1
00:00:00,813 --> 00:00:10,533
Solve for x, 5x - 3 is less than 12 "and" 4x plus 1 is greater than 25.

2
00:00:10,533 --> 00:00:13,867
So let's just solve for X in each of these constraints and keep in mind

3
00:00:13,867 --> 00:00:16,867
that any x has to satisfy both of them

4
00:00:16,867 --> 00:00:18,867
because it's an "and" over here

5
00:00:18,867 --> 00:00:20,600
so first we have this 5 x

6
00:00:20,600 --> 00:00:23,308
minus 3 is less than 12

7
00:00:23,308 --> 00:00:25,738
so if we want to isolate the x

8
00:00:25,738 --> 00:00:28,036
we can get rid of this negative 3

9
00:00:28,036 --> 00:00:29,200
here by adding 3 to both sides

10
00:00:29,200 --> 00:00:33,467
so let's add 3 to both sides of this inequality.

11
00:00:33,467 --> 00:00:38,000
The left-hand side, we're just left with a 5x, the minus 3 and the plus 3 cancel out.

12
00:00:38,000 --> 00:00:42,113
5x is less than 12 plus 3 is 15.

13
00:00:42,113 --> 00:00:44,600
Now we can divide both sides by positive 5,

14
00:00:44,600 --> 00:00:47,467
that won't swap the inequality since 5 is positive.

15
00:00:47,467 --> 00:00:50,600
So we divide both sides by positive 5

16
00:00:50,600 --> 00:00:53,046
and we are left with just from this constraint

17
00:00:53,046 --> 00:00:57,041
that x is less than 15 over 5, which is 3.

18
00:00:57,041 --> 00:00:59,333
So that constraint over here.

19
00:00:59,333 --> 00:01:01,667
But we have the second constraint as well.

20
00:01:01,667 --> 00:01:05,600
We have this one, we have 4x plus 1

21
00:01:05,600 --> 00:01:07,964
is greater than 25.

22
00:01:07,964 --> 00:01:11,800
So very similarly we can substract one from both sides

23
00:01:11,800 --> 00:01:14,467
to get rid of that one on the left-hand side.

24
00:01:14,467 --> 00:01:17,667
And we get 4x, the ones cancel out.

25
00:01:17,667 --> 00:01:22,267
is greater than 25 minus one is 24.

26
00:01:22,267 --> 00:01:25,000
Divide both sides by positive 4

27
00:01:25,000 --> 00:01:28,810
Don't have to do anything to the inequality since

28
00:01:28,810 --> 00:01:30,323
it's a positive number.

29
00:01:30,323 --> 00:01:32,667
And we get x is greater than

30
00:01:32,667 --> 00:01:35,600
24 over 4 is 6.

31
00:01:35,600 --> 00:01:37,800
And remember there was that "and" over here.

32
00:01:37,800 --> 00:01:39,800
We have this "and".

33
00:01:39,800 --> 00:01:46,533
So x has to be less than 3 "and" x has to be greater than 6.

34
00:01:46,533 --> 00:01:49,585
So already your brain might be realizing

35
00:01:49,585 --> 00:01:51,600
that this is a little bit strange.

36
00:01:51,600 --> 00:01:56,333
This first constraint says that x needs to be less than 3

37
00:01:56,333 --> 00:01:58,933
so this is 3 on the number line.

38
00:01:58,933 --> 00:02:00,867
We're saying x has to be less than 3

39
00:02:00,867 --> 00:02:05,000
so it has to be in this shaded area right over there.

40
00:02:05,000 --> 00:02:07,569
This second constraint says that x has to be

41
00:02:07,569 --> 00:02:08,882
greater than 6.

42
00:02:08,882 --> 00:02:11,698
So if this is 6 over here, it says that x has to greater than 6.

43
00:02:11,698 --> 00:02:14,200
It can't even include 6.

44
00:02:14,200 --> 00:02:16,200
And since we have this "and" here.

45
00:02:16,200 --> 00:02:19,467
The only x-es that are a solution for this compound inequality

46
00:02:19,467 --> 00:02:21,333
are the ones that satisfy both.

47
00:02:21,333 --> 00:02:23,133
The ones that are in the overlap

48
00:02:23,133 --> 00:02:24,667
of their solution set.

49
00:02:24,667 --> 00:02:26,267
But when you look at it right over here it's clear that

50
00:02:26,267 --> 00:02:27,800
there is no overlap.

51
00:02:27,800 --> 00:02:33,133
There is no x that is both greater than 6 "and" less than 3.

52
00:02:33,133 --> 99:59:59,999
So in this situation we no solution.