
so we divide both sides by 'a.'

70
00:04:12,343 --> 00:04:15,892
This becomes one and this becomes 'b' squared.

71
00:04:15,892 --> 00:04:23,192
So we're left with one is less than 'b' squared

72
00:04:23,192 --> 00:04:26,523
or another way we can think about this is,

73
00:04:26,538 --> 00:04:33,592
well we can say if one is less than 'b' squared that means that 'b,'

74
00:04:33,592 --> 00:04:37,879
let's be careful here, that means that the ABSOLUTE VALUE of 'b'

75
00:04:37,879 --> 00:04:40,244
is going to be greater than one.

76
00:04:40,244 --> 00:04:44,301
This means that the absolute value of 'b' is greater than one.

77
00:04:44,301 --> 00:04:50,169
So you say, "Wait Sal, how did you get from that to that right there?"

78
00:04:50,169 --> 00:04:52,222
Well just think about this a little bit,

79
00:04:52,222 --> 00:04:55,016
if I square something and if its greater than one,

80
00:04:55,031 --> 00:05:00,679
so that means that either 'b' is less than negative one because if it was negative one and

81
00:05:00,679 --> 00:05:02,193
you squared it you would get one

82
00:05:02,193 --> 00:05:04,154
and if it was greater than negative one

83
00:05:04,154 --> 00:05:06,022
or just directly greater than negative one.

84
00:05:06,022 --> 00:05:09,458
If it was negative 0.99 than when you squared it,

85
00:05:09,458 --> 00:05:13,213
it would also be something less than one.

86
00:05:13,213 --> 00:05:14,328
So that doesn't work.

87
00:05:14,328 --> 00:05:20,850
So 'b' has to be less than negative one or 'b' has to be greater than one

88
00:05:20,850 --> 00:05:24,684
and because for the same exact logic if it was exactly equal to one

89
00:05:24,684 --> 00:05:26,306
if you squared it, this would be equal.

90
00:05:26,306 --> 00:05:30,242
If it was 0.5 then 'b' squared would be 0.25 which wouldn't be greater than.

91
00:05:30,242 --> 00:05:32,924
So you know both of these are true

92
00:05:32,924 --> 00:05:37,843
This is another way of saying the absolute value of 'b' is greater than one.

93
00:05:37,843 --> 00:05:42,427
Now we have our other constraint that 'b' is less than zero.

94
00:05:42,427 --> 00:05:46,538
So since we know that 'b' is less than zero we can take this out of the picture

95
00:05:46,538 --> 00:05:48,760
and if the absolute value of 'b' is greater than one

96
00:05:48,760 --> 00:05:52,480
and 'b' is less than zero, then we know that 'b' has to be less than

97
00:05:52,480 --> 00:05:55,214
'b' has to be less than negative one.

98
00:05:55,352 --> 00:05:58,834
So, wherever negative one is 'b' is going to be less than that.

99
00:05:58,834 --> 00:06:00,580
And the interviewer's very impressed,

100
00:06:00,580 --> 00:06:03,366
and she says "Look you've done very good reasoning here

101
00:06:03,474 --> 00:06:07,000
I think you deserve the job."